The geometry of Tchebycheffian splines

Helmut Pottmann*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

115 Scopus citations

Abstract

This paper shows that many properties of Bézier and B-spline curves hold for a much wider class of curves. Using a "normal curve" associated with an extended Tchebycheff space, we derive a Bézier representation of Tchebycheffian spline curve segments. These are affine or projective images of segments of the normal curve. A generalization of the blossoming method allows us to study Tchebycheffian B-spline curves and their segments in a simple geometric way. The basic algorithms such as knot insertion and construction of the Bézier points are described. Whereas the generation of tensor product surfaces is straightforward, some preliminary studies indicate that a similarly natural generalization of Bézier triangles does not exist.

Original languageEnglish (US)
Pages (from-to)181-210
Number of pages30
JournalComputer Aided Geometric Design
Volume10
Issue number3-4
DOIs
StatePublished - Jan 1 1993

Keywords

  • B-spline
  • Free-form curve
  • Tchebycheffian spline
  • blossoming
  • geometric order
  • projective invariance
  • rational curve
  • subdivision algorithm
  • tensor product surface.
  • variation diminishing property

ASJC Scopus subject areas

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

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