Patent Publication Number: US-9891318-B2

Title: Location detection system

Description:
BACKGROUND 
     Many automobiles today include object detection as a feature when operating the automobile in potentially crowded areas like parking lots or driveways. The most common technique implements sound navigation and ranging (sonar). Sonar is relatively inexpensive but suffers from poor resolution. Another technique is radio detection and ranging (radar). Radar provides acceptable resolution but can be relatively expensive to implement. Yet another technique is light detection and ranging (lidar). Lidar provides excellent resolution but at significant cost. Accordingly, there may be a need for improved techniques to solve these and other problems. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  illustrates one embodiment of a location detection system. 
         FIG. 1B  illustrates another embodiment of a location detection system. 
         FIG. 2  illustrates one embodiment of location detection system integrated into an automobile. 
         FIG. 3  illustrates an example of light entering a lens. 
         FIG. 4  illustrates one example of a waveform being reflected off an object according to an embodiment of the invention. 
         FIG. 5  illustrates one example of the geometry pertaining to light reflected off an object according to an embodiment of the invention. 
         FIG. 6  illustrates another example of the geometry pertaining to light reflected off an object according to an embodiment of the invention. 
         FIG. 7A  illustrates one example of elliptical geometry that may be used to derive the equation of an ellipse according to an embodiment of the invention. 
         FIG. 7B  illustrates two intersecting ellipses that may be used to calculate the location of an object according to an embodiment of the invention. 
         FIG. 7C  illustrates another example of two intersecting ellipses that may be used to calculate the location of an object according to an embodiment of the invention. 
         FIG. 8  illustrates one embodiment of a logic flow. 
         FIG. 9  illustrates another embodiment of a logic flow. 
         FIG. 10  illustrates another embodiment of a logic flow. 
         FIG. 11  illustrates one embodiment of a computing architecture. 
     
    
    
     DETAILED DESCRIPTION 
     In various embodiments, a location detection system may address common deficiencies associated with current location detection systems. 
     The location detection system may utilize, in some embodiments, multiple light sources capable of emitting modulated light. The light sources may utilize multiple light emitting diodes (LEDs). The light sources may be in a fixed and known position. The location detection system may utilize, in some embodiments, a light receiver. The light receiver may be operative to detect light emitted by the light sources that has been reflected back off an object. The light receiver may also be in a fixed and known position. The location detection system may utilize, in some embodiments, a processor circuit. The processor circuit may be communicatively coupled with the light receiver and the light sources. The processor circuit may be operative to receive signals indicative of the detected reflected emitted light from the light receiver. The processor circuit may also be operative to process the signals to determine a location of the object that reflected the emitted light. 
     Reference is now made to the drawings, wherein like reference numerals are used to refer to like elements throughout. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding thereof. It may be evident, however, that the novel embodiments can be practiced without these specific details. In other instances, well known structures and devices are shown in block diagram form in order to facilitate a description thereof. The intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the claimed subject matter. 
       FIG. 1A  illustrates one embodiment  100 -A of a location detection system and its components. A first light source  110 -a is shown and may be communicatively coupled with a modulation component  140  under control of a processor circuit  130 . Similarly, a second light source  110 -b is shown and may be communicatively coupled with the modulation component  140  under control of the processor circuit  130 . The modulation component  140  may be operative to modulate the light emitted from the first and second light sources  110 -a,  110 -b to a specific pattern. The modulation may be accomplished by turning the first and second light sources  110 -a,  110 -b on and off thousands of times per second. This is termed on-off keying (OOK) and may be referred to as a ranging tone. The modulation component  140  may create a first ranging tone for the first light source  110 -a and a second ranging tome for the second light source  110 -b. The first and second light sources may be comprised of multiple light emitting diodes (LEDs)  115 . The embodiments are not limited in this context. 
       FIG. 1A  also illustrates a light receiver  120  communicatively coupled with the processor circuit  130  and a lensed detector array  170 . The processor circuit  130  may include a filtering component  160  that may be operative to filter signals indicative of light that has been reflected off an object. The filtering may be performed to ignore received light signals that do not match the emitted light that has been modulated according to either the first or second ranging tone. By emitting the light in a known pattern, reflected light that is detected can be filtered to remove environmental interference that may produce noise in the reflected light. Thus, only specific light emitted by the light sources  110 -a,  110 -b may be detected and acted upon by processor circuit  130 . The embodiments are not limited in this context. 
     The lensed detector array may be operative to map an azimuth angle of arrival for detected emitted light that has been reflected off an object and that matches the first or second OOK ranging tones to pixel locations on an image plane. A display  180  communicatively coupled with the lensed detector array  170  may be operative to display the mapped pixel locations to create an image of the object reflecting the emitted light. 
     The processor circuit  130  may be operative to determine the location of the object with respect to the known location of the first and second light sources  110 -a,  110 -b and the known location of the light receiver  120 . The location of the object may be calculated based on factors inherent in the geometry of the physical arrangement of the known location of the first and second light sources  110 -a,  110 -b and the known location of the light receiver  120 . Specifically, the object location may be calculated by a calculation application  165  under control of the processor circuit  130  using known principles of elliptical geometry. These calculations are described in more detail below. The embodiments are not limited in this context. 
     The processor circuit  130  may be communicatively coupled with other applications and components such as the display  180  and a speaker  185 . Based on the location of the object a visual alert may be displayed on display  180  to alert a human operator as to the existence and location of the object. Moreover, the distance of the object determined by the calculation application  165  may be displayed on the display along with a rendering of the object. Similarly, an audible alert may be emitted from speaker  185  to alert a human operator as to the existence of the object. The audible alert may change in pitch, frequency and/or tone as the distance to the object gets closer. The embodiments are not limited in this context. 
       FIG. 1B  illustrates another embodiment  100 -B of a location detection system. In this embodiment, the processor circuit  130  may be communicatively coupled with the first and second light sources  110 -a,  110 -b but not the light receiver  120 , the display  180 , or the speaker  185 . A second processor circuit  150  may be communicatively coupled with the light receiver  120 , the display  180 , or the speaker  185 . In this arrangement, the tasks and functions described with respect to the embodiment  100 -A of  FIG. 1A  may be split between the two processor circuits  130 ,  150 . Processor circuit  130  may be responsible for the modulation functions associated with the first and second light sources  110 -a,  110 -b (e.g., the transmission side). Processor circuit  150  may be responsible for the receiving and processing tasks and functions associated with the light receiver  120  (e.g., the reception side). The embodiments are not limited in this context. 
     It should be noted that the description above and corresponding figures disclose two light sources and a single light receiver which creates two light source/light receiver pairings. Adding a second light receiver would create two additional light source/light receiver pairings. 
       FIG. 2  illustrates one embodiment  200  of location detection system integrated into an automobile  210 . The first light source  110 -a may be integrated into a first taillight of the automobile  210 . Similarly, the second light source  110 -a may be integrated into a second taillight of the automobile  210 . The light receiver  120  may be mounted in a fixed and known location somewhere near the rear of the automobile  210 . An object  250  is shown that may be located behind the automobile  210 . The first light source  110 -a may emit modulated light  220 -a according to the first ranging tone. The emitted modulated light  220 -a may bounce off the object  250  and may be reflected  230  back to the light receiver  120  where it may be received and processed. Similarly, the second light source  110 -a may emit modulated light  220 -b according to the second ranging tone. The emitted modulated light  220 -b may bounce off the object  250  and may be reflected  230  back to the light receiver  120  where it may be received and processed. The embodiments are not limited in this context. 
       FIG. 3  illustrates an example of light entering a lensed detector array  170 . The lensed detector array  170  may be comprised of a lens  310  and a linear detector array  320 . Light reflected  230  off the object  250  may enter lens  310 . The lensed detector array  170  may be operative to map an azimuth angle of arrival for detected emitted light that has been reflected off an object and that matches the first or second OOK ranging tones to pixel locations on an image plane of the linear detector array  320 . A display  180  may be communicatively coupled with the lensed detector array  170  and may be operative to display the mapped pixel locations to create an image of the object  250  reflecting the emitted light. The embodiments are not limited in this context. 
       FIG. 4  illustrates one example of a waveform  400  being reflected off an object  250  according to an embodiment of the invention. The waveform  400  represents an OOK ranging tone. The waveform  400  may be high when on and low when off. The first light source  110 - a  located at a known and fixed position (x t , y t ) may emit the modulated waveform  400  which may strike the object  250  and be reflected to the light receiver  120  at a known and fixed position (x r , y r ). It may take a certain time τ t  for the emitted light to reach the object and another time τ r  to reach the light receiver  120 . The total time delay from light source  110 -a to light receiver  120  can be represented by τ=τ t +τ r . This time delay, τ, is graphically shown as the phase difference of the emitted waveform (e w ) as compared to the received waveform (r w ). In addition, the distance from the light source  110 -a to the object  250  to the light receiver  120  can be calculated as:
 
 D=c*τ 
 
in which c is the speed of light. While not shown, the same analysis would apply to light emitted from the second light source  110 -b. The embodiments are not limited in this context.
 
       FIG. 5  illustrates one example of the geometry pertaining to emitted light reflected off an object  250  according to an embodiment of the invention. Light emitted from the first light source  110 -a located at a known and fixed position (x t , y t ) may travel a distance D t  before reaching object  250 . This light may be reflected and received by light receiver  120  at a known and fixed position (x r , y r ) after travelling a second distance D r . The total distance travelled is D=D t +D r . For a given light source  110 -a and light receiver  120  pair, knowing the total time delay from first light source  110 -a to light receiver  120  means that the object  250  that reflected the light lies on an ellipse  510  in which the foci are the first light source  110 -a and the light receiver  120 . The origin, o, of the ellipse  510  is the midpoint between a line that connects the first light source  110 -a and the light receiver  120 . The embodiments are not limited in this context. 
       FIG. 6  illustrates another example of the geometry pertaining to light reflected off an object  250  according to an embodiment of the invention. In this example, there are two light sources  110 -a,  110 -b and a light receiver  120 . Thus, there are two light source and light receiver pairs meaning that two ellipses  610 ,  620  may result. Ellipse  610  may be the result of light emitted from the first light source  110 -a that is reflected off the object  250  and received by the light receiver  120 . Ellipse  620  may be the result of light emitted from the second light source  110 -b that is reflected off the object  250  and received by the light receiver  120 . As is shown in  FIG. 6  and which can be derived mathematically, the intersecting point of the two ellipses  610 ,  620  is at the location of the object  250  that reflected the emitted light. Because the first and second light sources  110 -a,  110 -b and the light receiver  120  are in fixed and known locations that happen to be the foci of the ellipses  610 ,  620 , the location of the object  250  can be solved for using the equation for an ellipse and the time delay for each light source and light receiver pairs. The embodiments are not limited in this context. 
       FIG. 7A  illustrates one example of elliptical geometry that may be used to derive the equation of an ellipse according to an embodiment of the invention. 
     Consider the situation where the foci are defined and the round time distance D has been measured. Referring to  FIG. 7A , the givens are the location of the foci and the bounce distance D. From this it can be defined that the origin “o” lies at point 
     
       
         
           
             
               
                 
                   o 
                   = 
                   
                     
                       ( 
                       
                         
                           f 
                           2 
                         
                         + 
                         
                           f 
                           2 
                         
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     The distance from the origin to the ellipse passing through one of the foci may be given as 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     D 
                     2 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     The distance from the origin to either foci may be given as 
     
       
         
           
             
               
                 
                   c 
                   = 
                   
                     abs 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             f 
                             2 
                           
                           - 
                           
                             f 
                             1 
                           
                         
                         2 
                       
                       ) 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     The distance from the origin to the ellipse, as measurement orthogonally to vector a, may be given as 
     
       
         
           
             
               
                 
                   b 
                   = 
                   
                     
                       
                         
                           D 
                           2 
                         
                         4 
                       
                       - 
                       
                         c 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   4 
                 
               
             
           
         
       
     
     The rotational angle may be the argument of the vector extending between the foci.
 
θ=∠( f   2   −f   1 )  Eq. 5
 
     The coordinates of the origin may be given as
 
 c   x =real( o )  Eq. 6
 
 c   y =imag( o )
 
     The canonical form of an ellipse (i.e. an ellipse centered at the origin with no rotational angle) may be given as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           x 
                           ^ 
                         
                         2 
                       
                       
                         a 
                         2 
                       
                     
                     + 
                     
                       
                         
                           y 
                           ^ 
                         
                         2 
                       
                       
                         b 
                         2 
                       
                     
                   
                   = 
                   1 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   7 
                 
               
             
           
         
       
     
     To change the canonical form to the general form the foci may be rotated then translated.
 
 {circumflex over (x)}+jŷ=xe   +jθ +( c   x   +jc   y )={ x  cos θ+ c   x}+j{x  sin θ+ c   y   }={tilde over (x)}+j{tilde over (y)}   Eq. 8
 
     To change the general form to the canonical form the translation bias may be subtracted off and the foci vector may be de-rotated as shown below. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                                       ~ 
                                     
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                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     The real (x axis) and imaginary (y axis) parts may be given by:
 
 x =( {tilde over (x)}−c   x )cos θ+( {tilde over (y)}−c   y )sin θ
 
 y =( {tilde over (y)}−c   y )cos θ−( {tilde over (x)}−c   x )sin θ  Eq. 11
 
     The canonical form as shown in Equation 7 may be modified as shown below. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           { 
                           
                             
                               
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                         2 
                       
                       
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                         2 
                       
                       
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                         2 
                       
                     
                   
                   = 
                   1 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
           
         
       
     
     A change of variable may be made by dropping the tilde over the variables. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                         2 
                       
                       
                         a 
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                         2 
                       
                       
                         b 
                         2 
                       
                     
                   
                   = 
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                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   13 
                 
               
             
           
         
       
     
     Expanding the square terms yields 
                         (                 [     x   -     c   x       ]     2     ⁢     cos   2     ⁢   θ     +       2   ⁡     [     x   -     c   x       ]       ⁡     [     y   -     c   y       ]                     cos   ⁢           ⁢   θ   ⁢           ⁢   sin   ⁢           ⁢   θ     +         [     y   -     c   y       ]     2     ⁢     sin   2     ⁢   θ             )       a   2       +       (                 [     x   -     c   x       ]     2     ⁢     sin   2     ⁢   θ     -       2   ⁡     [     x   -     c   x       ]       ⁡     [     y   -     c   y       ]                     cos   ⁢           ⁢   θ   ⁢           ⁢   sin   ⁢           ⁢   θ     +         [     y   -     c   y       ]     2     ⁢     cos   2     ⁢   θ             )       b   2         =   1           Eq   .           ⁢   14                     (               [       x   2     -     2   ⁢           ⁢     xc   x       +     c   x   2       ]     ⁢     cos   2     ⁢   θ     +     2   ⁡     [     xy   -       c   x     ⁢   y     -       c   y     ⁢   x     +       c   x     ⁢     c   y         ]                     cos   ⁢           ⁢   θ   ⁢           ⁢   sin   ⁢           ⁢   θ     +         [       y   2     -     2   ⁢           ⁢     c   y     ⁢   y     +     c   y   2       ]     2     ⁢     sin   2     ⁢   θ             )       a   2       +       (               [       x   2     -     2   ⁢           ⁢     xc   x       +     c   x   2       ]     ⁢     sin   2     ⁢   θ     -     2   ⁡     [     xy   -       c   x     ⁢   y     -       c   y     ⁢   x     +       c   x     ⁢     c   y         ]                     cos   ⁢           ⁢   θ   ⁢           ⁢   sin   ⁢           ⁢   θ     +         [       y   2     -     2   ⁢           ⁢     c   y     ⁢   y     +     c   y   2       ]     2     ⁢     cos   2     ⁢   θ             )       b   2         =   1           Eq   .           ⁢   15               
Collecting terms yields
 
     
       
         
           
             
               
                 
                   
                     
                       
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                                 cos 
                                 2 
                               
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                                 sin 
                                 2 
                               
                               ⁢ 
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                               b 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         x 
                         2 
                       
                     
                     + 
                     
                       
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                               2 
                             
                           
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                                 sin 
                                 2 
                               
                               ⁢ 
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                               a 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         y 
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
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                               ⁢ 
                               
                                   
                               
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                               b 
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                       ⁢ 
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                       ⁢ 
                       
                           
                       
                       ⁢ 
                       xy 
                     
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                         ( 
                         
                           
                             
                               
                                 
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                                 ⁢ 
                                 
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                               a 
                               2 
                             
                           
                           + 
                           
                             
                               
                                 
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                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
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                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
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                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                             
                               b 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       x 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   c 
                                   y 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 θ 
                               
                               + 
                               
                                 
                                   c 
                                   x 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                             
                               a 
                               2 
                             
                           
                           + 
                           
                             
                               
                                 
                                   + 
                                   
                                     c 
                                     y 
                                   
                                 
                                 ⁢ 
                                 
                                   cos 
                                   2 
                                 
                                 ⁢ 
                                 θ 
                               
                               - 
                               
                                 
                                   c 
                                   x 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                             
                               b 
                               2 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       y 
                     
                     + 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       c 
                                       x 
                                       2 
                                     
                                     ⁢ 
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     θ 
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       c 
                                       x 
                                     
                                     ⁢ 
                                     
                                       c 
                                       y 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   + 
                                   
                                     
                                       c 
                                       y 
                                       2 
                                     
                                     ⁢ 
                                     
                                       sin 
                                       2 
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                                 
                                   a 
                                   2 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     c 
                                     x 
                                     2 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   θ 
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     c 
                                     x 
                                   
                                   ⁢ 
                                   
                                     c 
                                     y 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   
                                     c 
                                     y 
                                     2 
                                   
                                   ⁢ 
                                   
                                     cos 
                                     2 
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               
                                 b 
                                 2 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   1 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   16 
                 
               
             
           
         
       
     
     Rearranging terms and equating to zero yields
 
( b   2  cos 2    θ+a   2  sin 2  θ) x   2 +( a   2  cos 2    θ+b   2  sin 2  θ) y   2 +( b   2  cos θ sin θ− a   2  cos θ sin θ)2 xy −( b   2 ( c   x  cos 2    θ+c   y  cos θ sin θ)+ a   2 ( c   x  sin 2    θ−c   y  cos θ sin θ))2 x −( b   2 ( c   y  sin 2    θ+c   x  cos θ sin θ)+ a   2 ( c   y  cos 2    θ−c   x  cos θ sin θ))2 y +( b   2 ( c   x   2  cos 2  θ+2 c   x   c   y  cos θ sin θ+ c   y   2  sin 2  θ)+ a   2 ( c   x   2  sin 2  θ−2 c   x   c   y  cos θ sin θ+ c   y   2  cos 2  θ))− a   2   b   2 =0  Eq. 17
 
     Now define the following parameters:
 
 A=b   2  cos 2    θ+a   2  sin 2  θ
 
 B=a   2  cos 2    θ+b   2  sin 2  θ
 
 C= 2( b   2  cos θ sin θ− a   2  cos θ sin θ)
 
 D=− 2[ b   2 (− c   x  cos 2    θ−c   y  cos θ sin θ)+ a   2 (− c   x  sin 2    θ+c   y  cos θ sin θ)]
 
 E=− 2[ b   2 (− c   y  sin 2    θ−c   y  cos θ sin θ)+ a   2 (− c   y  cos 2    θ+c   y  cos θ sin θ)]
 
 F =( b   2 ( c   x   2  cos 2  θ+2 c   x   c   y  cos θ sin θ+ c   y   2  sin 2  θ)+ a   2 ( c   x   2  sin 2  θ−2 c   x   c   y  cos θ sin θ+ c   y   2  cos 2  θ))− a   2   b   2   Eq. 18
 
     The following equation may be defined
 
 Ax   2   +By   2   +Cxy+Dx+Ey+F= 0  Eq. 19
 
     For a given value of x, Eq. 19 can be written as 
     
       
         
           
             
               
                 
                   
                     
                       
                         y 
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               Cx 
                               + 
                               E 
                             
                             B 
                           
                           ) 
                         
                         ⁢ 
                         y 
                       
                       + 
                       
                         ( 
                         
                           
                             
                               Ax 
                               2 
                             
                             + 
                             Dx 
                             + 
                             F 
                           
                           B 
                         
                         ) 
                       
                     
                     = 
                     0 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   Define 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   20 
                 
               
             
             
               
                 
                   
                     
                       
                         y 
                         2 
                       
                       + 
                       
                         α 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         y 
                       
                       + 
                       γ 
                     
                     = 
                     0 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   21 
                 
               
             
             
               
                 
                   α 
                   = 
                   
                     
                       Cx 
                       + 
                       E 
                     
                     B 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   22 
                 
               
             
             
               
                 
                   γ 
                   = 
                   
                     
                       
                         Ax 
                         2 
                       
                       + 
                       Dx 
                       + 
                       F 
                     
                     B 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   23 
                 
               
             
           
         
       
     
     Completing the square yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             y 
                             - 
                             h 
                           
                           ) 
                         
                         2 
                       
                       + 
                       k 
                     
                     = 
                     0 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   24 
                 
               
             
             
               
                 
                   h 
                   = 
                   
                     - 
                     
                       α 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   25 
                 
               
             
             
               
                 
                   
                     k 
                     = 
                     
                       γ 
                       - 
                       
                         
                           α 
                           2 
                         
                         4 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   then 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   26 
                 
               
             
             
               
                 
                   y 
                   = 
                   
                     h 
                     ± 
                     
                       
                         - 
                         k 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   27 
                 
               
             
           
         
       
     
     To find the maximum and minimum x value for a given offset and rotation
 
 x (θ)= c   x   +a  cos(θ)cos(θ)− b  sin(θ)sin(θ)  Eq. 28
 
     The inflection point may be found by taking the first derivative. 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       ⁡ 
                       
                         [ 
                         
                           x 
                           ⁡ 
                           
                             ( 
                             ϑ 
                             ) 
                           
                         
                         ] 
                       
                     
                     
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ϑ 
                     
                   
                   = 
                   
                     
                       
                         
                           - 
                           a 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             ϑ 
                             ) 
                           
                         
                       
                       - 
                       
                         b 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             ϑ 
                             ) 
                           
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   29 
                 
               
             
           
         
       
     
     The angle may be solved for the maximum. 
     
       
         
           
             
               
                 
                   
                     
                       tan 
                       ⁡ 
                       
                         ( 
                         
                           ϑ 
                           max 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         - 
                         
                           b 
                           a 
                         
                       
                       ⁢ 
                       
                         tan 
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       ϑ 
                       max 
                     
                     = 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               b 
                               a 
                             
                           
                           ⁢ 
                           
                             tan 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   30 
                 
               
             
           
         
       
     
     Using this angle, the x minimum and maximum may be determined.
 
 x   max   =c   x   +a [cos(θ)cos(θ max )− b  sin(θ)sin(θ max )]  Eq. 31
 
 x   min   =c   x   −[a  cos(θ)cos(θ max )− b  sin(θ)sin(θ max )]  Eq. 32
 
     Based upon equations 31 and 32, it may be known that the domain of the x values lie between x min  and x max  such that the resulting range falls on the ellipse. This domain may be used, in conjunction with equation 27, to plot out the resulting ellipse. 
       FIG. 7B  illustrates one example of elliptical geometry that may be used to calculate the location of an object at the intersection of two ellipses according to an embodiment of the invention. Consider the case of two ellipses, expressed in parametric values as shown in equation 19, and both converging at a common intersection point.
 
 A   1   x   2   +B   1   y   2   +C   1   xy+D   1   x+E   1   y+F   1 =0
 
 A   2   x   2   +B   2   y   2   +C   2   xy+D   2   x+E   2   y+F   2 =0  Eq. 33
 
     Following the method leading up to equation 27, two equations expressing the same y value may be written. 
     
       
         
           
             
               
                 
                   
                     
                       
                         y 
                         = 
                           
                         ⁢ 
                         
                           
                             h 
                             1 
                           
                           ± 
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             
                               
                                 α 
                                 1 
                               
                               2 
                             
                           
                           ± 
                           
                             
                               
                                 - 
                                 
                                   γ 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   α 
                                   1 
                                   2 
                                 
                                 4 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁢ 
                                   x 
                                 
                                 + 
                                 
                                   E 
                                   1 
                                 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ± 
                           
                             
                               
                                 - 
                                 
                                   
                                     
                                       
                                         A 
                                         1 
                                       
                                       ⁢ 
                                       
                                         x 
                                         2 
                                       
                                     
                                     + 
                                     
                                       
                                         D 
                                         1 
                                       
                                       ⁢ 
                                       x 
                                     
                                     + 
                                     
                                       F 
                                       1 
                                     
                                   
                                   
                                     B 
                                     1 
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   4 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             C 
                                             1 
                                           
                                           ⁢ 
                                           x 
                                         
                                         + 
                                         
                                           E 
                                           1 
                                         
                                       
                                       
                                         B 
                                         1 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   34 
                 
               
             
             
               
                 
                   
                     
                       
                         y 
                         = 
                           
                         ⁢ 
                         
                           
                             h 
                             2 
                           
                           ± 
                           
                             
                               - 
                               
                                 k 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             
                               
                                 α 
                                 2 
                               
                               2 
                             
                           
                           ± 
                           
                             
                               
                                 γ 
                                 2 
                               
                               - 
                               
                                 
                                   α 
                                   2 
                                   2 
                                 
                                 4 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁢ 
                                   x 
                                 
                                 + 
                                 
                                   E 
                                   2 
                                 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                           
                           ± 
                           
                             
                               
                                 - 
                                 
                                   
                                     
                                       
                                         A 
                                         2 
                                       
                                       ⁢ 
                                       
                                         x 
                                         2 
                                       
                                     
                                     + 
                                     
                                       
                                         D 
                                         2 
                                       
                                       ⁢ 
                                       x 
                                     
                                     + 
                                     
                                       F 
                                       2 
                                     
                                   
                                   
                                     B 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   4 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             C 
                                             2 
                                           
                                           ⁢ 
                                           x 
                                         
                                         + 
                                         
                                           E 
                                           2 
                                         
                                       
                                       
                                         B 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   35 
                 
               
             
           
         
       
     
     Equating equations 34 and 35 we get: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         
                           
                             
                               C 
                               1 
                             
                             ⁢ 
                             x 
                           
                           + 
                           
                             E 
                             1 
                           
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             B 
                             1 
                           
                         
                       
                     
                     ± 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       1 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     1 
                                   
                                 
                                 
                                   B 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             
                               C 
                               2 
                             
                             ⁢ 
                             x 
                           
                           + 
                           
                             E 
                             2 
                           
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             B 
                             2 
                           
                         
                       
                     
                     ± 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       2 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   36 
                 
               
             
           
         
       
     
     For the special case where there is no rotation and the light sources  110 -a,  110 -b and light receiver  120  are in a linear row it is observed that C 1 =E 1 =C 2 =E 2 =0. This can be termed the collinear solution. 
     Equation 36 may be rewritten as 
     
       
         
           
             
               
                 
                   
                     ± 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                         
                         = 
                       
                     
                   
                   ± 
                   
                     
                       - 
                       
                         
                           
                             
                               A 
                               2 
                             
                             ⁢ 
                             
                               x 
                               2 
                             
                           
                           + 
                           
                             
                               D 
                               2 
                             
                             ⁢ 
                             x 
                           
                           + 
                           
                             F 
                             2 
                           
                         
                         
                           B 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   37 
                 
               
             
           
         
       
     
     Squaring both sides yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           A 
                           1 
                         
                         ⁢ 
                         
                           x 
                           2 
                         
                       
                       + 
                       
                         
                           D 
                           1 
                         
                         ⁢ 
                         x 
                       
                       + 
                       
                         F 
                         1 
                       
                     
                     
                       B 
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           A 
                           2 
                         
                         ⁢ 
                         
                           x 
                           2 
                         
                       
                       + 
                       
                         
                           D 
                           2 
                         
                         ⁢ 
                         x 
                       
                       + 
                       
                         F 
                         2 
                       
                     
                     
                       B 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   38 
                 
               
             
           
         
       
     
     Collecting terms yields
 
 x   2 ( A   1   B   2   −A   2   B   1 )+ x ( D   1   B   2   −D   2   B   1 )+( F   1   B   2   −F   2   B   1 )=0  Eq. 39
 
Let
 
σ= A   1   B   2   −A   2   B   1  
 
ν= D   1   B   2   −D   2   B   1  
 
μ= F   1   B   2   −F   2   B   1   Eq. 40
 
     Equation 39 can be rewritten in a parametric quadratic equation form as
 
 x   2   σ+xν+μ= 0  Eq. 41
 
     This can be solved using the quadratic formula, yielding 
     
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       
                         - 
                         v 
                       
                       ± 
                       
                         
                           
                             v 
                             2 
                           
                           - 
                           
                             4 
                             ⁢ 
                             σμ 
                           
                         
                       
                     
                     
                       2 
                       ⁢ 
                       σ 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   42 
                 
               
             
           
         
       
     
     Equation 42 may be substituted back into equation 34 or 35 to solve for y. The object  250  location may now be known as the point (x,y) that was just calculated. 
     The general solution in which there may be rotation and the light sources  110 -a,  110 -b and light receiver  120  are not necessarily in a linear row can be solved starting with equation 36 (reproduced below). 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         
                           
                             
                               C 
                               1 
                             
                             ⁢ 
                             x 
                           
                           + 
                           
                             E 
                             1 
                           
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             B 
                             1 
                           
                         
                       
                     
                     ± 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       1 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     1 
                                   
                                 
                                 
                                   B 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             
                               C 
                               2 
                             
                             ⁢ 
                             x 
                           
                           + 
                           
                             E 
                             2 
                           
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             B 
                             2 
                           
                         
                       
                     
                     ± 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       2 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   36 
                 
               
             
           
         
       
     
     Collecting the radicals on the right side yields 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               C 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               C 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       ( 
                       
                         
                           
                             E 
                             2 
                           
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               B 
                               2 
                             
                           
                         
                         - 
                         
                           
                             E 
                             1 
                           
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               B 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∓ 
                       
                         
                           
                             - 
                             
                               
                                 
                                   
                                     A 
                                     1 
                                   
                                   ⁢ 
                                   
                                     x 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     D 
                                     1 
                                   
                                   ⁢ 
                                   x 
                                 
                                 + 
                                 
                                   F 
                                   1 
                                 
                               
                               
                                 B 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               1 
                               4 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         C 
                                         1 
                                       
                                       ⁢ 
                                       x 
                                     
                                     + 
                                     
                                       E 
                                       1 
                                     
                                   
                                   
                                     B 
                                     1 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                     ± 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       2 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   43 
                 
               
             
           
         
       
     
     Squaring both sides of the equation yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           x 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 C 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 C 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       2 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         x 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 C 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 C 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               E 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               E 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               E 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               E 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                   = 
                   
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 E 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     - 
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁢ 
                           
                             x 
                             2 
                           
                         
                         + 
                         
                           
                             D 
                             1 
                           
                           ⁢ 
                           x 
                         
                         + 
                         
                           F 
                           1 
                         
                       
                       
                         B 
                         1 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             - 
                             
                               
                                 
                                   
                                     A 
                                     1 
                                   
                                   ⁢ 
                                   
                                     x 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     D 
                                     1 
                                   
                                   ⁢ 
                                   x 
                                 
                                 + 
                                 
                                   F 
                                   1 
                                 
                               
                               
                                 B 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               1 
                               4 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         C 
                                         1 
                                       
                                       ⁢ 
                                       x 
                                     
                                     + 
                                     
                                       E 
                                       1 
                                     
                                   
                                   
                                     B 
                                     1 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             - 
                             
                               
                                 
                                   
                                     A 
                                     2 
                                   
                                   ⁢ 
                                   
                                     x 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     D 
                                     2 
                                   
                                   ⁢ 
                                   x 
                                 
                                 + 
                                 
                                   F 
                                   2 
                                 
                               
                               
                                 B 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               1 
                               4 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         C 
                                         2 
                                       
                                       ⁢ 
                                       x 
                                     
                                     + 
                                     
                                       E 
                                       2 
                                     
                                   
                                   
                                     B 
                                     2 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 E 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     - 
                     
                       
                         
                           
                             A 
                             2 
                           
                           ⁢ 
                           
                             x 
                             2 
                           
                         
                         + 
                         
                           
                             D 
                             2 
                           
                           ⁢ 
                           x 
                         
                         + 
                         
                           F 
                           2 
                         
                       
                       
                         B 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   44 
                 
               
             
           
         
       
     
     Moving the radicals to the right side of the equation yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           x 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 C 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 C 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       2 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         x 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 C 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 C 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               E 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               E 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               E 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               E 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 E 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁢ 
                           
                             x 
                             2 
                           
                         
                         + 
                         
                           
                             D 
                             1 
                           
                           ⁢ 
                           x 
                         
                         + 
                         
                           F 
                           1 
                         
                       
                       
                         B 
                         1 
                       
                     
                     - 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 E 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             A 
                             2 
                           
                           ⁢ 
                           
                             x 
                             2 
                           
                         
                         + 
                         
                           
                             D 
                             2 
                           
                           ⁢ 
                           x 
                         
                         + 
                         
                           F 
                           2 
                         
                       
                       
                         B 
                         2 
                       
                     
                   
                   = 
                   
                     
                       - 
                       2 
                     
                     ⁢ 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       1 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     1 
                                   
                                 
                                 
                                   B 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       2 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   45 
                 
               
             
           
         
       
     
     Expanding the left side of the equation yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           x 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 C 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 C 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       2 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         x 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 C 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 C 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               E 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               E 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               E 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               E 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             1 
                             2 
                           
                           ⁢ 
                           
                             x 
                             2 
                           
                         
                         
                           B 
                           1 
                           2 
                         
                       
                     
                     - 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             xC 
                             1 
                           
                           ⁢ 
                           
                             E 
                             1 
                           
                         
                         
                           B 
                           1 
                           2 
                         
                       
                     
                     - 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           E 
                           1 
                           2 
                         
                         
                           B 
                           1 
                           2 
                         
                       
                     
                     + 
                     
                       
                         
                           A 
                           1 
                         
                         ⁢ 
                         
                           x 
                           2 
                         
                       
                       
                         B 
                         1 
                       
                     
                     + 
                     
                       
                         
                           D 
                           1 
                         
                         ⁢ 
                         x 
                       
                       
                         B 
                         1 
                       
                     
                     + 
                     
                       
                         F 
                         1 
                       
                       
                         B 
                         1 
                       
                     
                     - 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             2 
                             2 
                           
                           ⁢ 
                           
                             x 
                             2 
                           
                         
                         
                           B 
                           2 
                           2 
                         
                       
                     
                     - 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             xC 
                             2 
                           
                           ⁢ 
                           
                             E 
                             2 
                           
                         
                         
                           B 
                           2 
                           2 
                         
                       
                     
                     - 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           E 
                           2 
                           2 
                         
                         
                           B 
                           2 
                           2 
                         
                       
                     
                     + 
                     
                       
                         
                           A 
                           2 
                         
                         ⁢ 
                         
                           x 
                           2 
                         
                       
                       
                         B 
                         2 
                       
                     
                     + 
                     
                       
                         
                           D 
                           2 
                         
                         ⁢ 
                         x 
                       
                       
                         B 
                         2 
                       
                     
                     + 
                     
                       
                         F 
                         2 
                       
                       
                         B 
                         2 
                       
                     
                   
                   = 
                   
                     
                       - 
                       2 
                     
                     ⁢ 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       1 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     1 
                                   
                                 
                                 
                                   B 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       2 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   46 
                 
               
             
           
         
       
     
     Collecting terms to form a quadratic equation on the left side yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         2 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       B 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     C 
                                     1 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       B 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               1 
                               4 
                             
                             ⁢ 
                             
                               
                                 C 
                                 1 
                                 2 
                               
                               
                                 B 
                                 1 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               A 
                               1 
                             
                             
                               B 
                               1 
                             
                           
                           - 
                           
                             
                               1 
                               4 
                             
                             ⁢ 
                             
                               
                                 C 
                                 2 
                                 2 
                               
                               
                                 B 
                                 2 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               A 
                               2 
                             
                             
                               B 
                               2 
                             
                           
                         
                         } 
                       
                     
                     + 
                     
                       x 
                       ⁢ 
                       
                         { 
                         
                           
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       B 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     C 
                                     1 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       B 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     E 
                                     2 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       B 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     E 
                                     1 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       B 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁢ 
                                 
                                   E 
                                   1 
                                 
                               
                               
                                 B 
                                 1 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               D 
                               1 
                             
                             
                               B 
                               1 
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁢ 
                                 
                                   E 
                                   2 
                                 
                               
                               
                                 B 
                                 2 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               D 
                               2 
                             
                             
                               B 
                               2 
                             
                           
                         
                         } 
                       
                     
                     + 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               
                                 
                                   E 
                                   2 
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     B 
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   E 
                                   1 
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     B 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               E 
                               1 
                               2 
                             
                             
                               B 
                               1 
                               2 
                             
                           
                         
                         + 
                         
                           
                             F 
                             1 
                           
                           
                             B 
                             1 
                           
                         
                         - 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               E 
                               2 
                               2 
                             
                             
                               B 
                               2 
                               2 
                             
                           
                         
                         + 
                         
                           
                             F 
                             2 
                           
                           
                             B 
                             2 
                           
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       - 
                       2 
                     
                     ⁢ 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       1 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     1 
                                   
                                 
                                 
                                   B 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       2 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   47 
                 
               
             
           
         
       
     
     Expressing equation 47 in terms of parametric parameters yields 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     G 
                     = 
                     
                       
                         
                           ( 
                           
                             
                               
                                 C 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 C 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       - 
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           
                             C 
                             1 
                             2 
                           
                           
                             B 
                             1 
                             2 
                           
                         
                       
                       + 
                       
                         
                           A 
                           1 
                         
                         
                           B 
                           1 
                         
                       
                       - 
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           
                             C 
                             2 
                             2 
                           
                           
                             B 
                             2 
                             2 
                           
                         
                       
                       + 
                       
                         
                           A 
                           2 
                         
                         
                           B 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   48 
                 
               
             
             
               
                 
                   H 
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               C 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               C 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               E 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               E 
                               1 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 B 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             1 
                           
                           ⁢ 
                           
                             E 
                             1 
                           
                         
                         
                           B 
                           1 
                           2 
                         
                       
                     
                     + 
                     
                       
                         D 
                         1 
                       
                       
                         B 
                         1 
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             E 
                             2 
                           
                         
                         
                           B 
                           2 
                           2 
                         
                       
                     
                     + 
                     
                       
                         D 
                         2 
                       
                       
                         B 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   49 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     I 
                     = 
                     
                       
                         
                           ( 
                           
                             
                               
                                 E 
                                 2 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 E 
                                 1 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       - 
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           
                             E 
                             1 
                             2 
                           
                           
                             B 
                             1 
                             2 
                           
                         
                       
                       + 
                       
                         
                           F 
                           1 
                         
                         
                           B 
                           1 
                         
                       
                       - 
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           
                             E 
                             2 
                             2 
                           
                           
                             B 
                             2 
                             2 
                           
                         
                       
                       + 
                       
                         
                           F 
                           2 
                         
                         
                           B 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   50 
                 
               
             
             
               
                 
                   
                     
                       
                         x 
                         2 
                       
                       ⁢ 
                       G 
                     
                     + 
                     xH 
                     + 
                     I 
                   
                   = 
                   
                     
                       - 
                       2 
                     
                     ⁢ 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       1 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     1 
                                   
                                 
                                 
                                   B 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       2 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   51 
                 
               
             
           
         
       
     
     Squaring both sides of equation 51 yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         4 
                       
                       ⁢ 
                       
                         G 
                         2 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         GHx 
                         3 
                       
                     
                     + 
                     
                       
                         x 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             GI 
                           
                           + 
                           
                             H 
                             2 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       xHI 
                     
                     + 
                     
                       I 
                       2 
                     
                   
                   = 
                   
                     4 
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   1 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 1 
                               
                             
                             
                               B 
                               1 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       1 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     1 
                                   
                                 
                                 
                                   B 
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       } 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           
                             
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   D 
                                   2 
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 F 
                                 2 
                               
                             
                             
                               B 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             4 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       C 
                                       2 
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   52 
                 
               
             
           
         
       
     
     Collecting terms yields 
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         4 
                       
                       ⁢ 
                       
                         G 
                         2 
                       
                     
                     + 
                     
                       
                         x 
                         3 
                       
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           GH 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         x 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             GI 
                           
                           + 
                           
                             H 
                             2 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       x 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           HI 
                         
                         ) 
                       
                     
                     + 
                     
                       I 
                       2 
                     
                   
                   = 
                   
                     4 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             x 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   C 
                                   1 
                                   2 
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     B 
                                     1 
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   A 
                                   1 
                                 
                                 
                                   B 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           x 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                   
                                   ⁢ 
                                   
                                     E 
                                     1 
                                   
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     B 
                                     1 
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   D 
                                   1 
                                 
                                 
                                   B 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           ( 
                           
                             
                               
                                 E 
                                 1 
                                 2 
                               
                               
                                 4 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   1 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 F 
                                 1 
                               
                               
                                 B 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             x 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   C 
                                   2 
                                   2 
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     B 
                                     2 
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           x 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     C 
                                     2 
                                   
                                   ⁢ 
                                   
                                     E 
                                     2 
                                   
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     B 
                                     2 
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   D 
                                   2 
                                 
                                 
                                   B 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           ( 
                           
                             
                               
                                 E 
                                 2 
                                 2 
                               
                               
                                 4 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   B 
                                   2 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 F 
                                 2 
                               
                               
                                 B 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   53 
                 
               
             
           
         
       
     
     Defining the following parameters yields 
     
       
         
           
             
               
                 
                   J 
                   = 
                   
                     
                       
                         C 
                         1 
                         2 
                       
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           1 
                           2 
                         
                       
                     
                     - 
                     
                       
                         A 
                         1 
                       
                       
                         B 
                         1 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   54 
                 
               
             
             
               
                 
                   K 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                         ⁢ 
                         
                           E 
                           1 
                         
                       
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           1 
                           2 
                         
                       
                     
                     - 
                     
                       
                         D 
                         1 
                       
                       
                         B 
                         1 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   55 
                 
               
             
             
               
                 
                   L 
                   = 
                   
                     
                       
                         E 
                         1 
                         2 
                       
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           1 
                           2 
                         
                       
                     
                     - 
                     
                       
                         F 
                         1 
                       
                       
                         B 
                         1 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   56 
                 
               
             
             
               
                 
                   M 
                   = 
                   
                     
                       
                         C 
                         2 
                         2 
                       
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           2 
                           2 
                         
                       
                     
                     - 
                     
                       
                         A 
                         2 
                       
                       
                         B 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   57 
                 
               
             
             
               
                 
                   N 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           C 
                           2 
                         
                         ⁢ 
                         
                           E 
                           2 
                         
                       
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           2 
                           2 
                         
                       
                     
                     - 
                     
                       
                         D 
                         2 
                       
                       
                         B 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   58 
                 
               
             
             
               
                 
                   O 
                   = 
                   
                     
                       
                         E 
                         2 
                         2 
                       
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           2 
                           2 
                         
                       
                     
                     - 
                     
                       
                         F 
                         2 
                       
                       
                         B 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   59 
                 
               
             
           
         
       
     
     Equation 53 may be rewritten as shown below.
 
 x   4   G   2   +x   3 (2 GH )+ x   2 (2 GI+H   2 )+ x (2 HI )+ I   2 =4{ x   2   J+xK+L}{x   2   M+xN+O}   Eq. 60
 
     Expanding the right side yields
 
 x   4   G   2   +x   3 (2 GH )+ x   2 (2 GI+H   2 )+ x (2 HI )+ I   2   =x   4 4 JM+x   3 4( JN+KM )+ x   2 4( JO+KN+LM )+ x 4( KO+LN )+4 LO   Eq. 61
 
     Collecting terms and equating to zero yields
 
 x   4 ( G   2 −4 JM )+ x   3 (2 GH −4 JN− 4 KM )+ x   2 (2 GI+H   2 −4 JO− 4 KN− 4 LM )+ x (2 HI− 4 KO− 4 LN )+( I   2 −4 LO )=0  Eq. 62
 
     Defining the following parameters yields 
     
       
         
           
             
               
                 
                   Q 
                   = 
                   
                     
                       G 
                       2 
                     
                     - 
                     
                       4 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       JM 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   63 
                 
               
             
             
               
                 
                   V 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         GH 
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         JN 
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         KM 
                       
                     
                     Q 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   64 
                 
               
             
             
               
                 
                   S 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         GI 
                       
                       + 
                       
                         H 
                         2 
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         JO 
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         KN 
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         LM 
                       
                     
                     Q 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   65 
                 
               
             
             
               
                 
                   W 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         HI 
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         KO 
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         LN 
                       
                     
                     Q 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   66 
                 
               
             
             
               
                 
                   U 
                   = 
                   
                     
                       
                         I 
                         2 
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         LO 
                       
                     
                     Q 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   67 
                 
               
             
           
         
       
     
     Equation 62 may be rewritten in a quartic form as
 
 x   4   +x   3   V+x   2   S+xW+U= 0  Eq. 68
 
     A quartic solver method in which the four roots of Equation 68 (r1, r2, r3 and r4) are solved by the following method may be used. 
     Defining the following parameters yields 
     
       
         
           
             
               
                 
                   
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     Based upon these defined parameters, the four roots of equation 68 may be solved for as according to
 
 r 1= T 1− R 4−√{square root over ( R 5− R 6)}  Eq. 80
 
 r 2= T 1− R 4+√{square root over ( R 5− R 6)}  Eq. 81
 
 r 3= T 1+ R 4−√{square root over ( R 5+ R 6)}  Eq. 82
 
 r 4= T 1+ R 4+√{square root over ( R 5+ R 6)}  Eq. 83
 
     Referring to  FIG. 7B , of the four roots that define an x value, the real roots may be kept. In general, the way the problem was defined in  FIG. 7B , two of the roots will be real and equal valued. However, in general this may not be the case as shown in  FIG. 7C  where the x values at the intersection are not necessarily equal depending upon how the reference plane has been defined. 
     In the case represented by  FIG. 7C  the goal may be to find the x value that yields a real and positive y value while also satisfying the measured bounce distance D b . To find that y value, we substitute the appropriate root from equations 80 through 83 back into either equation 34 or 35 and solve for y. 
     
       
         
           
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     Included herein are one or more flow charts representative of exemplary methodologies for performing novel aspects of the disclosed architecture. While, for purposes of simplicity of explanation, the one or more methodologies shown herein, for example, in the form of a flow chart or flow diagram, are shown and described as a series of acts, it is to be understood and appreciated that the methodologies are not limited by the order of acts, as some acts may, in accordance therewith, occur in a different order and/or concurrently with other acts from that shown and described herein. For example, those skilled in the art will understand and appreciate that a methodology could alternatively be represented as a series of interrelated states or events, such as in a state diagram. Moreover, not all acts illustrated in a methodology may be required for a novel implementation. 
       FIG. 8  illustrates one embodiment of a logic flow in which the distance to an object may be calculated. The logic flow  800  may be representative of some or all of the operations executed by one or more embodiments described herein. 
     In the illustrated embodiment shown in  FIG. 8 , the logic flow  800  may emit modulated light from first and second light sources  110 -a,  110 -b at block  810 . For example, the first and second light sources  110 -a,  110 -b may be integrated into the taillights of an automobile  210 . The first and second light sources  110 -a,  110 -b may be comprised of multiple LEDs  115 . A modulation component  140  under the control of a processor circuit  130  may modulate the first and second light sources  110 -a,  110 -b to pulse on and off at a specified frequency. This may be termed on-off keying (OOK) and creates a ranging tone for the light sources  110 -a,  110 -b. The ranging tone for the first light source  110 -a may be different from the ranging tone for the second light source  110 -b so as to be able to distinguish light emitted from each. The embodiments are not limited to this example. 
     The logic flow  800  may detect modulated light in a light receiver  120  that has been reflected off an object  250  at block  820 . For example, the modulated light emitted by the first and second light sources  110 -a,  110 -b may reflect or bounce off an object  250 . The reflected modulated light may then enter the light receiver  120  where it can undergo processing. The embodiments are not limited to this example. 
     The logic flow  800  may filter the incoming reflected light at block  830 . For example, the light received by the light receiver  120  may be converted to signals that can be filtered according to the modulation scheme implemented at the first and second light sources  110 -a,  110 -b. A filtering component  160  under control of a processor circuit  130  may be operative to screen out any signals that do not match either the first or second ranging tones that may be associated with light emitted from the first and second light sources  110 -a,  110 -b. The embodiments are not limited to this example. 
     The logic flow  800  may process the filtered signals to determine the location of the object  250  at block  840 . For example, the processing circuit  130  may be able to calculate the location of the object  250  knowing several pieces of information and using elliptical geometry. For instance, the first light source  110 -a and the light receiver  120  can be used as the foci for a first ellipse if the propagation delay for light emitted by the first light source  110 -a and received by the light receiver  120  after bouncing off the object  250  can be determined. Similarly, the first light source  110 -a and the light receiver  120  can be used as the foci for a second ellipse if the propagation delay for light emitted by the second light source  110 -b and received by the light receiver  120  after bouncing off the object  250  can be determined. Recall that the first and second light sources  110 -a,  110 -b and the light receiver  120  are all in fixed and known positions. The two ellipses will intersect at a point where the object  250  is located. Based on the known information, the location for the object can be calculated. The embodiments are not limited to this example. 
       FIG. 9  illustrates another embodiment of a logic flow  840  in which the distance of an object may be calculated. The logic flow  840  may be representative of some or all of the operations executed by one or more embodiments described herein. 
     In the illustrated embodiment shown in  FIG. 9 , the logic flow  840  may calculate a propagation delay (τ 1 ) for light that was emitted by the first light source  110 -a and detected by the light receiver  120  after reflecting off the object  250  at block  910 . For example, the first light source  110 -a located at a known and fixed position (x t , y t ) may emit modulated light which may strike the object  250  and be reflected to the light receiver  120  at a known and fixed position (x r , y r ). It may take a certain time τ t  for the emitted light to reach the object and another time τ r  to reach the light receiver  120 . The total time delay from the first light source  110 -a to light receiver  120  can be represented by τ 1 =τ 1 +τ r . The embodiments are not limited to this example. 
     The logic flow  840  may calculate a bounce distance (D 1 ) from the first light source  110 -a to the light receiver  120  after reflecting off the object  250  at block  920 . For example, the distance from the first light source  110 -a to the object  250  to the light receiver  120  can be calculated as:
 
 D   1   =c*τ   1  
 
in which c is the speed of light. The embodiments are not limited to this example.
 
     The logic flow  840  may calculate a propagation delay (τ 2 ) for light that was emitted by the second light source  110 -b and detected by the light receiver  120  after reflecting off the object  250  at block  930 . For example, the second light source  110 -b located at a known and fixed position (x t , y t ) may emit modulated light which may strike the object  250  and be reflected to the light receiver  120  at a known and fixed position (x r , y r ). It may take a certain time τ t  for the emitted light to reach the object and another time τ r  to reach the light receiver  120 . The total time delay from the second light source  110 -b to light receiver  120  can be represented by τ 2 =τ t +τ r . The embodiments are not limited to this example. 
     The logic flow  840  may calculate a bounce distance (D 1 ) from the second light source  110 -b to the light receiver  120  after reflecting off the object  250  at block  940 . For example, the distance from the second light source  110 -b to the object  250  to the light receiver  120  can be calculated as:
 
 D   2   =c*τ   2  
 
in which c is the speed of light. The embodiments are not limited to this example.
 
     The logic flow  840  may construct a first semi-ellipse in which the location of the first light source  110 -a and the location of the light receiver  120  represent the foci of the first semi-ellipse at block  950 . For example, the location of the first light source  110 -a and the location of the light receiver  120  are both known. The time delay for light emitted from the first light source  110 -a to the light receiver  120  after bouncing off object  250  is also known. The distance from the first light source  110 -a to the light receiver  120  after bouncing off object  250  has been calculated. Knowing these pieces of information allows for the construction of the semi-ellipse using the math described with respect to  FIG. 7A  above. The embodiments are not limited to this example. 
     The logic flow  840  may construct a first semi-ellipse in which the location of the first light source  110 -b and the location of the light receiver  120  represent the foci of the first semi-ellipse at block  960 . For example, the location of the second light source  110 -b and the location of the light receiver  120  are both known. The time delay for light emitted from the first light source  110 -b to the light receiver  120  after bouncing off object  250  is also known. The distance from the second light source  110 -b to the light receiver  120  after bouncing off object  250  has been calculated. Knowing these pieces of information allows for the construction of the semi-ellipse using the math described with respect to  FIG. 7A  above. In addition, the intersection point (P) of the two ellipses represents the location of object  250 . The embodiments are not limited to this example. 
     The logic flow  840  may solve for the intersection point (P) of the first and second semi-ellipses at block  970 . For example, the semi-ellipses have been constructed according to the math associated with  FIG. 7A  above and the intersection point (P) of the two ellipses represents the location of object  250 . The intersection point (P) can be represented in the Cartesian coordinate system as an (x,y) value. The equations for the two ellipses may have been derived as previously described. Because they intersect, the two ellipses share a common point which happens to be the object location. That point can be solved for using the math described above with reference to  FIGS. 7B-C . The embodiments are not limited to this example. 
       FIG. 10  illustrates one embodiment of a logic flow  1000  in which an image of the object  250  may be displayed on a display  180 . The logic flow  1000  may be representative of some or all of the operations executed by one or more embodiments described herein. 
     In the illustrated embodiment shown in  FIG. 10 , the logic flow  1000  may map an azimuth angle of arrival for detected emitted light matching the modulation scheme to pixel locations on an image plane at block  1010 . For example, light may enter a lensed detector array  170 . The lensed detector array  170  may be comprised of a lens  310  and a linear detector array  320 . Light reflected  230  of the object  250  may enter lens  310 . The lensed detector array  170  may be operative to map an azimuth angle of arrival for detected emitted light that has been reflected off an object and that matches the first or second OOK ranging tones to pixel locations on an image plane of the linear detector array  320 . 
     The logic flow  1000  may display an image of the object  250  at block  1020 . For example, a display  180  may be communicatively coupled with the lensed detector array  170  and may be operative to display the mapped pixel locations to create an image of the object  250  reflecting the emitted light. The embodiments are not limited to this example. 
     The processor circuit  130  may also cause an alert to be issued based on the location of the object  250 . For instance, the processor circuit  130  may be communicatively coupled with other applications and components such as the display  180  and a speaker  185 . Based on the location of the object a visual alert may be displayed on display  180  to alert a human operator as to the existence and location of the object. Moreover, the distance of the object determined by the calculation application  165  may be displayed on the display along with a rendering of the object. The distance may be used as a threshold value when determining whether to initiate an alert. Similarly, an audible alert may be emitted from speaker  185  to alert a human operator as to the existence of the object. The audible alert may change in pitch, frequency and/or tone as the distance to the object gets closer and/or crosses a threshold distance. The embodiments are not limited in this context. 
     It should be noted that the number of light sources and light receivers can vary so long as there are a combined number equal to or greater than three (3). Each distinct light source/light receiver pairing can create an ellipse based on light bounced of an object  250 . Thus, two ellipses bouncing light off the same object will intersect at a common point which happens to be the location of the object  250 . The location of the object  250  can only be solved for if there at least two intersecting ellipses. However, there can be more intersecting ellipses. For instance, a second light receiver may be added which would double the number of intersecting ellipses created to four (4). The math is still the same. 
     The present description has illustrated two light sources and a single light receiver which will generate two ellipses. This configuration was chosen because the light sources may be integrated into an automobile&#39;s  210  existing taillights of which there are two. Another configuration could be one light source and two light receivers. Such a configuration would also generate two ellipses intersecting at the location of the object  250 . This configuration may be implemented on a motorcycle. For instance, the light source may be integrated into the headlight and the light receivers may be integrated into the handle bars of a motorcycle. The embodiments are not limited to these examples. 
       FIG. 11  illustrates an embodiment of an exemplary computing architecture  1100  suitable for implementing various embodiments as previously described. As used in this application, the terms “system” and “device” and “component” are intended to refer to a computer-related entity, either hardware, a combination of hardware and software, software, or software in execution, examples of which are provided by the exemplary computing architecture  1100 . For example, a component can be, but is not limited to being, a process running on a processor, a processor, a hard disk drive, multiple storage drives (of optical and/or magnetic storage medium), an object, an executable, a thread of execution, a program, and/or a computer. By way of illustration, both an application running on a server and the server can be a component. One or more components can reside within a process and/or thread of execution, and a component can be localized on one computer and/or distributed between two or more computers. Further, components may be communicatively coupled to each other by various types of communications media to coordinate operations. The coordination may involve the uni-directional or bi-directional exchange of information. For instance, the components may communicate information in the form of signals communicated over the communications media. The information can be implemented as signals allocated to various signal lines. In such allocations, each message is a signal. Further embodiments, however, may alternatively employ data messages. Such data messages may be sent across various connections. Exemplary connections include parallel interfaces, serial interfaces, and bus interfaces. 
     In one embodiment, the computing architecture  1100  may comprise or be implemented as part of an electronic device. Examples of an electronic device may include without limitation a mobile device, a personal digital assistant, a mobile computing device, a smart phone, a cellular telephone, a handset, a one-way pager, a two-way pager, a messaging device, a computer, a personal computer (PC), a desktop computer, a laptop computer, a notebook computer, a handheld computer, a tablet computer, a server, a server array or server farm, a web server, a network server, an Internet server, a work station, a mini-computer, a main frame computer, a supercomputer, a network appliance, a web appliance, a distributed computing system, multiprocessor systems, processor-based systems, consumer electronics, programmable consumer electronics, television, digital television, set top box, wireless access point, base station, subscriber station, mobile subscriber center, radio network controller, router, hub, gateway, bridge, switch, machine, or combination thereof. The embodiments are not limited in this context. 
     The computing architecture  1100  includes various common computing elements, such as one or more processors, co-processors, memory units, chipsets, controllers, peripherals, interfaces, oscillators, timing devices, video cards, audio cards, multimedia input/output (I/O) components, and so forth. The embodiments, however, are not limited to implementation by the computing architecture  1100 . 
     As shown in  FIG. 11 , the computing architecture  1100  comprises a processing unit  1104 , a system memory  1106  and a system bus  1108 . The processing unit  1104  can be any of various commercially available processors. Dual microprocessors and other multi processor architectures may also be employed as the processing unit  1104 . The system bus  1108  provides an interface for system components including, but not limited to, the system memory  1106  to the processing unit  1104 . The system bus  1108  can be any of several types of bus structure that may further interconnect to a memory bus (with or without a memory controller), a peripheral bus, and a local bus using any of a variety of commercially available bus architectures. 
     The computing architecture  1100  may comprise or implement various articles of manufacture. An article of manufacture may comprise a computer-readable storage medium to store various forms of programming logic. Examples of a computer-readable storage medium may include any tangible media capable of storing electronic data, including volatile memory or non-volatile memory, removable or non-removable memory, erasable or non-erasable memory, writeable or re-writeable memory, and so forth. Examples of programming logic may include executable computer program instructions implemented using any suitable type of code, such as source code, compiled code, interpreted code, executable code, static code, dynamic code, object-oriented code, visual code, and the like. 
     The system memory  1106  may include various types of computer-readable storage media in the form of one or more higher speed memory units, such as read-only memory (ROM), random-access memory (RAM), dynamic RAM (DRAM), Double-Data-Rate DRAM (DDRAM), synchronous DRAM (SDRAM), static RAM (SRAM), programmable ROM (PROM), erasable programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), flash memory, polymer memory such as ferroelectric polymer memory, ovonic memory, phase change or ferroelectric memory, silicon-oxide-nitride-oxide-silicon (SONOS) memory, magnetic or optical cards, or any other type of media suitable for storing information. In the illustrated embodiment shown in  FIG. 6 , the system memory  1106  can include non-volatile memory  1110  and/or volatile memory  1112 . A basic input/output system (BIOS) can be stored in the non-volatile memory  1110 . 
     The computer  1102  may include various types of computer-readable storage media in the form of one or more lower speed memory units, including an internal hard disk drive (HDD)  1114 , a magnetic floppy disk drive (FDD)  1116  to read from or write to a removable magnetic disk  1118 , and an optical disk drive  1120  to read from or write to a removable optical disk  1122  (e.g., a CD-ROM or DVD). The HDD  1114 , FDD  1116  and optical disk drive  1120  can be connected to the system bus  1108  by a HDD interface  1124 , an FDD interface  1126  and an optical drive interface  1128 , respectively. The HDD interface  1124  for external drive implementations can include at least one or both of Universal Serial Bus (USB) and IEEE 1394 interface technologies. 
     The drives and associated computer-readable media provide volatile and/or nonvolatile storage of data, data structures, computer-executable instructions, and so forth. For example, a number of program modules can be stored in the drives and memory units  1110 ,  1112 , including an operating system  1130 , one or more application programs  1132 , other program modules  1134 , and program data  1136 . 
     A user can enter commands and information into the computer  1102  through one or more wire/wireless input devices, for example, a keyboard  1138  and a pointing device, such as a mouse  1140 . Other input devices may include a microphone, an infra-red (IR) remote control, a joystick, a game pad, a stylus pen, touch screen, or the like. These and other input devices are often connected to the processing unit  1104  through an input device interface  1142  that is coupled to the system bus  1108 , but can be connected by other interfaces such as a parallel port, IEEE 1394 serial port, a game port, a USB port, an IR interface, and so forth. 
     A monitor  1144  or other type of display device is also connected to the system bus  1108  via an interface, such as a video adaptor  1146 . In addition to the monitor  1144 , a computer typically includes other peripheral output devices, such as speakers, printers, and so forth. 
     The computer  1102  may operate in a networked environment using logical connections via wire and/or wireless communications to one or more remote computers, such as a remote computer  1148 . The remote computer  1148  can be a workstation, a server computer, a router, a personal computer, portable computer, microprocessor-based entertainment appliance, a peer device or other common network node, and typically includes many or all of the elements described relative to the computer  1102 , although, for purposes of brevity, only a memory/storage device  1150  is illustrated. The logical connections depicted include wire/wireless connectivity to a local area network (LAN)  1152  and/or larger networks, for example, a wide area network (WAN)  1154 . Such LAN and WAN networking environments are commonplace in offices and companies, and facilitate enterprise-wide computer networks, such as intranets, all of which may connect to a global communications network, for example, the Internet. 
     When used in a LAN networking environment, the computer  1102  is connected to the LAN  1152  through a wire and/or wireless communication network interface or adaptor  1156 . The adaptor  1156  can facilitate wire and/or wireless communications to the LAN  1152 , which may also include a wireless access point disposed thereon for communicating with the wireless functionality of the adaptor  1156 . 
     When used in a WAN networking environment, the computer  1102  can include a modem  1158 , or is connected to a communications server on the WAN  1154 , or has other means for establishing communications over the WAN  1154 , such as by way of the Internet. The modem  1158 , which can be internal or external and a wire and/or wireless device, connects to the system bus  1108  via the input device interface  1142 . In a networked environment, program modules depicted relative to the computer  1102 , or portions thereof, can be stored in the remote memory/storage device  1150 . It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers can be used. 
     The computer  1102  is operable to communicate with wire and wireless devices or entities using the IEEE 802 family of standards, such as wireless devices operatively disposed in wireless communication (e.g., IEEE 802.11 over-the-air modulation techniques) with, for example, a printer, scanner, desktop and/or portable computer, personal digital assistant (PDA), communications satellite, any piece of equipment or location associated with a wirelessly detectable tag (e.g., a kiosk, news stand, restroom), and telephone. This includes at least Wi-Fi (or Wireless Fidelity), WiMax, and Bluetooth™ wireless technologies. Thus, the communication can be a predefined structure as with a conventional network or simply an ad hoc communication between at least two devices. Wi-Fi networks use radio technologies called IEEE 802.11x (a, b, g, n, etc.) to provide secure, reliable, fast wireless connectivity. A Wi-Fi network can be used to connect computers to each other, to the Internet, and to wire networks (which use IEEE 802.3-related media and functions). 
     Some embodiments may be described using the expression “one embodiment” or “an embodiment” along with their derivatives. These terms mean that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment. Further, some embodiments may be described using the expression “coupled” and “connected” along with their derivatives. These terms are not necessarily intended as synonyms for each other. For example, some embodiments may be described using the terms “connected” and/or “coupled” to indicate that two or more elements are in direct physical or electrical contact with each other. The term “coupled,” however, may also mean that two or more elements are not in direct contact with each other, but yet still co-operate or interact with each other. 
     It is emphasized that the Abstract of the Disclosure is provided to allow a reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. In addition, in the foregoing Detailed Description, it can be seen that various features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the claimed embodiments require more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed embodiment. Thus the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separate embodiment. In the appended claims, the terms “including” and “in which” are used as the plain-English equivalents of the respective terms “comprising” and “wherein,” respectively. Moreover, the terms “first,” “second,” “third,” and so forth, are used merely as labels, and are not intended to impose numerical requirements on their objects. 
     What has been described above includes examples of the disclosed architecture. It is, of course, not possible to describe every conceivable combination of components and/or methodologies, but one of ordinary skill in the art may recognize that many further combinations and permutations are possible. Accordingly, the novel architecture is intended to embrace all such alterations, modifications and variations that fall within the spirit and scope of the appended claims.