Cone Spline Surfaces and Spatial Arc Splines – a Sphere Geometric Approach

Basic sphere geometric principles are used to analyze approximation schemes of developable surfaces with cone spline surfaces, i.e., G¹-surfaces composed of segments of right circular cones. These approximation schemes are geometrically equivalent to the approximation of spatial curves with G¹-arc splines, where the arcs are circles in an isotropic metric. Methods for isotropic biarcs and isotropic osculating arc splines are presented that are similar to their Euclidean counterparts. Sphere geometric methods simplify the proof that two sufficiently close osculating cones of a developable surface can be smoothly joined by a right circular cone segment. This theorem is fundamental for the construction of osculating cone spline surfaces. Finally, the analogous theorem for Euclidean osculating circular arc splines is given.