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Zbl 0872.65011
Pottmann, Helmut
Rational curves and surfaces with rational offsets. (English)
[J] Comput. Aided Geom. Des. 12, No.2, 175-192 (1995). ISSN 0167-8396

Summary: Given a rational algebraic surface in the rational parametric representation $s\rightarrow (u,v)$ with unit normal vectors $$n\rightarrow (u,v)=(s\rightarrow_{u} \times s\rightarrow_{v})/\parallel s\rightarrow_{u} \times s\rightarrow_{v}\parallel ,$$ the offset surface at distance $d$ is $$s\rightarrow_{d}(u,v)=s\rightarrow (u,v)+dn\rightarrow (u,v).$$ This is in general not a rational representation, since $\parallel s\rightarrow_{u} \times s\rightarrow_{v} \parallel $ is in general not rational. We present an explicit representation of all rational surfaces with a continuous set of rational offsets $s\rightarrow_{d}(u,v)$. 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 $c\rightarrow (t)$ 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.

MSC 2000:: *65D17 Computer aided design (modeling of curves and surfaces)
68U07 Computer aided design
53A07 Higher-dimension and -codimension surfaces in Euclidean n-space

Keywords: rational curve; rational surface; offset curve; offset surface; rational Bézier representation; dual Bézier curves and surfaces; spherical Bézier patch; isophote

Cited in: Zbl 1156.65016 Zbl 0906.65011

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