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Coordinates, Calculators, and Intersections by Earl F. Burkholder Abstract. Programmable calculators have become quite indispensable to anyone performing surveying calculations. ‘Trigonometrie formulas used in plane coordinate computations are universally understood and many have programmed them for various calculators; some efficiently and correetly, 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 polar-rectangular 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. Additionaily, the pro- fessional surveyor is responsible for correct- ness of the result and should know what a “canned” program is doing with the data. This paper presents formulas for coordinate geometry computations which arc 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 (i.e., state plane coordinates} in the intersection formulas. Greater efficiency is achioved through use of the “surveyor's reference system” in the polarrectangular conversions and by exploiling similarities found in various intersection problems. Goal The goal here is to present rigorous, efficient caleulator and programming procedures for the following computations: « Forward (Traverse) * Inverse * Line-tine intersection (bearing-bearing) * Line-circle intersection (bearing-distance) * Circle-circle intersection (distance-distance) + Perpendicular offset It is possible to program each problem the way it would be solved longhand. How- ever, it is more efficient to use built-in fane- 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 specili- cally. There must be no ambiguity in the programmer's mind or the user’s under- standing as to the meaning or use of any ele- ment in the solulion 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: * A set of mutually perpendicular axes con- sisting of: a. The abscissa, a horizontal line along which the X distance is measured and, b. The ordinate, a vertical line along which the ¥ distance is measured. Professor Burkholder is a registered P.L.S, and P.E. and teaches upper-division 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. 29-39 29 
