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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 \(s\rightarrow (u,v)\) 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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