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Rational curves and surfaces with rational offsets. (English) Zbl 0872.65011

Summary: Given a rational algebraic surface in the rational parametric representation $s\to \left(u,v\right)$ with unit normal vectors

$n\to \left(u,v\right)=\left(s{\to }_{u}×s{\to }_{v}\right)/\parallel s{\to }_{u}×s{\to }_{v}\parallel ,$

the offset surface at distance $d$ is

$s{\to }_{d}\left(u,v\right)=s\to \left(u,v\right)+dn\to \left(u,v\right)·$

This is in general not a rational representation, since $\parallel s{\to }_{u}×s{\to }_{v}\parallel$ is in general not rational. We present an explicit representation of all rational surfaces with a continuous set of rational offsets $s{\to }_{d}\left(u,v\right)$. 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 $c\to \left(t\right)$ whose arc length parameter $s\left(t\right)$ 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:
 65D17 Computer aided design (modeling of curves and surfaces) 68U07 Computer aided design 53A07 Higher-dimensional and -codimensional surfaces in Euclidean $n$-space