PATENT DOCUMENT

Abstract:
An on-board vehicle control system and method employ a storage device that stores information representing a plurality of boundary points of a boundary circumscribing an area of interest in which the boundary points are defined by two prescribed parameters, and a controller that obtains at least one condition point defined by current values of the prescribed parameters, determines a first boundary point of the boundary points that is closest to the condition point, and generates geometric data representing a geometric relationship between the first boundary point, the condition point and a second boundary point of the boundary points. The controller performs further calculations taking into account the geometric data to generate coordinate condition data, and determines whether a warning condition exists by determining whether the condition point lies within the area of interest based on a comparison between coordinates of the condition point and the coordinate condition data.

Full Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The present invention generally relates to an on-board vehicle control system and method. More specifically, the present invention relates to an on-board vehicle control system and method for determining whether a condition point lies within an area of interest and controlling a host vehicle based on the determination to, for example, suppress extraneous warning information. 
         [0003]    2. Background Information 
         [0004]    Recently, vehicles are being equipped with a variety of informational systems such as navigation systems, satellite radio systems, two-way satellite services, built-in cell phones, DVD players and the like. Various informational systems have been proposed that use wireless communications between vehicles and between infrastructures, such as roadside units. These wireless communications have a wide range of applications ranging from safety applications to entertainment applications. Also vehicles are sometimes equipped with various types of systems, such as global positioning systems (GPS), which are capable of determining the location of the vehicle and identifying the location of the vehicle on a map for reference by the driver. The type of wireless communications to be used depends on the particular application. Some examples of wireless technologies that are currently available include digital cellular systems, Bluetooth systems, wireless LAN systems and dedicated short range communications (DSRC) systems. 
         [0005]    Also, vehicles can be equipped with a collision warning system that identifies the location of the vehicle and the locations of other nearby vehicles to determine whether the vehicle may come into contact with any of the other vehicles. The possibility of contact between vehicles can be particularly high at road intersections in which the travel paths of the vehicle and other nearby vehicles may intersect. If the possibility of contact exists, the system can issue a warning to the driver so that the driver can take the appropriate action. 
       SUMMARY OF THE INVENTION 
       [0006]    As can be appreciated from the above, a need exists for an improved on-board vehicle control system for identifying the location of a vehicle or vehicles of interest for use in various vehicle applications such as in collision warning systems, braking systems, mapping systems and so on. 
         [0007]    In accordance with one aspect of the present invention, an on-board vehicle control system and method are provided which employ a storage device and a controller. The storage device stores information representing a plurality of boundary points of a boundary that circumscribes an area of interest in which the boundary points are defined by two prescribed parameters, and the controller obtains at least one condition point defined by current values of the prescribed parameters, determines a first boundary point of the boundary points that is closest to the condition point, and generates geometric data representing a geometric relationship between the first boundary point, the condition point and a second boundary point of the boundary points. The geometric relationship includes a first straight line connecting the first boundary point and the condition point, a second straight line connecting the second boundary point and the condition point and a third straight line connecting the first boundary point and the second boundary point. The controller calculates reference point data representing a reference point based on the geometric data, determines coordinate condition data based on an angle between a predetermined direction and a reference line connecting the first boundary point and the reference point, and determines whether a warning condition exists by determining whether the condition point lies within the area of interest based on a comparison between coordinates of the condition point and the coordinate condition data. 
         [0008]    These and other objects, features, aspects and advantages of the present invention will become apparent to those skilled in the art from the following detailed description, which, taken in conjunction with the annexed drawings, discloses a preferred embodiment of the present invention. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]    Referring now to the attached drawings which form a part of this original disclosure: 
           [0010]      FIG. 1  is a schematic diagram illustrating an example of a host vehicle equipped with an on-board vehicle control system according to embodiments disclosed herein in relation to a remote vehicle and components of a global positioning system (GPS); 
           [0011]      FIG. 2  is a block diagram of exemplary components of the host vehicle equipped with an on-board vehicle control system according to embodiments disclosed herein; 
           [0012]      FIG. 3  is a diagrammatic view illustrating an example of an area of interest that represents a relationship between a plurality of vehicle-related parameters and a vehicle condition, and which is evaluated according to the embodiments described herein; 
           [0013]      FIG. 4  is a diagrammatic view illustrating an example of an area of interest that represents a geographic location in relation to a location of a vehicle of interest as evaluated according to the embodiments described herein; 
           [0014]      FIG. 5  is diagrammatic view illustrating an example of the locations of the area of interest as shown in  FIG. 4  with respect to a host vehicle; 
           [0015]      FIG. 6  is diagrammatic view illustrating an example of the locations of the area of interest as shown in  FIG. 4  with respect to a host vehicle; 
           [0016]      FIG. 7  is a diagrammatic view illustrating exemplary relationships between areas of interest and jurisdictional boundaries; 
           [0017]      FIG. 8  is a flowchart illustrating examples of operations performed by the on-board vehicle control system according to the embodiments described herein to determine whether determined values, such as those representing a vehicle location or a vehicle condition, lies inside or outside an area of interest; and 
           [0018]      FIGS. 9-22  are diagrammatic views illustrating examples of the relationships between the determined values and boundary points defining the area of interest as used in accordance with the process shown in the flowchart of  FIG. 8 . 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0019]    Selected embodiments of the present invention will now be explained with reference to the drawings. It will be apparent to those skilled in the art from this disclosure that the following descriptions of the embodiments of the present invention are provided for illustration only and not for the purpose of limiting the invention as defined by the appended claims and their equivalents. 
         [0020]    Referring initially to  FIG. 1 , a two-way wireless communications network is illustrated that includes vehicle to vehicle communication and vehicle to base station communication. In  FIG. 1 , a host vehicle (HV)  10  is illustrated that is equipped with an on-board vehicle control system  12  according to a disclosed embodiment, and two remote vehicles (RV)  14  that also includes the on-board vehicle control system  12 . As discussed herein, the host vehicle  10  can also be referred to as a subject vehicle (SV). The remote vehicle  14  can also be referred to as a target or threat vehicle (TV). While the host vehicle (HV)  10  and the remote vehicles  14  are illustrated as having the same on-board vehicle control system  12 , it will be apparent from this disclosure that each of the remote vehicles  14  can include another type of two-way communication system that is capable of communicating information about at least the location and speed of the remote vehicle  14  to the host vehicle  10 . 
         [0021]    The on-board vehicle control system  12  of the host vehicle  10  and the remote vehicle  14  communicates with the two-way wireless communications network. As seen in  FIG. 1 , for example, the two-way wireless communications network can include one or more global positioning satellites  16  (only one shown), and one or more roadside (terrestrial) units  18  (only one shown), and a base station or external server  20 . The global positioning satellites  16  and the roadside units  18  send and receive signals to and from the on-board vehicle control system  12  of the host vehicle  10  and the remote vehicles  14 . The base station  20  sends and receives signals to and from the on-board vehicle control system  12  of the host vehicle  10  and the remote vehicles  14  via a network of the roadside units  18 , or any other suitable two-way wireless communications network. 
         [0022]    As shown in more detail in  FIG. 2 , the on-board vehicle control system  12  includes an application controller  22  that can be referred to simply as a controller  22 . The controller  22  preferably includes a microcomputer with a control program that controls the components of the on-board vehicle control system  12  as discussed below. The controller  22  includes other conventional components such as an input interface circuit, an output interface circuit, and storage devices such as a ROM (Read Only Memory) device and a RAM (Random Access Memory) device. The microcomputer of the controller  22  is at least programmed to control the on-board vehicle control system  12  in accordance with the flow chart of  FIG. 8  as discussed below. It will be apparent to those skilled in the art from this disclosure that the precise structure and algorithms for the controller  22  can be any combination of hardware and software that will carry out the functions of the present invention. Furthermore, the controller  22  can communicate with the other components of the on-board vehicle control system  12  discussed herein via, for example a controller area network (CAN) bus or in any other suitable manner as understood in the art. 
         [0023]    As shown in more detail in  FIG. 2 , the on-board vehicle control system  12  can further include a wireless communication system  24 , a global positioning system (GPS)  26 , a storage device  28 , a plurality of in-vehicle sensors  30  and a human-machine interface unit  32 . The human-machine interface unit  32  includes a screen display  32 A, an audio speaker  32 B and various manual input controls  32 C that are operatively coupled to the controller  22 . The screen display  32 A and the audio speaker  32 B are examples of interior warning devices that are used to alert a driver. Of course, it will be apparent to those skilled in the art from this disclosure that interior warning devices include anyone of or a combination of visual, audio and/or tactile warnings as understood in the art that can be perceived inside the host vehicle  10 . The host vehicle  10  also includes a pair of front headlights  34  and rear brake lights  36 , which constitutes examples of exterior warning devices of the on-board vehicle control system  12 . These components can communicate with each other and, in particular, with the controller  22  in any suitable manner, such as wirelessly or via a vehicle bus  38 . 
         [0024]    The wireless communications system  24  can include an omni-directional antenna and a multi-directional antenna, as well as communication interface circuitry that connects and exchanges information with a plurality of the remote vehicles  14  that are similarly equipped, as well as with the roadside units  20  through at least a portion of the wireless communications network within the broadcast range of the host vehicle  10 . For example, the wireless communications system  24  can be configured and arranged to conduct direct two way communications between the host and remote vehicles  10  and  14  (vehicle-to-vehicle communications) and the roadside units  18  (roadside-to-vehicle communications). Moreover, the wireless communications system  24  can be configured to periodically broadcast a signal in the broadcast area. The wireless communication system  24  can be any suitable type of two-way communication device that is capable of communicating with the remote vehicles  14  and the two-way wireless communications network. In this example, the wireless communication system  24  can include or be coupled to a dedicated short range communications (DSRC) antenna to receive, for example, 5.9 GHz DSRC signals from the two-way wireless communications network. These DSRC signals can include basic safety messages (BSM) defined by current industry recognized standards that include information which, under certain circumstances, can be analyzed to warn drivers of a potential problem situation or threat in time for the driver of the host vehicle  10  to take appropriate action to avoid the situation. For instance, the DSRC signals can also include information pertaining to weather conditions, adverse driving conditions and so on. In the disclosed embodiments, a BSM includes information in accordance with SAE Standard J2735 as can be appreciated by one skilled in the art. Also, the wireless communication system  24  and the GPS  26  can be configured as a dual frequency DSRC and GPS devices as understood in the art. 
         [0025]    The GPS  26  can be a conventional global positioning system that is configured and arranged to receive global positioning information of the host vehicle  10  in a conventional manner. Basically, the global positioning system  26  receives GPS signals from the global positioning satellite  16  at regular intervals (e.g. one second) to detect the present position of the host vehicle  10 . The OPS  26  has an accuracy in accordance with industry standards and thus, can indicate the actual vehicle position of the host vehicle  10  within a few meters or less (e.g., 10 meters less). The data representing the present position of the host vehicle  10  is provided to the controller  22  for processing as discussed herein. For example, the controller  22  can include or be coupled to navigation system components that are configured and arranged to process the GPS information in a conventional manner as understood in the art. 
         [0026]    The storage device  28  can store road map data as well as other data that can be associated with the road map data such as various landmark data, fueling station locations, restaurants, weather data, traffic information and so on. Furthermore, the storage device  28  can store other types of data, such as data pertaining to vehicle-related parameters and vehicle conditions. For example, the vehicle-related parameters can include predetermined data indicating relationships between vehicle speed, vehicle acceleration, yaw, steering angle, etc. when a vehicle is preparing to make a turn. In this event, the storage device  28  can further store data pertaining to vehicle conditions, which can represent a determined vehicle condition of a vehicle of interest, such as the host vehicle  10 , a remote vehicle  14 , or both. This determined vehicle condition can represent, for example, a vehicle speed and acceleration that is determined for the vehicle of interest at a moment in time. Accordingly, the embodiments disclosed herein can evaluate whether the vehicle condition lies within the area of interest, as represented by the vehicle-related parameters, to determine, for example, whether the vehicle of interest is preparing to make a turn. The storage device  28  can include, for example, a large-capacity storage medium such as a CD-ROM (Compact Disk-Read Only Memory) or IC (Integrated Circuit) card. The storage device  28  permits a read-out operation of reading out data held in the large-capacity storage medium in response to an instruction from the controller  22  to, for example, acquire the map information and/or the vehicle condition information as needed or desired for use in representing the location of the host vehicle  10 , the remote vehicle  14  and other location information and/or vehicle condition information as discussed herein for route guiding, map display, turning indication, and so on as understood in the art. For instance, the map information can include at least road links indicating connecting states of nodes, locations of branch points (road nodes), names of roads branching from the branch points, place names of the branch destinations, and so on. The information in the storage device  28  can also be updated by the controller  22  or in any suitable manner as discussed herein and as understood in the art. 
         [0027]    The in-vehicle sensors  30  are configured to monitor various devices, mechanisms and systems within the host vehicle  10  and provide information relating to the status of those devices, mechanisms and systems to the controller  22 . For example, the in-vehicle sensors  30  can be connected to a traction control system, a windshield wiper motor or wiper motor controller, a headlight controller, a steering system, a speedometer, a braking system and so on as understood in the art. 
         [0028]    As will now be discussed with reference to  FIGS. 3 to 7 , the on-board vehicle control system  12  can operate to determine whether a condition is inside (condition  104 A) or outside (condition  104 B) the area of interest  100 . The condition can represent any type of condition, such as those discussed herein. For instance, in one exemplary embodiment, the on-board vehicle control system  12  can evaluate whether a vehicle condition representing a speed and acceleration of a vehicle of interest, such as the host vehicle  10  or the remote vehicle  14  is inside or outside the area of interest  100 , which in this exemplary embodiment represents predetermined vehicle speed and acceleration data, to determine whether the vehicle of interest is preparing to make a turn. Also, the on-board vehicle control system  12  can determine whether the location of a vehicle of interest, such as the host vehicle  10 , the remote vehicle  14 , or both, is present within an area of interest  100 , which represents a geographical area. In other words, either the host vehicle  10  and/or one or more the remote vehicles  14  can be considered to be a vehicle of interest by the on-board vehicle control system  12 . The on-board vehicle control system  12  controls an aspect of the host vehicle  10  differently upon a determination of the vehicle of interest (i.e., the host vehicle  10  and/or one or more the remote vehicles  14 ) is located within the area of interest  100  from a determination of the vehicle of interest (i.e., the host vehicle  10  and/or one or more the remote vehicles  14 ) is located outside of the area of interest. The term “aspect” as used herein with respect to the host vehicle  10  refers to any component, application, and/or application parameter of the host vehicle  10 . 
         [0029]    The area of interest  100  can be defined by a plurality of boundary points  102  as shown in  FIG. 3 . For instance, the area of interest  100  can be a particularly complex area defined by scattered boundary points that define a complex boundary that can vary in an irregular, non-symmetrical manner as defined by the boundary points  102 . The boundary points  102  can be represented by, for example, a series of experimental and/or historical data points. The boundary points  102  can be represented by, for example, data sets (e.g., x, y coordinates) that represent a relationship between vehicle-related parameters, such as vehicle speed and acceleration. The data sets are stored in the database of the storage device  28  or otherwise provided to the on-board vehicle control system  12  via the wireless communications network in any suitable manner. The vehicle condition shown in  FIG. 3  can represent a determined speed and acceleration of the vehicle of interest at a moment in time. In another example, such as that shown in  FIGS. 4 through 7 , the boundary points  102  can be represented by, for example, longitude and latitude data sets (e.g., x, y coordinates) that are stored in the database of the storage device  28  or otherwise provided to the on-board vehicle control system  12  via the wireless communications network  16  on in any suitable manner. For example, the boundary points  102  can represent terrestrial points on the earth, which can correspond to the locations of the roadside units  20  or any other suitable locations. Also the area of interest  100  can be either a dynamic area that changes as shown in  FIGS. 4 to 6 , or a static area that remains stationary as shown in  FIG. 7 . The number of boundary points  102  and the distance between the boundary points  102  for any given one of the areas of interest  100  can vary as needed and/or desired to accomplish the desired result of the application in which the system is being used. 
         [0030]    As can be appreciated from  FIG. 4 , the area of interest  100  can move, for example, in accordance with movement of the host vehicle  10  or due to other reasons, such as changes in environmental conditions, change in traffic conditions and so on as discussed herein. Thus, as the host vehicle  10  moves from a first location (shown in solid lines) to a second location (shown in broken lines), the area of interest  100  can shift to become area of interest  100 ′ represented in phantom lines. As seen in  FIG. 4 , the shape of the area of interest  100  not only shifted but also changed in shape in accordance with map data. 
         [0031]    As can further be appreciated from  FIGS. 5 and 6 , the area of interest  100  can be defined proximate to the host vehicle  10  (e.g., in front of the host vehicle  10 ) as shown in  FIG. 5 , or can encompass the host vehicle  10  as shown in  FIG. 6 . Furthermore, as shown in  FIG. 7 , the area of interest  100  can represent particular jurisdictions (Jurisdictions  1  to  4 ) that are each governed by respective traffic laws, etc. that may be different from each other. Thus, the on-board vehicle control system  12  can control one or more aspects of the host vehicle upon determining that the host vehicle  10  is located within the area of interest  100 . In the example shown in  FIG. 7 , four areas of interest are represented as areas of interest  100 - 1  through  100 - 4  corresponding to Jurisdictions  1  through  4 , respectively. As described herein, the on-board vehicle control system  12  can determine whether the host vehicle  10  lies in a particular one of the Jurisdictions  1  to  4 . For example, if one of the Jurisdictions  1  to  4  requires headlights to be turned “on” while on highways, then the controller  22  can turn on the headlights  34  upon determining that the host vehicle  10  is located within that Jurisdiction and on a highway based on the navigation system. Also, one or more of the Jurisdictions  1  through  4  may permit hands-free telephone use only. Accordingly, the controller  22  can issue a warning to the driver of the host vehicle  10  about this requirement. Furthermore, the Jurisdictions  1  through  4  may have different regulations with regard to the location of a vehicle with respect to an intersection when the light at the intersection is turning from green to amber and then to red. Therefore, the controller  22  can control the vehicle intersection warning system to operate in compliance with the requirements of the Jurisdiction  1  through  4  in which the vehicle of interest is present. 
         [0032]    An example of operations that are performed by the on-board vehicle control system  12  to determine whether a vehicle of interest (e.g., the host vehicle  10  and/or the remote vehicle  14 ) are present within an area of interest will now be described with reference to the flowchart in  FIG. 8  and the diagrams in  FIGS. 9-22 . 
         [0033]    In step S 11 , the processing stores information representing a plurality of boundary points  102 - 1  through  102 - 8  of a boundary that circumscribes the area of interest  100 . In step S 12 , the processing obtains a vehicle condition of a vehicle of interest, which is represented by a condition point p k . The vehicle condition can be determined in any suitable manner. For example, if the vehicle condition represents speed and acceleration of the vehicle of interest, the data representing the vehicle speed and acceleration can be determined based on information that can represent braking and accelerator pedal information, steering wheel information and so on provided to the controller  22  via the sensors  30 , which can include signals from, for example, vehicle speed and vehicle accelerator sensors. If the vehicle condition represents, for example, the location of the vehicle of interest, the vehicle location can be determined in any suitable manner, such as by using GPS information representing the longitude and latitude of the vehicle of interest. 
         [0034]    As shown in  FIG. 9 , the area of interest  100  for purposes of these examples can be any two-dimensional area, regardless of shape, that is represented by a 360-degree path P where the start and end points are the same. In this example, the start and end points are represented by boundary point  102 - 1 , with the other exemplary boundary points being represented by points  102 - 2  through  102 - 8 . The processing performed by the on-board vehicle control system  12  determines whether the vehicle condition (e.g., the determined speed and acceleration of the vehicle of interest, or the location of a vehicle of interest as discussed herein) as represented by the condition point p k , falls inside or outside the area of interest  100 . The number of the points  102 - 1  through  102 - 8  used to define the path P is immaterial, as any suitable number of the points  102  can be used as needed to accurately define the area of interest  100 . Likewise, the distance between the points  102  can vary as needed to accurately define the area of interest  100 . 
         [0035]    In this example, the processing is performed by starting at one of the points  102 , such as point  102 - 1 , and continuing clockwise around the path P from sequential point to point. However, the processing can start at any of the points  102 , and can proceed in a clockwise or counterclockwise manner as can be understood by one skilled in the art. It should be noted that the processing should be performed either in the clockwise manner or counterclockwise manner while completing the calculations discussed below to make one determination whether the vehicle condition lies within or outside the area of interest  100 . 
         [0036]    A line segment between two consecutive points on the path P can be characterized in one of eight ways as can be appreciated from  FIG. 10 . For instance, a line segment can be characterized by an angle β 1  between the line segment and a predetermined direction. 
         [0037]    Mathematically, the angle β 1  can be expressed as follows: 
         [0000]    
       
         
           
             
               β 
               1 
             
             = 
             
               
                 π 
                  
                 
                   ( 
                   
                     
                       
                         
                           x 
                           m 
                         
                         - 
                         
                           x 
                           
                             m 
                             + 
                             1 
                           
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               x 
                               m 
                             
                             - 
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               - 
               
                 
                   
                     cos 
                     
                       - 
                       1 
                     
                   
                   ( 
                   
                     
                       
                         y 
                         
                           m 
                           + 
                           1 
                         
                       
                       - 
                       
                         y 
                         m 
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 x 
                                 
                                   m 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 x 
                                 m 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 y 
                                 
                                   m 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 y 
                                 m 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         x 
                         m 
                       
                       - 
                       
                         x 
                         
                           m 
                           + 
                           1 
                         
                       
                       + 
                       σ 
                     
                     
                       
                          
                         
                           
                             x 
                             m 
                           
                           - 
                           
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                               + 
                               1 
                             
                           
                         
                          
                       
                       + 
                       σ 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0038]    The length of the line l k  between two consecutive points is a straight line defined as follows: 
         [0000]        l   k =√{square root over (( x   m+1    . . . x   m ) 2 +( y   m+1   −y   m ) 2 )}{square root over (( x   m+1    . . . x   m ) 2 +( y   m+1   −y   m ) 2 )}
 
         [0039]    The following exemplary calculations can be made to determine whether a condition point p k  lies within or outside the boundary defined by the circumferential path P. This determination can be made by using the following steps: 
         [0040]    First, in step S 13 , the processing determines via the on-board vehicle controller  22  which of the boundary points is closest to the vehicle condition (e.g. the point representing the vehicle speed and acceleration, or the point representing the vehicle location), and the processing further generates geometric data in step S 14 , as described below. As shown in  FIG. 10 , the point p m  (with coordinates x m  and y m ) on the path P that is closest to the condition point p k  (with coordinates x k  and y k ) can be determined by sequentially, in a predetermined direction (e.g. clockwise or counterclockwise), calculating the straight-line distance between each boundary point along the path P and the condition point p k , and then choosing the boundary point at the shortest straight-line distance. This distance l m+1  is defined as follows: 
         [0000]        l   m+1 =√{square root over (( x   k   −x   m ) 2 +( y   k   −y   m ) 2 )}{square root over (( x   k   −x   m ) 2 +( y   k   −y   m ) 2 )}
 
         [0041]    After the point on the path P that is closest to the condition point p k  has been identified, the processing generates via the on-board vehicle controller  22  geometric data representing a geometric relationship between the first boundary point p m , the condition point p k  and a second boundary point of the boundary points as will now be described. That is, the next consecutive point p m+1  in the predetermined direction along the path is chosen and a triangle is defined as shown in  FIG. 10 , where: 
         [0000]        l   k =√{square root over (( x   m+1   −x   m ) 2 +( y   m+1   −y   m ) 2 )}{square root over (( x   m+1   −x   m ) 2 +( y   m+1   −y   m ) 2 )}
 
         [0000]        l   m =√{square root over (( x   m+1   −x   k ) 2 +( y   m+1   −y   k ) 2 )}{square root over (( x   m+1   −x   k ) 2 +( y   m+1   −y   k ) 2 )}
 
         [0000]        l   m+1 =√{square root over (( x   m   −x   k ) 2 +( y   m   −y   k ) 2 )}{square root over (( x   m   −x   k ) 2 +( y   m   −y   k ) 2 )}
 
         [0000]    and from the Law of Cosines 
         [0000]    
       
         
           
             
               cos 
                
               
                   
               
                
               
                 α 
                 k 
               
             
             = 
             
               
                 
                   l 
                   m 
                   2 
                 
                 + 
                 
                   l 
                   
                     m 
                     + 
                     1 
                   
                   2 
                 
                 - 
                 
                   l 
                   k 
                   2 
                 
               
               
                 2 
                  
                 
                   l 
                   m 
                 
                  
                 
                   l 
                   
                     m 
                     + 
                     1 
                   
                 
               
             
           
         
       
       
         
           
             
               cos 
                
               
                   
               
                
               
                 α 
                 m 
               
             
             = 
             
               
                 
                   l 
                   k 
                   2 
                 
                 + 
                 
                   l 
                   
                     m 
                     + 
                     1 
                   
                   2 
                 
                 - 
                 
                   l 
                   m 
                   2 
                 
               
               
                 2 
                  
                 
                   l 
                   k 
                 
                  
                 
                   l 
                   
                     m 
                     + 
                     1 
                   
                 
               
             
           
         
       
       
         
           
             
               cos 
                
               
                   
               
                
               
                 α 
                 
                   m 
                   + 
                   1 
                 
               
             
             = 
             
               
                 
                   
                     l 
                     k 
                     2 
                   
                   + 
                   
                     l 
                     m 
                     2 
                   
                   - 
                   
                     l 
                     
                       m 
                       + 
                       1 
                     
                     2 
                   
                 
                 
                   2 
                    
                   
                     l 
                     k 
                   
                    
                   
                     l 
                     m 
                   
                 
               
               . 
             
           
         
       
     
         [0042]    Different Cases 1 through 4 will now be described with reference to  FIGS. 11 through 14  which indicate different possible relationships between the lines l k , l m  and l m+1  identified in  FIG. 10 . 
         [0000]        l   k   2   +l   m+1   2    . . . l   m   2 ≧0  Case 1
 
         [0043]    Referring to  FIG. 11 , if l k   2 +l m+1   2 −l m   2 ≧0, cos α m  is greater than 0 and the coordinates x q  and y q  are calculated as follows: 
         [0044]    The value of x q  is determined follows: 
         [0045]    First: 
         [0000]    
       
         
           
             
               l 
               k 
               ′ 
             
             = 
             
               
                 
                   l 
                   
                     m 
                     + 
                     1 
                   
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 
                   α 
                   m 
                 
               
               = 
               
                 
                   
                     l 
                     
                       m 
                       + 
                       1 
                     
                   
                    
                   
                     
                       
                         l 
                         k 
                         2 
                       
                       + 
                       
                         l 
                         
                           m 
                           + 
                           1 
                         
                         2 
                       
                       - 
                       
                         l 
                         m 
                         2 
                       
                     
                     
                       2 
                        
                       
                         l 
                         k 
                       
                        
                       
                         l 
                         
                           m 
                           + 
                           1 
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       l 
                       k 
                       2 
                     
                     + 
                     
                       l 
                       
                         m 
                         + 
                         1 
                       
                       2 
                     
                     - 
                     
                       l 
                       m 
                       2 
                     
                   
                   
                     2 
                      
                     
                       l 
                       k 
                     
                   
                 
               
             
           
         
       
       
         
           
             Now 
              
             
               : 
             
           
         
       
       
         
           
             
               
                 x 
                 q 
               
               - 
               
                 x 
                 m 
               
             
             = 
             
               
                 l 
                 k 
                 ′ 
               
                
               sin 
                
               
                   
               
                
               
                 β 
                 1 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               sin 
                
               
                   
               
                
               
                 β 
                 1 
               
             
             = 
             
               
                 
                   x 
                   
                     m 
                     + 
                     1 
                   
                 
                 - 
                 
                   x 
                   m 
                 
               
               
                 l 
                 k 
               
             
           
         
       
     
         [0000]    and substitutions are made to obtain: 
         [0000]    
       
         
           
             
               
                 x 
                 q 
               
               - 
               
                 x 
                 m 
               
             
             = 
             
               
                 
                   
                     l 
                     k 
                     2 
                   
                   + 
                   
                     l 
                     
                       m 
                       + 
                       1 
                     
                     2 
                   
                   - 
                   
                     l 
                     m 
                     2 
                   
                 
                 
                   2 
                    
                   
                     l 
                     k 
                   
                 
               
                
               
                 ( 
                 
                   
                     
                       x 
                       
                         m 
                         + 
                         1 
                       
                     
                     - 
                     
                       x 
                       m 
                     
                   
                   
                     l 
                     k 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 x 
                 q 
               
               - 
               
                 x 
                 m 
               
             
             = 
             
               
                 ( 
                 
                   1 
                   + 
                   
                     
                       
                         l 
                         
                           m 
                           + 
                           1 
                         
                         2 
                       
                       - 
                       
                         l 
                         m 
                         2 
                       
                     
                     
                       l 
                       k 
                       2 
                     
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     
                       x 
                       
                         m 
                         + 
                         1 
                       
                     
                     - 
                     
                       x 
                       m 
                     
                   
                   2 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 x 
                 q 
               
               - 
               
                 x 
                 m 
               
             
             = 
             
               
                 
                   
                     x 
                     
                       m 
                       + 
                       1 
                     
                   
                   - 
                   
                     x 
                     m 
                   
                 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       
                         l 
                         
                           m 
                           + 
                           1 
                         
                         2 
                       
                       - 
                       
                         l 
                         m 
                         2 
                       
                     
                     
                       l 
                       k 
                       2 
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         x 
                         
                           m 
                           + 
                           1 
                         
                       
                       - 
                       
                         x 
                         m 
                       
                     
                     2 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               x 
               q 
             
             = 
             
               
                 
                   
                     x 
                     
                       m 
                       + 
                       1 
                     
                   
                   + 
                   
                     x 
                     m 
                   
                 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       
                         l 
                         
                           m 
                           + 
                           1 
                         
                         2 
                       
                       - 
                       
                         l 
                         m 
                         2 
                       
                     
                     
                       l 
                       k 
                       2 
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         x 
                         
                           m 
                           + 
                           1 
                         
                       
                       - 
                       
                         x 
                         m 
                       
                     
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0046]    which are expanded to obtain: 
         [0000]    
       
         
           
             
               x 
               q 
             
             = 
             
               
                 
                   
                     x 
                     
                       m 
                       + 
                       1 
                     
                   
                   + 
                   
                     x 
                     m 
                   
                 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       
                         
                           ( 
                           
                             
                               x 
                               m 
                             
                             - 
                             
                               x 
                               k 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               y 
                               m 
                             
                             - 
                             
                               y 
                               k 
                             
                           
                           ) 
                         
                         2 
                       
                       - 
                       
                         
                           ( 
                           
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               x 
                               k 
                             
                           
                           ) 
                         
                         2 
                       
                       - 
                       
                         
                           ( 
                           
                             
                               y 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               y 
                               k 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     
                       
                         
                           ( 
                           
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               x 
                               m 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               y 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               y 
                               m 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         x 
                         
                           m 
                           + 
                           1 
                         
                       
                       - 
                       
                         x 
                         m 
                       
                     
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    The value of y q  is determined as follows: 
         [0000]        y   q   −y   m   =l   k ′ cos β 1  
 
         [0047]    where: 
         [0000]    
       
         
           
             
               cos 
                
               
                   
               
                
               
                 β 
                 1 
               
             
             = 
             
               
                 
                   y 
                   
                     m 
                     + 
                     1 
                   
                 
                 - 
                 
                   y 
                   m 
                 
               
               
                 l 
                 k 
               
             
           
         
       
     
         [0000]    and substitutions are made to obtain: 
         [0000]    
       
         
           
             
               
                 y 
                 q 
               
               - 
               
                 y 
                 m 
               
             
             = 
             
               
                 
                   
                     l 
                     k 
                     2 
                   
                   + 
                   
                     l 
                     
                       m 
                       + 
                       1 
                     
                     2 
                   
                   - 
                   
                     l 
                     m 
                     2 
                   
                 
                 
                   2 
                    
                   
                     l 
                     k 
                   
                 
               
                
               
                 ( 
                 
                   
                     
                       y 
                       
                         m 
                         + 
                         1 
                       
                     
                     - 
                     
                       y 
                       m 
                     
                   
                   
                     l 
                     k 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 y 
                 q 
               
               - 
               
                 y 
                 m 
               
             
             = 
             
               
                 ( 
                 
                   1 
                   + 
                   
                     
                       
                         l 
                         
                           m 
                           + 
                           1 
                         
                         2 
                       
                       - 
                       
                         l 
                         m 
                         2 
                       
                     
                     
                       l 
                       k 
                       2 
                     
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     
                       y 
                       
                         m 
                         + 
                         1 
                       
                     
                     - 
                     
                       y 
                       m 
                     
                   
                   2 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 y 
                 q 
               
               - 
               
                 y 
                 m 
               
             
             = 
             
               
                 
                   
                     y 
                     
                       m 
                       + 
                       1 
                     
                   
                   - 
                   
                     y 
                     m 
                   
                 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       
                         l 
                         
                           m 
                           + 
                           1 
                         
                         2 
                       
                       - 
                       
                         l 
                         m 
                         2 
                       
                     
                     
                       l 
                       k 
                       2 
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         y 
                         
                           m 
                           + 
                           1 
                         
                       
                       - 
                       
                         y 
                         m 
                       
                     
                     2 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               y 
               q 
             
             = 
             
               
                 
                   
                     y 
                     
                       m 
                       + 
                       1 
                     
                   
                   + 
                   
                     y 
                     m 
                   
                 
                 2 
               
               + 
               
                 
                   ( 
                   
                     
                       
                         l 
                         
                           m 
                           + 
                           1 
                         
                         2 
                       
                       - 
                       
                         l 
                         m 
                         2 
                       
                     
                     
                       l 
                       k 
                       2 
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         y 
                         
                           m 
                           + 
                           1 
                         
                       
                       - 
                       
                         y 
                         m 
                       
                     
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    which are expanded to obtain: 
         [0000]              y   q     =           y     m   +   1       +     y   m       2     +       (           (       x   m     -     x   k       )     2     +       (       y   m     -     y   k       )     2     -       (       x     m   +   1       -     x   k       )     2     -       (       y     m   +   1       -     y   k       )     2             (       x     m   +   1       -     x   m       )     2     +       (       y     m   +   1       -     y   m       )     2         )          (         y     m   +   1       -     y   m       2     )                 l   k   2   +l   m+1   2   −l   m   2 &lt;0 and  l   k   2   +l   m−1   2   −l   m   2 ≧0  Case 2
 
         [0048]    Referring to  FIG. 12 , if l k   2 +l m+1   2 −l m   2 &lt;0 (i.e. cos α m1 &lt;0) but l k   2 +l m−1   2 −l m   2 ≧0 (i.e. cos α m2 ≧0), becomes p m+1  and p m−1  becomes p m  and x q  and y q  are calculated in the same way as previously. 
       Thus: 
       [0049]              x   q     =           x     m   +   1       +     x   m       2     +       (           (       x   m     -     x   k       )     2     +       (       y   m     -     y   k       )     2     -       (       x     m   +   1       -     x   k       )     2     -       (       y     m   +   1       -     y   k       )     2             (       x     m   +   1       -     x   m       )     2     +       (       y     m   +   1       -     y   m       )     2         )          (         x     m   +   1       -     x   m       2     )                                And                 y   q     =           y     m   +   1       +     y   m       2     +       (           (       x   m     -     x   k       )     2     +       (       y   m     -     y   k       )     2     -       (       x     m   +   1       -     x   k       )     2     -       (       y     m   +   1       -     y   k       )     2             (       x     m   +   1       -     x   m       )     2     +       (       y     m   +   1       -     y   m       )     2         )          (         y     m   +   1       -     y   m       2     )                 l   k   2   +l   m+1   2   −l   m   2 &lt;0 and  l   k   2   +l   m−1   2   −l   m   2 &lt;0  Case 3
 
         [0050]    Referring to  FIG. 13 , it is possible that l k   2 +l m+1   2 −l m   2 &lt;0 and l k   2 +l m−1   2 −l m   2 &lt;0 (i.e. cos α m1  and cos α m2 &lt;0). In this case, p m  becomes p q    
       Thus: 
       [0051]    
       
      
       x 
       q 
       =x 
       m  
      
     
         [0000]      And 
         [0000]        y   q   =y   m . 
         [0000]        l   k   2   +l   m+1   2   −l   m   2 ≧0 and  l   k   2   +l   m−1   2   −l   m   2 ≧0  Case 4
 
         [0052]    Referring to  FIG. 14 , it is possible that l k   2 +l m+1   2 −l m   2 ≧0 and l k   2 +l m−1   2 −l m   2 ≧0 (i.e. cos α m1  and cos α m2 ≧0). In this case, Case 1 applies. 
         [0053]    Accordingly, the geometric relationship including a first straight line l m+1  connecting the first boundary point p m  and the condition point (e.g. the vehicle speed/acceleration or the vehicle location) p k , a second straight line l m  connecting the second boundary point p m+1  and the condition point p k  and a third straight line l k  connecting the first and second boundary points p m  and p m+1 . Furthermore, the calculations described above are performed in step S 15  to calculate via the on-board vehicle controller  22  reference point data representing a reference point p q  based on the geometric data. 
         [0054]    Referring back to  FIG. 9 , expressions to determine if the condition point p k  is inside or outside the area defined by the circumferential path P can be determined for each of the eight characteristic configurations shown. With the coordinates of p q (x q , y q ) known, the on-board vehicle control system  12  can perform the following calculations to determine whether the condition point p k  lies within or outside the boundary defined by the circumferential path P. Thus, as described below, the processing performed in step S 16  determine via the on-board vehicle controller  22  coordinate condition data based on an angle between a predetermined direction and a reference line connecting between the first boundary point p m  and the reference point p q . 
         [0055]    Angle β 1  Greater than or Equal to 0 and Less than π/2 
         [0056]    For the case where the angle β 1  is equal to or greater than zero and less than π/2 as illustrated in  FIG. 15 , it can be seen that as long as x k  is greater than or equal to x q  and y k  is less than or equal to y q , the condition point p k  falls within the defined boundary. The following expressions can be used to define this case mathematically where: 
         [0000]    
       
         
           
             
               
                 f 
                 1 
               
                
               
                 ( 
                 
                   β 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           π 
                           / 
                           2 
                         
                         - 
                         
                           β 
                           1 
                         
                       
                       
                         
                            
                           
                             
                               π 
                               / 
                               2 
                             
                             - 
                             
                               β 
                               1 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               1 
             
           
         
       
       
         
           if 
         
       
       
         
           
             0 
             ≤ 
             
               β 
               1 
             
             ≤ 
             
               π 
               2 
             
           
         
       
     
         [0000]    otherwise f 1 (β 1 )=0. 
       And 
       [0057]    
       
         
           
             
               
                 f 
                 1 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           x 
                           k 
                         
                         - 
                         
                           x 
                           q 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               x 
                               k 
                             
                             - 
                             
                               x 
                               q 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           y 
                           q 
                         
                         - 
                         
                           y 
                           k 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               y 
                               q 
                             
                             - 
                             
                               y 
                               k 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               1 
             
           
         
       
     
         [0000]    if the condition point p k  lies below and to the right of the reference point p q  otherwise f 1 (x,y)=0. 
         [0058]    Angle β 1  Equal to π/2 
         [0059]    For the case where the angle β 1  is equal to π/2 as illustrated in  FIG. 16 , it can be seen that as long as y k  is less than or equal to y q , the condition point p k  falls within the defined boundary. The following expressions can be used to define this case mathematically where: 
         [0000]    
       
         
           
             
               
                 f 
                 2 
               
                
               
                 ( 
                 
                   β 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           β 
                           1 
                         
                         - 
                         
                           π 
                           2 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               β 
                               1 
                             
                             - 
                             
                               π 
                               2 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           π 
                           2 
                         
                         - 
                         
                           β 
                           1 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               π 
                               2 
                             
                             - 
                             
                               β 
                               1 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               
                 
                   1 
                    
                   
                       
                   
                    
                   if 
                    
                   
                       
                   
                    
                   
                     β 
                     1 
                   
                 
                 = 
                 
                   π 
                   2 
                 
               
             
           
         
       
     
         [0000]    otherwise f 2 (β 1 )=0. 
       And 
       [0060]    
       
         
           
             
               
                 f 
                 2 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           y 
                           q 
                         
                         - 
                         
                           y 
                           k 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               y 
                               q 
                             
                             - 
                             
                               y 
                               k 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               1 
             
           
         
       
     
         [0000]    if the condition point p k  lies below the reference
       point p q  otherwise f 2 (x,y)=0.       
 
         [0062]    Angle β 1  Greater than or Equal to π/2 and Less than π 
         [0063]    For the case where the angle β 1  is equal to or greater than π/2 and less than π as illustrated in  FIG. 17 , it can be seen that as long as x k  is less than or equal to x q  and y k  is less than or equal to y q , the condition point p k  falls within the defined boundary. The following expressions can be used to define this case mathematically where: 
         [0000]    
       
         
           
             
               
                 f 
                 3 
               
                
               
                 ( 
                 
                   β 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           β 
                           1 
                         
                         - 
                         
                           π 
                           2 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               β 
                               1 
                             
                             - 
                             
                               π 
                               2 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         π 
                         - 
                         
                           β 
                           1 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             π 
                             - 
                             
                               β 
                               1 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               
                 
                   1 
                    
                   
                       
                   
                    
                   if 
                    
                   
                       
                   
                    
                   
                     
                       f 
                       3 
                     
                      
                     
                       ( 
                       
                         β 
                         1 
                       
                       ) 
                     
                   
                 
                 = 
                 0. 
               
             
           
         
       
     
         [0000]    if π/2≦β 1 &lt;π otherwise 
       And 
       [0064]    
       
         
           
             
               
                 
                   f 
                   3 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     1 
                     4 
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             x 
                             q 
                           
                           - 
                           
                             x 
                             k 
                           
                           + 
                           σ 
                         
                         
                           
                              
                             
                               
                                 x 
                                 q 
                               
                               - 
                               
                                 x 
                                 k 
                               
                             
                              
                           
                           + 
                           σ 
                         
                       
                       + 
                       1 
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             y 
                             q 
                           
                           - 
                           
                             y 
                             k 
                           
                           + 
                           σ 
                         
                         
                           
                              
                             
                               
                                 y 
                                 q 
                               
                               - 
                               
                                 y 
                                 k 
                               
                             
                              
                           
                           + 
                           σ 
                         
                       
                       + 
                       1 
                     
                     ) 
                   
                 
                 = 
                 1 
               
             
              
             
                 
             
           
         
       
     
         [0000]    if condition point p k  lies below and to the left of reference point p q  otherwise f 3 (x,y)=0. 
         [0065]    Angle β 1  Equal to π 
         [0066]    For the case where the angle β 1  is equal to π as illustrated in  FIG. 18 , it can be seen that as long as x k  is less than or equal to x q , the condition point p k  falls within the defined boundary. The following expressions can be used to define this case mathematically where: 
         [0000]    
       
         
           
             
               
                 f 
                 4 
               
                
               
                 ( 
                 
                   β 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           β 
                           1 
                         
                         - 
                         π 
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               β 
                               1 
                             
                             - 
                             π 
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         π 
                         - 
                         
                           β 
                           1 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             π 
                             - 
                             
                               β 
                               1 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               1 
             
           
         
       
     
         [0000]    when β 1 =π otherwise it equals 0. 
       And 
       [0067]    
       
         
           
             
               
                 f 
                 4 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           x 
                           q 
                         
                         - 
                         
                           x 
                           k 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               x 
                               q 
                             
                             - 
                             
                               x 
                               k 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               1 
             
           
         
       
     
         [0000]    if the condition point p k  lies to the left of the reference point p q  otherwise f 4 (x,y)=0. 
         [0068]    Angle β 1  Greater than or Equal to π and Less than 3π/2 
         [0069]    For the case where the angle β 1  is equal to or greater than π and less than 3π/2 as illustrated in  FIG. 19 , it can be seen that as long as x k  is less than or equal to x k  and y k  is greater than or equal to y q , the condition point p k  falls within the defined boundary. The following expressions can be used to define this case mathematically where: 
         [0000]    
       
         
           
             
               
                 f 
                 5 
               
                
               
                 ( 
                 
                   β 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           β 
                           1 
                         
                         - 
                         π 
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               β 
                               1 
                             
                             - 
                             π 
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             3 
                              
                             π 
                           
                           2 
                         
                         - 
                         
                           β 
                           1 
                         
                       
                       
                         
                            
                           
                             
                               
                                 3 
                                  
                                 π 
                               
                               2 
                             
                             - 
                             
                               β 
                               1 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               
                 
                   1 
                    
                   
                       
                   
                    
                   if 
                    
                   
                       
                   
                    
                   π 
                 
                 ≤ 
                 
                   β 
                   1 
                 
                 &lt; 
                 
                   
                     3 
                     2 
                   
                    
                   π 
                 
               
             
           
         
       
     
         [0000]    otherwise f 5 (β 1 )=0. 
       And 
       [0070]    
       
         
           
             
               
                 
                   f 
                   5 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     1 
                     4 
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             x 
                             q 
                           
                           - 
                           
                             x 
                             k 
                           
                           + 
                           σ 
                         
                         
                           
                              
                             
                               
                                 x 
                                 q 
                               
                               - 
                               
                                 x 
                                 k 
                               
                             
                              
                           
                           + 
                           σ 
                         
                       
                       + 
                       1 
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             y 
                             k 
                           
                           - 
                           
                             y 
                             q 
                           
                           + 
                           σ 
                         
                         
                           
                              
                             
                               
                                 y 
                                 k 
                               
                               - 
                               
                                 y 
                                 q 
                               
                             
                              
                           
                           + 
                           σ 
                         
                       
                       + 
                       1 
                     
                     ) 
                   
                 
                 = 
                 1 
               
             
              
             
                 
             
           
         
       
     
         [0000]    if the condition point p k  lies above and to the left of the reference point p q  otherwise f 5 (x,y)=0. 
         [0071]    Angle β 1  Equal to 3π/2 
         [0072]    For the case where the angle β 1  is equal to 3π2 as illustrated in  FIG. 20 , it can be seen that as long as y k  is greater than or equal to y q , the condition point p k  falls within the defined boundary. The following expressions can be used to define this case mathematically where: 
         [0000]    
       
         
           
             
               
                 f 
                 6 
               
                
               
                 ( 
                 
                   β 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           β 
                           1 
                         
                         - 
                         
                           
                             3 
                              
                             π 
                           
                           2 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               β 
                               1 
                             
                             - 
                             
                               
                                 3 
                                  
                                 π 
                               
                               2 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             3 
                              
                             π 
                           
                           2 
                         
                         - 
                         
                           β 
                           1 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               
                                 3 
                                  
                                 π 
                               
                               2 
                             
                             - 
                             
                               β 
                               1 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               
                 
                   1 
                    
                   
                       
                   
                    
                   if 
                    
                   
                       
                   
                    
                   
                     β 
                     1 
                   
                 
                 = 
                 
                   
                     3 
                     2 
                   
                    
                   π 
                 
               
             
           
         
       
     
         [0000]    otherwise f 6 (β 1 )=0. 
       And 
       [0073]    
       
         
           
             
               
                 f 
                 6 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           y 
                           k 
                         
                         - 
                         
                           y 
                           q 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               y 
                               k 
                             
                             - 
                             
                               y 
                               q 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               1 
             
           
         
       
     
         [0000]    if the condition point p k  lies above the reference point p q  otherwise f 6 (x,y)=0. 
         [0074]    Angle β 1  Greater than or Equal to 3/π2 and Less than 2π 
         [0075]    For the case where the angle β 1  is equal to or greater than 3π/2 and less than 2π as illustrated in  FIG. 21 , it can be seen that as long as x k  is greater than or equal to x q  and y k  is greater than or equal to y q , the condition point p f  falls within the defined boundary. The following expressions can be used to define this case mathematically where: 
         [0000]    
       
         
           
             
               
                 f 
                 7 
               
                
               
                 ( 
                 
                   β 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           β 
                           1 
                         
                         - 
                         
                           
                             3 
                              
                             π 
                           
                           2 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               β 
                               1 
                             
                             - 
                             
                               
                                 3 
                                  
                                 π 
                               
                               2 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           2 
                            
                           π 
                         
                         - 
                         
                           β 
                           1 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               2 
                                
                               π 
                             
                             - 
                             
                               β 
                               1 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               
                 
                   1 
                    
                   
                       
                   
                    
                   if 
                    
                   
                       
                   
                    
                   
                     β 
                     1 
                   
                 
                 = 
                 
                   
                     
                       3 
                       2 
                     
                      
                     π 
                   
                   ≤ 
                   
                     β 
                     1 
                   
                   &lt; 
                   
                     2 
                      
                     π 
                   
                 
               
             
           
         
       
     
         [0000]    otherwise f 7 (β 1 )=0. 
       And 
       [0076]    
       
         
           
             
               
                 
                   f 
                   7 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     1 
                     4 
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             x 
                             k 
                           
                           - 
                           
                             x 
                             q 
                           
                           + 
                           σ 
                         
                         
                           
                              
                             
                               
                                 x 
                                 k 
                               
                               - 
                               
                                 x 
                                 q 
                               
                             
                              
                           
                           + 
                           σ 
                         
                       
                       + 
                       1 
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             y 
                             k 
                           
                           - 
                           
                             y 
                             q 
                           
                           + 
                           σ 
                         
                         
                           
                              
                             
                               
                                 y 
                                 k 
                               
                               - 
                               
                                 y 
                                 q 
                               
                             
                              
                           
                           + 
                           σ 
                         
                       
                       + 
                       1 
                     
                     ) 
                   
                 
                 = 
                 1 
               
             
              
             
                 
             
           
         
       
     
         [0000]    if the condition point p k  lies above and to the right of the reference point p q  otherwise f 7 (x,y)=0. 
         [0077]    Angle β 1  Equal to 0 
         [0078]    For the case where the angle β 1  is equal to zero as illustrated in  FIG. 22 , it can be seen that as long as x k  is greater than or equal to x q , the condition point p k  falls within the defined boundary. The following expressions can be used to define this case mathematically where: 
         [0000]    
       
         
           
             
               
                 f 
                 8 
               
                
               
                 ( 
                 
                   β 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   4 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           β 
                           1 
                         
                         - 
                         0 
                         + 
                         σ 
                       
                       
                         
                            
                           
                             
                               β 
                               1 
                             
                             - 
                             0 
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         0 
                         - 
                         
                           β 
                           1 
                         
                         + 
                         σ 
                       
                       
                         
                            
                           
                             0 
                             - 
                             
                               β 
                               1 
                             
                           
                            
                         
                         + 
                         σ 
                       
                     
                     + 
                     1 
                   
                   ) 
                 
               
               = 
               1 
             
           
         
       
     
         [0000]    when β 1 =0 otherwise f 6 (β 1 )=0. 
       And 
       [0079]    
       
         
           
             
               
                 f 
                 8 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                 
                 ) 
               
             
             = 
             
               
                 1 
                 2 
               
                
               
                 ( 
                 
                   
                     
                       
                         x 
                         k 
                       
                       - 
                       
                         x 
                         q 
                       
                       + 
                       σ 
                     
                     
                       
                          
                         
                           
                             x 
                             k 
                           
                           - 
                           
                             x 
                             q 
                           
                         
                          
                       
                       + 
                       σ 
                     
                   
                   + 
                   1 
                 
                 ) 
               
             
           
         
       
     
         [0000]    if the condition point p k  lies to the right of the reference point p q  otherwise f 8 (x,y)=0. 
         [0080]    Using the above information, the on-board vehicle control system  12  determines in step S 17  via the on-board vehicle controller  22  whether the vehicle condition lies within the area of interest  100  based on a comparison between coordinates of the condition point p k  and the coordinate condition data. In particular, the controller  22  determines the following: 
         [0000]    
       
         
           
             
               if 
                
               
                   
               
                
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   8 
                 
                  
                 
                   Q 
                   i 
                 
               
             
             = 
             1 
           
         
       
     
         [0081]    where:
       Q 1 =f 1 (β 1 )×f 1 (x,y)   Q 2 =f 2 (β 1 )×f 2 (x,y)   Q 3 =f 3 (β 1 )×f 3 (x,y)   Q 4 =f 4 (β 1 )×f 4 (x,y)   Q 5 =f 5 (β 1 )×f 5 (x,y)   Q 6 =f 6 (β 1 )×f 6 (x,y)   Q 7 =f 7 (β 1 )×f 7 (x,y)   Q 8 =f 8 (β 1 )×f 8 (x,y)       
 
         [0090]    then the condition point p k  lies within the region defined by the circumferential path P otherwise it falls outside. 
         [0091]    Thus, the processing can control the on-board vehicle controller  22  differently upon determination of the vehicle condition being located within the area of interest  100  (step S 18 ) from a determination of the vehicle condition being located outside of the area of interest  100  (step S 19 ). For example, as discussed above, the vehicle of interest can be a remote vehicle  14  which is different from a host vehicle  10  on which the on-board vehicle controller  22  is disposed, and the vehicle condition can represent a location of the remote vehicle  14 . Thus, the processing can control an aspect of the host vehicle  10  differently upon a determination of the remote vehicle  14  being located within the area of interest  100  (step S 18 ) from a determination of the remote vehicle  14  being located outside of the area of interest  100  (step S 19 ). For example, when performing a contact warning operation, the controller  22  can classify a message from the remote vehicle  14  as irrelevant upon determination of the remote vehicle  14  being located outside of the area of interest  100 . The processing can also perform any suitable vehicle control process, such as controlling a braking process, emitting a warning or warnings to be perceived by the driver of the host vehicle  10  and/or the driver of the remote vehicle  14 , and so on, upon determining that the remote vehicle  14  is located within the area of interest  100 , and different operations upon determining that remote vehicle  14  is located outside of the area of interest  100 . 
         [0092]    Alternatively, the vehicle of interest can be the host vehicle  10  on which the on-board vehicle controller  22  is disposed, and the vehicle condition can represent a location of the host vehicle  10 . Thus, the processing can control an aspect of the host vehicle  10  differently upon a determination of the host vehicle  14  being located within the area of interest  100  (step S 18 ) from a determination of the host vehicle  14  being located outside of the area of interest  100  (step S 19 ). For example, the controller  22  can control the host vehicle  10  to operate in accordance with the regulations of Jurisdiction  1  in which the host vehicle  10  is present, until the controller determines the host vehicle  10  moved to be outside of Jurisdiction  1  and inside of Jurisdiction  2 , with different regulations than that of Jurisdiction  1 . Therefore, the controller controls the host vehicle  10  in a first manner (in accordance with the regulations of Jurisdiction  1 ) while inside Jurisdiction  1 , and in a second manner (in accordance with the regulations of Jurisdiction  2 ) that is different from the first manner while inside Jurisdiction  2 . 
         [0093]    Also, as discussed above, if the vehicle condition represents, for example, a speed and acceleration of the remote vehicle  14 , the controller  22  can determine, for example, that the remote vehicle  14  is preparing to execute a turn when the vehicle condition is determined to lie within the area of interest  100  which represents predetermined relationships between vehicle speed and acceleration as discussed above. Accordingly, the controller  22  can control the host vehicle  10  to perform contact avoidance processes, warning processes and so on if it is determined that the remote vehicle  14  is about to execute a turn. Likewise, if the vehicle condition represents, for example, a speed and acceleration of the host vehicle  10 , the controller  22  can determine, for example, that the host vehicle  10  is preparing to execute a turn when the vehicle condition is determined to lie within the area of interest  100  which represents predetermined relationships between vehicle speed and acceleration as discussed above. Accordingly, the controller  22  can control the host vehicle  10  to perform contact avoidance processes, warning processes and so on if it is determined that the host vehicle  10  is about to execute a turn. 
         [0094]    It can further be appreciated that the above process shown in  FIG. 9  can be performed repeatedly to determine, for example, whether a vehicle condition pertaining to at least one additional remote vehicle  14  is located within the area of interest  100 . For example, if the vehicle condition represents a location of an additional remote vehicle  14 , the processing can perform a comparison between geographic coordinates of an additional remote vehicle location of the at least one additional remote vehicle  14  and additional coordinate condition data, with the additional coordinate condition data of the at least one additional remote vehicle  14  being obtained by the above process. That is, the processing determines via the on-board vehicle controller  22  an additional first boundary point of the boundary points for the at least one additional remote vehicle  14  that is closest to the additional remote vehicle location, generating via the on-board vehicle controller  22  additional geometric data representing an additional geometric relationship between the additional first boundary point, the additional remote vehicle location and an additional second boundary point of the boundary points for the at least one additional remote vehicle  14 . In this case, the geometric relationship includes an additional first straight line connecting the additional first boundary point and the additional remote vehicle location, an additional second straight line connecting the additional second boundary point and the additional vehicle location and an additional third straight line connecting the additional first and second boundary points. The processing thus calculates via the on-board vehicle controller  22  additional reference point data representing an additional reference point for the at least one additional remote vehicle  14  based on the additional geometric data, and determines via the on-board vehicle controller  22  the additional coordinate condition data based on an angle between a predetermined direction and a reference line connecting between the additional first boundary point and the additional reference point. 
         [0095]    Accordingly, the processing can perform a collision warning process differently upon determination of the remote vehicle  14  and the at least one additional remote vehicle  14  being located within the area of interest  100  than when at least one of the remote vehicle  14  and the at least one additional remote vehicle  14  are located outside of the area of interest  100 . Moreover, the processing can perform, for example, a vehicle navigation process differently upon determination that the remote vehicle  14  and the at least one additional remote vehicle  14  are located within the area of interest  100  as opposed to at least one of the remote vehicle  14  and the at least one additional remote vehicle  14  being located outside of the area of interest  100 . 
         [0096]    In addition, processing similar to that discussed above can be performed for any type of vehicle condition as discussed herein that is determined with respect to an additional remote vehicle  14 . 
         [0097]    Moreover, when the host vehicle  10  is moving and the area of interest  100  is changing and/or moving as shown in  FIGS. 4 and 5  above, the processing shown in  FIG. 8  and discussed above can repeat to recalculate the locations and information discussed above with respect to the new location of the host vehicle  10  and the boundary points of the area of interest  100 . The controller  22  can update or otherwise store that information in the database of the storage device  28  or in any other suitable location. Moreover, if the host vehicle  10  moves from one jurisdiction to another as shown in  FIG. 5 , the processing performed by the controller  22  can control the host vehicle  10  to operate in accordance with the requirements of that new jurisdiction. 
         [0098]    As can be appreciated from the above, this methodology can be employed to determine whether a point lies within a defined region whether the region is geographic in nature or something completely abstract, such as a region defined by engine revolutions per minute (rpms) and fuel consumption that is used to optimize or at least improve operations of the host vehicle  10  to improve fuel efficiency as understood in the art. Thus, the methodology can be applicable to any situation involving multiple parameters and a selected, complexly-shaped, region that is defined by a plurality of data points for indicating a situation in which, for example, and indication such as a warning should be provided. The methodology is particularly useful for evaluating a complex area of interest defined by scattered boundary points, which is much more difficult to evaluate than a well-defined area that can be analyzed by, for example, merely determining if a value of interest is above or below a simple threshold. The methodology can thus provide a much more precise evaluation to provide a much more precise indication, such as a precise warning, than can be achieved by simple threshold-based determination as understood in the art. 
         [0099]    While only selected embodiments have been chosen to illustrate the present invention, it will be apparent to those skilled in the art from this disclosure that various changes and modifications can be made herein without departing from the scope of the invention as defined in the appended claims. The functions of one element can be performed by two, and vice versa. The structures and functions of one embodiment can be adopted in another embodiment. It is not necessary for all advantages to be present in a particular embodiment at the same time. Every feature which is unique from the prior art, alone or in combination with other features, also should be considered a separate description of further inventions by the applicant, including the structural and/or functional concepts embodied by such feature(s). Thus, the foregoing descriptions of the embodiments according to the present invention are provided for illustration only, and not for the purpose of limiting the invention as defined by the appended claims and their equivalents.