Abstract:
This device facilitates field marking an arc of a circle having an obstructed center. It comprises a first clamp having a slot to engage a standard tape measure and a second clamp having another slot to engage a color-coded tape measure. A detachable swivel joint snap connects the first clamp to the second clamp, and a handle is connected to the first clamp. 
     A method is provided to use the device, a standard tape measure, a color-coded tape measure, a rod, and a scribe to mark out such an arc. The method comprises identifying coordinates of an eccentric ellipse, whose foci are field accessible, that will substantially overlap the arc. The two tape measures are anchored to the two foci. The standard tape measure is stretched tight between the two foci while the color-coded tape measure provides a string line for marking out a closely fitting elliptical arc segment.

Description:
RELATED APPLICATIONS 
   This application is a continuation-in-part of and claims benefit to U.S. patent application Ser. No. 10/061,482, filed on Feb. 4, 2002, now abandoned, entitled “A Device for and Method of Field Laying Out of a Radius When a Radius Point is Not Available.” 

   FIELD OF THE INVENTION 
   Applicant&#39;s invention relates to a device for and method of marking out an arc segment when a focal point is not available. 
   BACKGROUND OF THE INVENTION 
   Oftentimes when a contractor is asked to design a configuration such as an addition to a building or a swimming pool having a curved profile, the focal point of the arc segment defining the curved portion is available and accessible to mark out the curved portion. Under such circumstances, the contractor can access the focal point to plot out the curved portion of the new structure and design accordingly. Unfortunately, there are many instances where an obstacle obstructs access to the focal point. In these cases the obstacle presents difficulties in accurately laying out the desired arc segment. Presently, there are no satisfactory devices or procedures to facilitate the geometric construction of an arc segment when the focal point is unavailable, such as where a building or other obstacle is in the way of reaching the focal point. 
   SUMMARY OF THE INVENTION 
   It is an object of the present invention to provide a novel device for field laying out of an arc segment when the center of the circle of which the arc is a segment is unavailable. 
   It is another object of the present invention to provide a novel method of field laying out of an arc segment when the center of the circle of which the arc is a segment is unavailable. 
   In satisfaction of these and related objectives, Applicant&#39;s present invention provides a device and method for field laying out of an arc segment. Applicant&#39;s invention permits its practitioner to lay out an arc segment when the center of the circle of which the arc is a segment is unavailable. 
   In particular, a method is provided to approximately field mark an arc of a circle having an obstructed center. An eccentric ellipse is identified that will substantially overlap the arc, the ellipse having a first focus and a second focus, both of which are field accessible. Two line indicating means are obtained comprising, for example, a standard tape measuring instrument and a color-coded tape measure, two standard tape measuring instruments, or two strings. A reference point on a first line is placed at or near the first focus. The reference point may be anchored to the first focus with a rod. A second point on the first line, separated from the reference point by a distance equal to the distance between the first focus and the second focus of the ellipse, is placed at or near the second focus. A reference point on a second line is also placed at or near the first focus. A second point on the second line, separated from the second line&#39;s reference point by a distance equal to the major axis length of the ellipse is also placed at or near the second focus. The second line is drawn tight at a plurality of points on the ellipse where the ellipse substantially overlaps the arc; and these points are marked with a marking instrument such as a scribe. 
   This method may further comprise a calibration step involving pulling the second line toward an endpoint of the arc, and then toward the opposite endpoint, but not at the same time; and moving the first line toward or away from the chord of the arc, while keeping the first line centered and parallel to the chord, until the first line is in a position where the step of pulling the second line toward either the first or second endpoint will pull the second line tight. 
   A device is also provided to facilitate field marking an arc of a circle having an obstructed center. The device comprises a first clamp having a first slot to engage a first measuring instrument and a second clamp having a second slot to engage a second measuring instrument; wherein the second clamp is connected to the first clamp. A handle is preferably connected to the first clamp. A detachable swivel joint, such as a snap, preferably connects the first clamp to the second clamp. 
   One of the measuring instruments is preferably a standard tape measure with conventional metric or English-system markings and a length of 100 feet while the other measuring instrument is preferably a color-coded tape measure with a length of 100 feet. On each side of the color-coded tape measure there are set intervals which correspond to the predetermined distances for the major axis length of the ellipse for circular arc segments of various radii. 
   Furthermore, a method is provided to approximately mark an arc on a field using two measuring instruments (e.g., a standard tape measure and a color-coded tape measure, two standard tape measures, or two strings), where the arc is delimited by two endpoints, and the arc is part of a circle having an obstructed center. The method comprises identifying coordinates of an eccentric ellipse that will substantially overlap the arc, the ellipse having two foci, both of which are field accessible; anchoring the starting point on each of the two measuring instruments to a first field location (i.e., the first focus of the two ellipse foci); extending the first of the two measuring instruments to a second field location (i.e., the second focus of the two ellipse foci) separated from the first field location by a distance equal to the distance between the two foci of the ellipse, wherein the first and second field locations define points on a line that runs parallel to a chord which connects the endpoints of the arc; extending the second of the two measuring instruments to a length that is equal to the distance of the ellipse&#39;s major axis; clamping the second measurement instrument to the second field location; and pulling a marking instrument along and against the second measuring instrument to mark out the arc. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a plan view illustrating a design situation where the center of the circle of which a desired circular arc is a segment is obstructed by an obstacle in the field. 
       FIG. 2  is a plan view illustrating an ellipse, whose foci are field accessible, which can be used to approximate the desired circular arc segment of FIG.  1 . 
       FIG. 3  is a schematic representation of the dimensions and coordinates of an ellipse that closely fits a 60-foot chord length arc segment of a circle of radius 60 feet. 
       FIG. 4  is a schematic representation of the dimensions and coordinates of an ellipse that closely fits a 45-foot chord length arc segment of a circle of radius 60 feet. 
       FIG. 5  depicts a table of elliptical coordinates that may be used to provide a close fit to circular arc segments of various radii, assuming that the chord length equals the radius. 
       FIG. 6  depicts a table of formulas used to compute the elliptical coordinates of FIG.  5 . 
       FIG. 7  depicts a table of elliptical coordinates that may be used to provide a close fit to circular arc segments of various radii, assuming that the chord length equals 75% of the radius. 
       FIG. 8  depicts a table of formulas used to compute the elliptical coordinates of FIG.  7 . 
       FIG. 9  is a flow chart of a method to approximately lay or mark out a desired circular arc segment in the field using the principles of this invention. 
       FIG. 10  depicts a device for field laying out an arc segment when a focal point for the arc segment is unavailable. 
       FIG. 11  is a flow chart of a method for using the device of  FIG. 10  to mark out an elliptical arc segment. 
       FIG. 12  illustrates the rod-threading step of FIG.  11 . 
       FIG. 13  illustrates the clamping steps of FIG.  11 . 
       FIG. 14  illustrates the clamp-connecting step of FIG.  11 . 
       FIG. 15  illustrates the arc-marking step of FIG.  11 . 
   

   DETAILED DESCRIPTION 
     FIG. 1  illustrates the difficulty contractors frequently face in laying out a circular arc segment  10  having a focal point  120  that is obstructed by an object such as a house  150 . For such situations, Applicant has invented a method and device to lay out an elliptical arc segment  130  (see  FIG. 2 ) that closely approximates the desired circular arc segment  110 . Applicants&#39; invention capitalizes on the fact that a significant arc portion of an ellipse is approximately circular, as illustrated in  FIGS. 2-4 . 
   For example, assume that a contractor desires to lay out a 60-foot radius circular arc segment having a chord length of 60 feet. (A “chord” is commonly defined as “a line segment that joins two points on a curve”). Those of ordinary skill in the art will appreciate that such an arc segment would span 60 degrees of a 60-foot radius circle. Also assume that the center of such a 60-foot circle  115  is in the middle of a house  150 , thereby obstructing a contractor from connecting a string line to the center or focal point  120  to plot out the desired arc segment  110 . 
   In accordance with one embodiment of the invention, this problem is overcome by plotting an elliptical arc segment  130  that closely approximates the desired circular arc segment  110 . In  FIG. 2 , it can be seen that the foci  135  of the ellipse  140 , unlike the center point  120  of the circle  115 , are not obstructed by the house  150 . In accordance with Applicants&#39; invention, the foci  135  of the ellipse  140 , rather than the center point  120  of the circle  115 , are used to approximate the desired circular arc segment  110 . 
     FIGS. 3-4  illustrate the dimensions of elliptical arc segments  330 ,  430  that closely overlap a desired circular arc segment  310 ,  410  along the long sides of the ellipse  340 ,  440 . In each of  FIGS. 3-4 , the ellipse  340 ,  440  intersects the desired circular arc segment  310 ,  410  approximately at the midpoint  312 ,  412  of the desired circular arc segment  310 ,  410 . Also, in each of  FIGS. 3-4 , the major axis  342 ,  442  of the ellipse  340 ,  440  lies parallel to the chord of the desired circular arc segment  310 ,  410 . 
   It was found that the arc  330  of a 60-foot chord of an ellipse  340  with a minor axis  344  of 75% of the 60-foot radius  316  (45 feet), and a major axis  342  of 1.3053633 times the 60-foot radius  316  (78.3218 feet) would closely fit the arc  310  of a 60-foot chord  311  on a circle  315  having a 60-foot radius  316 . Assuming that the ellipse  340  intersects the circle  315  at the midpoint  312  of the circular arc segment  310 , the endpoints of such an elliptical arc segment  330  would be spaced only 0.2727 feet (3.27 inches) apart from the endpoints  313 ,  314  of the desired circular arc segment  310 . It will be appreciated that if the midpoint of elliptical arc segment  330  is set 1.64 inches away from the midpoint  312  of the circular arc segment.  310 , the overall error of the elliptical curve fit is ±1.64 inches.  FIG. 3  depicts such an ellipse  340  superimposed on a 60-foot-radius circle  315 .  FIGS. 5 and 6 , which are explained further below, show spreadsheets depicting the formulas used and results yielded by extrapolating these relationships to a whole range of different arc radii. 
   It was also found that the arc  430  of a 45-foot chord  411  on an ellipse  440  with a minor axis  444  of 75% of the 60-foot radius  416  (45 feet), and a major axis  442  of 1.2653116 times the 60-foot radius  416  (75.9187 feet) would closely fit the arc  410  of a 45-foot chord  411  on a circle  415  having a 60-foot radius  416 . (Those of ordinary skill in the art will appreciate that such an arc segment would span approximately 44 degrees of a 60-foot radius circle). Assuming that the ellipse  440  intersects the circle  415  at the midpoint  412  of the circular arc segment  410 , the endpoints of such an elliptical arc segment  430  would be spaced only 0.0737 feet (0.88 inches) apart from the endpoints  413 ,  414  of the desired circular arc segment  410 . It will be appreciated that if the midpoint of elliptical arc segment is set 0.44 inches away from the midpoint  412  of the circular arc segment, the overall error of the elliptical curve fit is ±0.44 inches.  FIG. 4  depicts such an ellipse  440  superimposed on a 60-foot-radius circle  415 .  FIGS. 7 and 8 , which are explained further below, show spreadsheets depicting the formulas used and results yielded by extrapolating these relationships to a whole range of different arc radii. 
   As evident from  FIGS. 3 and 4 , an ellipse whose minor axis has a length equal to 75% of the radius of the circle of which the arc is a segment will be suitable for many construction tasks.  FIGS. 5-8  extrapolate this relationship by depicting preferred baseline and string length dimensions for approximating circular arc segments for a range of different arc radii and chord lengths.  FIG. 5  depicts the dimensions preferred for 60-degree arc segments having radii ranging from 5 to 100 feet.  FIG. 7  depicts the dimensions preferred for 44-degree arc segments having a chord length that is equal to 75% of the radius length which ranges from 5 to 100 feet.  FIGS. 6 and 8  depict the spreadsheet formulas used to compute the values depicted in  FIGS. 5 and 7 , respectively. 
   In addition,  FIGS. 5 and 7  also illustrate the difference in accuracy between the coordinates of the 60-degree arc segment when assuming the chord length equals the length of the radius as compared to the coordinates of the 44-degree arc segment when assuming the chord length is equal to 75% of the radius length. As seen from  FIGS. 5 and 7 , when using a chord length that is equal to 75% of the radius length the 44-degree arc segment coordinates are four times as accurate as they are when using a 60-degree arc segment with a chord length that is equal to the length of the radius. Persons with ordinary skill in the art will appreciate that the dimensions of the 60-degree arc segments can be used for exterior site layout work while the dimensions of the 44-degree arc segments with a chord length equal to 75% of the radius length can be used for interior work such as archways in buildings when a high degree of accuracy is required. 
     FIG. 9  sets forth a flow chart of a method to approximately lay or mark out a desired circular arc segment in the field using the principles of this invention. In block  910 , the chord length and radius of the desired circular arc segment are determined. In block  920   a,  a computation or identification is made of the coordinates of an ellipse, whose foci will be field accessible, which will provide the closest curve fit to the desired circular arc segment. 
   Those of ordinary skill in the art of computer programming or geometric calculus will understand how to implement iterative techniques or derive formulas to determine which elliptical coordinates (e.g., major axis length, focal coordinates) will provide the best fitting elliptical arc segment assuming that the following variables are known: the chord length of the desired arc segment, the radius of the circle of which the desired arc is a segment, and the set of possible ellipses whose foci would be field accessible. Using such techniques, the disclosed invention can be used to find the best elliptical coordinates for any set of arc radii and chord lengths. Indeed, a computer program may be provided to enable those in the field to quickly find the best-fitting ellipse whose foci are field accessible. 
   It is not necessary that brute-force calculations be performed in every case. In many field situations, an ellipse with a fixed minor-axis-length-to-major-axis-length ratio or a fixed minor-axis-length-to-circle-radius ratio will provide an adequate curve fit. In such situations, an alternative way to identify curve-fitting elliptical coordinates may be provided. For example, in block  920   b,  elliptical coordinates are identified based on the assumption that an ellipse whose minor axis has a length equal to 75% of the radius of the circle of which the arc is a segment will provide an adequate elliptical curve fit. Applying this assumption, persons in the field can identify the coordinates of an adequate curve-fitting ellipse by looking up the values in a table like those depicted in  FIG. 5  or  7 , or by using formulas like those displayed in  FIGS. 6 and 8 . 
   In block  930 , the approximate locations of the ellipse&#39;s foci are marked out in the field. This can be accomplished, for example, by marking or placing a baseline equal to the distance between the foci at a distance from the circular arc&#39;s midpoint equal to approximately one-half of the ellipse&#39;s minor axis length. Of course, the baseline should also be laid out approximately parallel to the chord of the desired arc segment, and the minor axis of the ellipse should pass through the midpoint of the baseline (this is done to ensure that the baseline is centered). 
   In block  940 , the ends of the measuring instrument whose length is equal to the major axis length of the ellipse, is anchored to the two foci (or the ends of the baseline). 
   In blocks  950  and  960 , the position of the baseline is calibrated by pulling a scribe against the string line toward one endpoint of the desired arc segment, and then the other endpoint. While at all times keeping the baseline centered and parallel to the chord, the baseline is moved toward or away from the chord until the baseline is in a position that will cause the measuring instrument to be pulled tight by the scribe at either endpoint of the desired arc segment. 
   In block  960 , the scribe is pulled along and against the measuring instrument to lay or mark out an elliptical arc segment that closely approximates the desired circular arc segment. 
     FIG. 10  depicts a device  1000  for field laying out an arc segment when a focal point for the arc segment is unavailable. Device  1000  comprises first and second clamps  1010  and  1020  for clamping the first and second measuring instruments  1030  and  1040 . First and second clamps  1010  and  1020  each have slots  1011 ,  1021  for receiving the measuring instruments  1030  and  1040 . Protruding through slots  1011  and  1021  are threaded screw holes  1012  and  1022 , respectively, for receiving screws  1013  and  1023 . Each screw  1013 ,  1023  can be tightened to hold, or loosened to disengage, the corresponding measuring instrument  1030  or  1040 . 
   A handle  1015 , designed and operable to be hand-held by a contractor in the field, is attached to the first clamp  1010 . The second clamp  1020  is pivotally mated to the first clamp  1010  using a swivel joint  1050  such as a snap, wherein the female portion  1051  of the joint  1050  is connected to the first clamp  1010 , and the male portion  1052  of the joint  1050  is connected to the second clamp  1020 . The swivel joint  1050  allows the second measuring instrument  1040  to move freely in relation to the first measuring instrument  1030 . 
   In operation, the first measuring instrument  1030  is preferably used to mark the baseline of the ellipse that is used to approximate the circular arc segment. The second measuring instrument  1040  is preferably used as a string line to mark out the elliptical arc segment (see discussion on FIG.  9 ). 
   The first measuring instrument  1030  is preferably a standard tape measure with a length of up to 100 feet and conventional metric or English-system markings. The second measuring instrument  1040  is preferably color-coded with each side having intervals that correspond to the predetermined distances for the major axis length of the ellipse for circular arc segments of various radii. The intervals on each side of the color-coded tape measure identify the various desired circle radii. For example, an interval marked as R25′-2″ will correspond to the predetermined distance for the major axis length of the ellipse for a desired circular arc segment having a radius of 25 feet and 2 inches. 
   The color-coded tape measure is preferably red on one side and green on the other. The green side represents the distances that are predetermined for an elliptical arc segment when assuming that the chord length is equal to the radius of the arc segment. The red side represents the distances that are predetermined for an elliptical arc segment when assuming that the chord length is equal to 75% of the arc segment radius. The green side would be used for exterior site layout work when only a normal degree of accuracy is required. The red side would be used for interior work such as archways in buildings when a high degree of accuracy is required. 
   The length of the first measuring instrument  1030  and the second measuring instrument  1040  are preferably 100 feet each. However, when using Applicant&#39;s device and method for field marking an arc of a circle where the radius of the circle is greater than 76 feet, the first measuring instrument  1030  and the second measuring instrument  1040  must each be greater than 100 feet in length. For example, as seen in  FIG. 7 , the stringline length (i.e., the distance of the major axis of the ellipse) needed for field marking an arc of a circle with a radius of 100 feet is 130.53633 feet and the baseline length (i.e., the distance between the two foci of the ellipse) is 106.8397559 feet. Thus, it is necessary that the first measuring instrument  1030  and the second measuring instrument  1040  used for field marking an arc of a circle having a radius of 100 feet must have lengths of at least 107 feet and 131 feet, respectively. Such adjustments in the length of the measuring instruments  1030  and  1040  should be considered as one of the various embodiments of the Applicant&#39;s device. 
     FIG. 11  is a flow chart of a method for using device  1000  to mark out an elliptical arc segment.  FIGS. 12-15  illustrate these steps. In step  1110 , a rod  1060  is threaded through the starting points of measuring instruments  1030  and  1040  as depicted in FIG.  12 . In step  1120 , measuring instrument  1030  is pulled to a length equal to the distance between the foci of the ellipse. Clamp  1010  is clamped to that point, as illustrated in FIG.  13 . In step  1130 , measuring instrument  1040  is pulled to a length equal to the length of the ellipse&#39;s major axis. Clamp  1020  is clamped to that point, also as illustrated in FIG.  13 . In step  1140 , clamp  1010  is joined to clamp  1020 , as illustrated in FIG.  14 . In step  1150 , the rod  1060  is placed at one of two foci of the ellipse, as illustrated in FIG.  15 . In step  1160 , device  1000  is positioned at the other of the two foci of the ellipse, as illustrated in FIG.  15 . Because the beginning-to-clamped portion of measuring instrument  1040  is longer than beginning-to-clamped portion of measuring instrument  1030 , it initially lies loose between the rod  1060  and device  1000 . In step  1170 , the measuring instrument  1040  is drawn tight with an instrument  1070  such as a scribe, thereby positioning the instrument  1070  at a point on the ellipse. In step  1180 , a scribe  1070  is moved along and against measuring instrument  1040 , while at all times holding measuring instrument  1040  tight, to mark out the desired elliptical arc segment. 
   It will of course be appreciated that the steps set forth in  FIG. 11  do not necessarily have to be done in the described order. It will also be appreciated that additional steps, such as the calibration steps described with respect to  FIG. 9 , might also be performed in the method of FIG.  12 . Because such variations are well within the understanding of those of ordinary skill in the art, the invention should, in all fairness, be understood to encompass such variations. 
   Although the invention has been described with reference to specific embodiments, this description is not meant to be construed in a limited sense. Various modifications of the disclosed embodiments, as well as alternative embodiments of the inventions will become apparent to persons skilled in the art upon the reference to the description of the invention. It is, therefore, contemplated that the appended claims will cover such modifications that fall within the scope of the invention.