Algorithms for Computing Conic Splines

Abstract

Conic splines play an important role in engineering geometry and computer graphics, in particular, in geometric and solids modeling where the majority of objects are bounded by simple surfaces such as natural quadrics (plane, cone, cylinder, and sphere). The integration of analytic and free‐form curves into solids modeling systems is one of the most important tasks in today's solids modelers. This paper presents algorithms that represent conic splines in nonuniform rational B‐spline (NURBS) form, thus incorporating them into a system based on free‐form geometry. The algorithms take as input the end points and end tangents of the segment, and one arbitrary point somewhere along the arc, and return the precise NURBS representation of the curve. It was originally developed for supporting Boolean operations on trimmed surfaces, i.e., to compute conic arc surface‐surface intersections in true 3‐D space. Since during Boolean operations all kinds of segments can pop up, special attention is paid to such cases as parallel end tangents and arcs contained outside the convex hull. Knot insertion is applied to obtain a representation that is numerically as stable as possible. The input required to generate a curve is very general, therefore the spline conversion has different kinds of applications apart from surface‐surface intersection. Some examples are font design, shape generation, geometry conversion and processing, and curve approximation.