Cone spline surfaces and spatial arc splines

Abstract

The design of developable surfaces is of fundamental importance for many applications in Computer Aided Geometric Design. The aim of this thesis is to derive different algorithms to approximate developable surfaces by cone spline surfaces, which are G"1 surfaces composed of segments of right circular cones. With a Hermite-like scheme a discrete set of generators plus tangent planes of a given developable surface is interpolated with smoothly joining cone pairs. In a second method a discrete set of osculating cones of the given developable surface is joined to an osculating cone spline surface. The approximation quality of the proposed methods is analyzed and the results are discussed for several examples. From the standpoint of higher geometry the use of the isotropic model of 3-dimensional Euclidean Laguerre geometry is natural. Cone spline surfaces appear as isotropic arc splines in this model. This gives valuable insight into the cone spline approximation algorithms presented above. Also, in this model it is possible to give an easy proof for an important theorem on osculating cone splines. Finally, the generalization of the planar osculating arc splines of Meek and Walton to 3-dimensional Euclidean 3-space is presented. An algorithm is given how to segment the given curve in order to minimize approximation errors. This segmentation algorithm is also illustrated by several examples.