*(English)*Zbl 0872.65011

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

the offset surface at distance $d$ is

This is in general not a rational representation, since $\parallel s{\to}_{u}\times 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 $s{\to}_{d}(u,v)$. 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 $c\to \left(t\right)$ 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 |