RATIONAL CURVES AND SURFACES WITH RATIONAL OFFSETS

Given a rational algebraic surface in the rational parametric representation s(u, v) with unit normal vectors n(u, v) = (s(u) x s(v))/parallel to s(u) x s(v) parallel to, the offset surface at distance d is s(d)(u, v) = s(u, v) + dn(u, v). This is in general not a rational representation, since parallel to s(u) x s(v) parallel to is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets s(d)(u, v). The analogous question is solved for curves, which is an extension of Farouki's Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c(t) whose are length parameter s(t) is a rational function of t. Offsets arise in the mathematical description of milling processes and in the representation of thick plates, such that the presented curves and surfaces possess a very attractive property for practical use.