Osculating Conic Biarcs

Anton Gfrerrer*, Gunter Weiss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A circular biarc can be defined by using two points K and L with their (oriented) tangents g K and g L as input. It is well-known that one can determine a one parametric set of circular arc pairs k,ℓ such that k starts at K with tangent g K, ℓ ends at L with tangent g L and k and ℓ meet with a common tangent in an intermediate point P. In this paper we investigate a similar construction where we replace the circle biarcs by pairs of conic arcs. It turns out that in this case we can prescribe a conic k 0 with a point K on it, another conic ℓ 0 with a point L on it, and moreover an intermediate point P to obtain a unique pair k,ℓ of conics such that k osculates k 0 in K, ℓ osculates ℓ 0 in L and k and ℓ osculate each other in P. This also confirms a result of H. Pottmann from 1991. We use our method to solve an interpolation task of Hermite type whose input consists of a series of points with their curvature circles and another series of intermediate points. The output is a GC 2 spline curve with conic arc segments.

Original languageEnglish
Article number101904
JournalComputer Aided Geometric Design
Publication statusPublished - Aug 2020


  • Biarcs
  • Conic arcs
  • GC -spline
  • Hermite interpolation
  • Osculation
  • Projective map

ASJC Scopus subject areas

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


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