On the parallel lines for nondegenerate conics

Abstract

Computation of parallel lines (envelopes) to parabolas, ellipses, and hyperbolas is of importance in structure engineering and theory of mechanisms. Homogeneous polynomials that implicitly define parallel lines for the given offset to a conic are found by computing Groebner bases for an elimination ideal of a suitably defined affine variety. Singularity of the lines is discussed and their singular points are explicitly found as functions of the offset and the parameters of the conic. Critical values of the offset are linked to the maximum curvature of each conic. Application to a finite element analysis is shown. Keywords: Affine variety, elimination ideal, Groebner basis, homogeneous polynomial, singularity, family of curves, envelope, pitch curve, undercutting, cam surfaceComment: 40 pages, 10 figures, TOC, 3 appendices, short version of this paper was presented at the 5th Annual Hawaii International Conference on Statistics, Mathematics and Related Fields, January 16 - 18, 2006, Honolulu Hawaii, US