Fitting Plane Curves to Three-Dimensional Points

Abstract

Fitting plane curves to the three-dimensional (3D) points defining those curves can be performed by ordinary general least squares adjustment. To this, a conformal transformation of the initial 3D system to a two-dimensional system in the best-estimated plane of the curve is used. The kind and the number of the unknown transformation parameters are selected according to one’s needs. In this adjustment, the coordinates of the defining points are considered observed parameters while the parameters of the aforementioned transformation and the ones defining the curve unobserved parameters. Finally, the curve—with respect to the initial system—is fully determined. Coordinate determination, together with least squares fitting of certain curves to these coordinates, is usually performed by advanced theodolite intersection systems. The straight line is a special case of curve lying in infinite number of planes, therefore, the suitable selection of the plane in which the adjustment is held is indicated, each time, by current convenience. Fitting a straight line to 3D points, a process hardly noticed in least squares textbooks, is extremely useful in high-precision deformation check and a lot of industrial survey applications.