Rational curves and surfaces with rational offsets

Abstract Given a rational algebraic surface in the rational parametric representation s (u,v) with unit normal vectors n (u,v) = ( s u × s v ) t | s u × s v t | the offset surface at distance d is s d (u,v) = s (u,v) + d n (u,v) . This is in general not a rational representation, since t | s u × s v 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 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.

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