A Laguerre geometric approach to rational offsets

Martin Peternell, Helmut Pottmann*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

92 Scopus citations


Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be constructed from arbitrary rational curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.

Original languageEnglish (US)
Pages (from-to)223-249
Number of pages27
JournalComputer Aided Geometric Design
Issue number3
StatePublished - Mar 1998
Externally publishedYes


  • Geometrical optics
  • Laguerre geometry
  • NC milling
  • Offset
  • Principal curvature line
  • Principal patch
  • Rational curve
  • Rational offset: Pythagorean-hodograph curve
  • Rational surface

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Geometry and Topology
  • Modeling and Simulation


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