Computation of Circular Curve Elements Passing through Multiple Points by Using Cosine Theorem

Studies have suggested various solutions for determining the center coordinates and radius of a circle curve that are desired to pass through points whose coordinates are known. One solution is via adjustment according to the least-squares method based on the radius length. Another is based on the indirect measurement method, where the condition equation between the unknowns depends on the polar coordinate values of the points given in the radius equation. The center of the orthogonal coordinate system forms a triangle with the points that the circular curve is to pass through. Here, the distance between the center of the curve and the point where the circular curve is to be passed is the sought radius ($R$). From the points whose orthogonal coordinates are given, the approximate coordinates of the curve center can be calculated by the geodetic positioning method. In this case, the edge length between the center of the orthogonal coordinate system and the center of the curve and the point where the curve is to be passed is calculated. The break angle between the two sides is determined by means of the azimuth angles calculated according to the center of the Cartesian coordinate system of these two points. The radius $R$ can be calculated using the cosine theorem in plane trigonometry. If the number of points that the curve is desired to pass is more than three, the determination can also be performed through adjustment computation. In this study, how to apply the cosine theorem to determine the circular curve elements is first explained. Then, two new algorithms are presented based on the cosine theorem, being inspired from the proposals given in two previous works. Both algorithms were applied to the numerical data given in the previous works, and the results were given comparatively. Finally, a new numerical example was solved, and the findings were examined. Comparing the solution method according to the proposed alternative cosine theorem with the Ghilani & Wolf and Niemeier approaches for curves with a radius of 2 to 3 km, it is seen that the curve parameters difference is a few decimeters in the Gauss–Markov adjustment computation method, while it is a few centimeters in the Gauss–Helmert adjustment computation method. However, for 100 m radius of the curves, the differences were determined to be a few centimeters in both adjustment computation methods. The precision criterion for the parameters determined by the surveying engineers by calculation is determined in centimeter size independent from the radius size with the proposed cosine theorem solution, thus increasing the reliability of the parameters.