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

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:

 65D17 Computer-aided design (modeling of curves and surfaces) 68U07 Computer science aspects of computer-aided design 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces
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### References:

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