Surveying Computation SURE 215
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Date Created: 10/12/15
Coordinates Calculators and Intersections by Earl F Burkholder Abstract Programmable calculators have become quite indispensable to anyone performing surveying calculations Trigonometric formulas used in plane coordinate computations are universally understood and many have programmed them for various calculators some efficiently and correctly others not so This paper presents formulas and calculator procedures for coordinate geometry and intersection com putations which are superior in accuracy and efficiency to those appearing in recent surveying texts Greater accuracy is obtained by utilizing coordinate differences in the intersection formulas Greater effi ciency is achieved through use of polarrectangular conversions and by exploiting similarities found in the solutions of various intersection problems introduction Programmable calculators have become an indispenable tool for anyone performing sur veying calculations Although tedium of look ing up trigonometric functions and recording numerous intermediate values has been elim inated performing computations efficiently is still desirable Additionally the pro fessional surveyor is responsible for correct ness of the result and sh0uld know what a canned program is doing with the data This paper presents formulas for coordinate geometry computations which are superior in accuracy and efficiency to many being used Greater accuracy is obtained by using coordi nate differences rather than the entire coor dinate value ie state plane coordinates in the intersection formulas Greater efficiency is achieved through use of the surveyor s reference systemquot in the polarrectangular conversions and by exploiting similarities found in various intersection problems Goal The goal here is to present rigorous efficient calculator and programming procedures for the following computations 0 Forward Traverse 0 Inverse 0 Lineline intersection bearingbearing 0 Linecircle intersection bearingdistance 0 Circlecircle intersection distancedistance 0 Perpendicular offset It is possible to program each problem the way it would be solved longhand How ever it is more efficient to use builtin func tions for the Forward and Inverse and to solve the intersections symbolically before programming them Definitions and Conventions Although redundant for most definitions and conventions to be followed are stated specifi cally There must be no ambiguity in the programmer s mind or the user39s under standing as to the meaning or use of any ele ment in the solution of a problem A com puter does only and exactly what it is told to do Surveyor s Reference System A two dimen sional plane cartesian coordinate system is used for surveying computations and in cludes 0 A set of mutually perpendicular axes con sisting of a The abscissa a horizontal line along which the X distance is measured and bThe ordinate a vertical line along which the Y distance is measured Professor Burkholder is a registered PLS and PE and teaches upperdivision surveying courses in cluding state plane coordinate theory and applications adjustment by least squares astronomy and geod esy at the Oregon Institute of Technology His mailing address is Oregon Institute of Technology Oretech Branch Post Office Klamath Falls Oregon 97601 Surveying and Mapping Vol 46 No 1 pp 2939 30 SURVEYING AND MAPPING March 1986 0 Labeling and use of map directions as follows a North the positive Y axis direction b East the positive X axis direction c South the negative Y axis direction d West the negative Y axis direction 0 Use of North as the reference direction 000 003900 39 A positive clockwise rotation measured in degrees minutes and seconds from 0 to 3600 azimuths 0 Quadrant 39abeling as a Northeast Quadrant I b Southeast Quadrant II c Southwest Quadrant III d Northwest Quadrant IV MathScience Reference System Practically all calculators are built or hardwired con ventionally as follows 0 The trigonometric functions normally oper ate in decimal degrees Radians or grads can be specified 0 The polarrectangular conversions are bas ed upon the mathscience coordinate system It is the same as the surveyor reference sys tem except a No map directions are used b The reference direction is along the X axis c Positive rotation is counterclockwise NORTH Y quadrant I quadrant IV XY NEST EAST 13X X quadrant quadrant III 11 SOUTH SURVEYOR39 S SYSTEM d Quadrants are labeled counterclock wise Fig 1 Each reader is responsible to reconcile the differences between the coordinate sys tem hardwired into the particular calculator and that used for surveying computations The following should minimize confusion caused by the differences X and Y coordinates are the same in both systems 0 Values of the trigonometric functions re main unchanged a Quadrant I sin cos b Quadrant II sin cos c Quadrant III sin cos d Quadrant IV sin cos 0 If the direction is alpha 1 in the surveyor s system and theta G in the mathscience sys tem they are related by a90 9 and 990 a sina cosG and cosa sine The polarrectangular PR conversion in most calculators is hardwired to give D cose change in X departure and D sine change in Y latitude The same result departure and latitude is obtained in the surveyor s system by using D sina change in X departure and D cosa change in Y latitude quadrant l quadrant 11 XY AYI 6 l X XX quadrant quadrant III IV MATHSCIENCE SYSTEM Figure 1 Comparison of coordinate systems Since the calculator does not know the dif ference between a and 8 the only change re quired of the user is to switch the latitude and departure designators associated with polarrectangular conversion For example to go from polar to rectangular coordinates the calculator manual may say departure is displayed as the product of distance times co sine of direction entered If the direction were entered as an azimuth in the surveyor39s system the same product is really the course latitude rather than the departure A similar switch is made going from rectangular to pol ar If one inputs the departurelatitude where the manual asks for latitudedeparture math science system the resulting azimuth will be correct in the surveyor s reference system The coordinate computation elements used throughout this paper and shown in Fig ure 2 are Xl amp Y1 X and Y coordinates of beginning point occupied X2 amp Y2 X and Y coordinates of ending point Xp amp Yp X and Y coordinates of intermedi ate point defined by the intersec tion of a two lines lineJine b a line with a circle linecircle c two circles circlecircle a0 Direction azimuth from point 1 to SURVEYING AND MAPPING March 1986 31 393 Direction from intersection point to point 2 D0 Distance from point 1 to point 2 D1 Distance from point 1 to intersection point D2 Distance from intersection point to point 2 AX X2 X1 departure of course 1 to 2 AY Y2 Y1 latitude of course 1 to 2 Y Angle formed at point 1 by Do and D1 always Assumptions and Approach The following assumptions and philosophy are critical to understanding derivation and use of equations listed in the Summary of Coordinate Computation Formulas later in this paper Coordinates of a point are considered pri mary data If coordinates for a point are not available the direction and distance to it from some known point are the defining data for that point However once established the coordinates are primary data and all other quantities are derived from the coordinates Uncertainty random errors positional tolerance and standard deviation are not con sidered This paper deals only with consisten cy of geometrical elements of a problem and redundancy is used only to check correctness point 2 of a solution a Generic direction from point 1 to any Inasmuch as state plane coordinates have point large magnitudes it is desirable to use coor NORTH Y 00 Figure 2 Elements of coordinate computation 32 SURVEYING AND MAPPING March 1986 dinate differences Certain problems with sig nificant figures and calculator capacity are avoided if a trigonometric function is multi plied by a coordinate difference rather than a very large number 0 The approach for the intersection solutions is to write the forward computation symbol ically once for each course The resulting equations are solved for an unknown direc tion or distance as required to compute coor dinates of the intersection point from point 1 using the forward computation An inverse from there to point 2 will give a direction and distance which can be compared with given data on the same course If the check fails an error was made and the computation must be repeated 0 More steps than might be necessary are in cluded in an effort to make the derivation easy to follow Basic Formulas Forward computation formulas are very bas ic but are the basis of intersection formula derivation Referring to Figure 2 and follow ing conventions previously adopted X2 X1DosinaoX1AX 1 Y2 Y1D0cos aoY1AY 2 When one uses the PR polarrectangular key on a calculator it computes AX and AY using direction and distance provided by the user Note however if the calculator is hard wired to the mathscience system it gives AX Do cos direction and AY D0 Sin direction If data were input in the surveyor s system azimuth from north the desired computa tion is still performed but result is given as AY Do cos azimuth and AX D0 sin azimuth Thus if one switches latitudedeparture designations the FIR key can be very useful When programming use of a summation key makes the PR even more powerful if the pro grammer and0r user is willing to keep track of which registers are accumulated as lati tudes and which are accumulated as depar tures The inverse computations are also basic formulas which are hardwired into most cal culators Given coordinates of two points equations 1 and 2 are used as AX D0 sina0 X2 X1 3 AY D0 cosa0 Y2 Y1 4 The inverse computation uses equations 3 and 4 to find direction and distance between two points The distance is obtained by squar ing and adding equations 3 and 4 AX2 AY2 D sin2a0 coszaol X2 X12 Y2 Y12 D0 V AX AY2 X2 X1 Y2 Y239 5 Dividing equation 3 by 4 will give azimuth point 1 to point 2 AXAY D0 sinaoDosinao tanaO 6 The relationship given in equation 6 is al ways true but will not yield a unique azimuth 0 to 360 due to the repetitive nature of the tangent function Another problem in a long hand solution is that computing an azimuth of due east or west is undefined when AY is zero These problems are handled in the long hand solution by adding a very small value 0000001 to AY before dividing and by using bearings for direction However a unique azi muth can be found efficiently if one is willing to use the following tests 39 If AY is negative then 10 180 arctanAXAY 0 If test 1 fails and if AX is negative then 10 360 arctanAXAY 0 If test 1 and test 2 both fail then 10 arctanAXAY It is rarely necessary to use the preced ing test as most calculators have the RIP rectangularpolar conversion builtin Given AX and AY the RP key will provide a dis tance and a unique direction even if AY is zero If an azimuth in the surveyor s system is desired one must be careful to input AX where the calculator expects latitude and AY for the departure Otherwise if hardwired in the mathscience system the calculator will give a counterclockwise azimuth from east If a negative azimuth is encountered one can execute the mod function found on some SURVEYING AND MAPPING March 1986 33 NORTH v X2Y2 ltyL g 0 vY x1Y1 p p 1x41 00 EAST Figure 3 General case for intersection computation calculators or simply add 360 to put the azi muth in the proper range Intersections So far only two points have been considered Intersections involve three points the begin ning and ending point plus the intermediate intersection point Figure 3 illustrates the general intersection case written with the forward computation formulas as Xp X1 D1 sina 7 Yp Y1 D1 cosa 8 X2 Xp D2 sin 9 Y2 Yp D2 cos3 10 This system of four equations can be solved for any combination of four unknowns For in tersections point 1 and point 2 are always known and the coordinates of the intersection are always unknown Different intersection problems are defined by various combina tions of unknowns as shown in Table 1 Un knowns XpYp are eliminated from the set of four equations by solving 9 for Xp and equat ing to 7 and solving 10 for Yp and equating to 8 X X1D1 sinaX2 D2 sinB 11 Y Y1 D1 cosa Y2 D2 cosB 12 Utilizing coordinate differences the equations are written as AX X2 X1 D1 sina D2 sin 13 AY Y2 Y1 D1 cosa D2 cost 14 The problem is now reduced to two equations which can be solved to find that pair of un knowns required by the particular inter section P P LineLine Intersection Given X1Y1 X2Y2 a and Find XpYp D1 and D2 The approach is to solve 14 for D2 substitute into 13 and solve for D1 Knowing D1 and a coordinates XpYp are computed us ing forward position formula given by equa tions 7 and 8 Knowing coordinates of the intersection point distance D2 can be com puted using the inverse computation The in verse direction from the intersection point to point 2 should agree identically within signif icant digit capacity of calculator with B the given azimuth for course 2 If the inverse di Table 1 Different intersection problems defined by various combinations of unknowns Known Always Unknown Unique Unknown Intersection a amp B Xp amp Yp D1 amp D2 lineline a amp D2 Xp amp Yp D1 amp B linecircle D1 amp D2 Xp amp Yp a amp B circlecircle 34 SURVEYING AND MAPPING March 1986 NORTH Y AX EAST Figure 4 Example of negative distance in lineline intersection rection does not agree an error was made in the computations From equation 14 D2 AY D1 cos acos3 Substituting into equation 13 and solv ing for D1 AX D1 sina AY D1 cosacos sin D1 sina AX cosB AY sin3 D1 cosa sin lcosB D1 sinacos cosasinB AX cosBAY sinB D1 AX cosB AY sinBsina 6 15 Equation 15 is an expression for D1 in terms of coordinate differences between points 1 and 2 and the directions azimuths of the two lines The only restriction on the solu tion is that the two lines not be parallel If they are parallel they will never intersect and no solution can be found for D1 due to dividing by zero Note that D1 may be either NORTH Y o o positive or negative If D1 is negative as shown in Figure 4 it means the intersection occurs behind you in the sense of forward be ing in the direction a In summary AX X2 X1 ampAYY2 Y1 and D1 AX cosB AY sinBsina 3 15 Xp X1 D1 sina 7 Yp Y1 D1 cosa 8 D2 V X2 Xp2 Y2 Yp2 5 tan X2 XpY2 Y to check given value 6 LineCircle Intersection Given XY1 X2Y2 a and D2 Find XpYp D1 and B The approach in this case is to solve equation 14 for c053 and to use a form of it in equation 13 to solve for D1 As shown in Fig ure 5 two values of D1 are expected Thus it EAST Figure 5 Elements of lineltcircle intersection is no surprise that the solution involves a quadratic equation From equation 14 cosB AY Dl cosaD2 c0526 AY2 ZAY D1 cosa D cosza Dg 16 Recall the trigonometric identity sin 6 V 1 cosZB 17 Now substitute equation 16 into 17 then in to 13 to get AX D1 sina D2 and AX D1 sina2 D3 1 AY2 2AY D1 cosa D coszaVDg from which AX2 2AXD1sina D39f sinza D3 AY2 2AYD1cosa chosza 0 Collect D and D1 terms in quadratic form am bD1 c 0 D sin2a cosza D12AX sina 2AY cosa AX2AY2 D 0 18 Equating coefficients in equation 18 with those of the quadratic a 1 b 2AX sina 2AY cosa and cAX2AY2 D Y 1 49 xm SURVEYING AND MAPPING March 1986 35 Substituting values into the quadratic equa tion solution gives D1 AX sina AY cosa V b24 c 19 The terms under the radical are b24 c 4AX2 sinza SAX sina AY cosa 4AY2 cos2a4 AX2 AY2 D AX2sin2a 1 AY2cosza 1 2AX sinaAY cosa D Now recall that sinZG 1 cosZG and c0329 1 sinZO Therefore b24 c 1AX2 cosZa 2AX cosaAY sina AYZSinZa D D3 AX cosa AY sina2 20 Combining equations 19 and 20 we get D1 AX sina AY cosa i V D AX cosa AY sina2 21 Equation 21 is an expression for D1 in terms of coordinate differences from point 1 to point 2 and the direction of the line a from point 1 to the intersection point Note that two values of D1 were obtained as expected In addition to efficiency enjoyed by using equation 21 there is an unexpected bonus Obtained from an analysis of the value under the radical If the value under the radical is negative the line does not intersect the circle and no intersection can be computed If the Case 1 Case 11 Case lIl Two Positive Values Two Negative Values One Positive Value 0 of D1 and One Negative Value of D1 Figure 6 Example of positive and negative values of D1 in linecircle intersection 36 SURVEYING AND MAPPING March 1986 value under the radical is exactly 0 there is only one solution the tangent case All posi tive values under the radical yield two possi ble values of D1 one for each intersection Note in Figure 6 that values of D1 which are negative are just as legitimate as positive values The formula for D1 is very powerful in that it tells if there is no solution one solu tion or two solutions Additionally the val ues of D1 can be examined to see if one or both of the intersections are behind us There are no restrictions other than making sure the line intersects the circle and being careful if the exact tangent solution is desir ed In that case the perpendicular offset for mula will give a specific value of D2 given two points and the direction of the line In sum mary Ax X2 X1ampAYY2 Y1 3and4 D1 AX sina AY cosa i V D AX cosa AY sina2 21 Xp X1 D1 sina 7 Yp Y1 D1 cosa 8 tanB X2 XpY2 Yp 6 D2 V X2 Xp2 Y2 Yp2 used to check computation 5 CircleCircle Intersection Given X1Y19 X2Y2 D1 amp D2 Find XPYP a amp B The simple elegance of the lineline and linecircle intersection solutions justified the laborious algebra required to obtain them That is not so with the circlecircle intersec NORTH Y 00 tion If we were to start with equations 13 and 14 eliminate as we did to get equation 18 and try to solve it for a the derivation gets very messy However our goal can be met by using the longhand approach and programming the result The solution is di rect simple rigorous and efficient Refer to Figure 7 showing two intersect ing circles one with its center at point 1 the other at point 2 The approach will be to in verse from point 1 to point 2 to obtain the di rection a0 and distance D0 The angle Y at point 1 is computed using three side lengths in the law of cosines The azimuth to the in tersection points is obtained by adding angle Y to or subtracting it from the inverse direc tion 10 Coordinates of each intersection are then computed using the forward computa tion An inverse from the intersection point to point 2 will give the direction and dis tance D2 which can be used as a check In summary 10 inverse direction from point 1 to point 2 5 D0 inverse distance from point 1 to point 2 6 C057 D 2 D02 DlDo a a0 7 two solutions 23 Xp X1 Dlsina 7 Yp Y1 D1 0050 tanB X2 XpY2 Yp 6 D2 V X2 XID2 Y2 Yp2 used to check computation 5 Consider possible alternatives If the EAST Figure 7 Elements of circle circle intersection b Left SURVEYING AND MAPPING March 1986 37 x1v1 01 a Above 01 greater than 00 02 00 greater than 01 02 Figure 8 Failure of circles to intersect two circles do not intersect as shown in Fig ure 8a or 8b cos Y will be greater than 10 for which 7 does not exist If the two circles are tangent at exactly one point cos Y will be either 10 or 10 and 7 will be exactly 0 or 180 Otherwise two solutions exist The angle 7 is added to or subtracted from the inverse direction 10 to give an azimuth to be used with D1 and the forward computation formula The inverse is then used to give 3 and to check the given distance D2 In some cases it is desired to know only the perpendicular distance from a line to a given point A lineline intersection with a90 will give the complete solution but if coordinates of the intersection are not needed and distance D2 as shown in Figure 9 is the only item of interest a simple equation can be used The approach is to solve equations 13 and 14 for D2 using 3 a 90 Recall trigo nometric identities Perpendicular Offset sin9 90 cosO amp cos9 90 sin9 Given X1Y1 X2Y2 and a Therefore equations 13 and 14 can be writ Find D2 ten as NORTH l X1Y1 4 X 0 0 EAST Figure 9 Elements of perpendicular offset 38 SURVEYING AND MAPPING March 1986 AX D1 sina D2 sina 90 D1 sina D2 cosa 24 AY D1 cosa D2 cosa 90 D1 cosa D2 sina 25 Solve equation 25 for D1 and substitute into equation 24 to solve for D2 D1 AY DQSinacosa AX AY Dzsinacosa D2 cosa AX cosa AY sina D2sin2a cosza D2 AX cosa AY sina 26 Equation 26 for a perpendicular offset distance is elegant rigorous and simple Not only does it give the offset distance but which side of the line it is on is given by whether it comes out positive or negative If point 2 lies right of the line as assumed in the derivation D2 comes out positive However if the point lies left of the line D2 comes out negative This feature can be particularly useful when computing offset from a random traverse line to a section line for clearing and marking One final item about the perpendicular offset Note that the perpendicular offset distance given by equation 26 also appears under the radical of equation 21 as one of the legs of a right triangle within the circle Thus equation 26 might be programmed as a sub routine to be called as required Since programmable calculators have be come available the author has encountered several inadequate intersection programs which are or have been available on a com mercial basis In one specific case the pro did not consider restrictions imposed by his assumptions In other cases the accuracy of the solution suffers because a large state plane coordinate value is multiplied by a trigonometric function rather than a coor dinate difference as presented herein Who is responsible for integrity of sur veying computations Is it the technician pushing the buttons as directed by the boss Is it the person who signs off on the computa tions or plat Is it the person who writes andor markets the programs Or is it those who teach Assuming all share that responsi bility it is hoped this systematic approach to coordinate computation and use of program mable calculators will improve our collective professional efforts Summary of Coordinate Computation Formulas Forward 1 X2 X1 D0 sina 2 Y2 Yl Docosa0 Inverse Figure 10 3AX X2 X1 4AY Y2 Yl 5 D0 m 6 tanaJ AXAY a0 arctan AXAY Quadrant I a0 180 arctan AXAY Quadrants II amp III a o 360 arctan AXAY Quadrant gram failed entirely because the programmer IV NORTH Y X 0 O EAST Figure 10 Elements of coordinate computation
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