Rational curves and surfaces with rational offsets

Helmut Pottmann*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

177 Scopus citations


Given a rational algebraic surface in the rational parametric representation s(u,v) with unit normal vectors n(u,v) = (su × sv) t ́|su × sv t ́| the offset surface at distance d is sd(u,v) = s(u,v) + dn(u,v). This is in general not a rational representation, since t ́|su × sv is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets sd(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(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.

Original languageEnglish (US)
Pages (from-to)175-192
Number of pages18
JournalComputer Aided Geometric Design
Issue number2
StatePublished - 1995
Externally publishedYes


  • Dual Bézier curves and surfaces
  • Isophote
  • Offset curve
  • Offset surface
  • Rational Bézier representation
  • Rational curve
  • Rational surface
  • Spherical Bézier patch

ASJC Scopus subject areas

  • Modeling and Simulation
  • Automotive Engineering
  • Aerospace Engineering
  • Computer Graphics and Computer-Aided Design


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