Rational curves and surfaces with rational offsets Original Research Article

Given a rational algebraic surface in the rational parametric representation s(u,v) with unit normal vectors n(u,v) = (su × sv)t́|su × svt́| the offset surface at distance d is sd(u,v) = s(u,v) + dn(u,v). This is in general not a rational representation, since t́|su × sv is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets sd(u,v). The analogous question is solved for curves, which is an extension of Faroukiʹs Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c(t) whose arc 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.