Mathematics in Computer Science

, Volume 8, Issue 2, pp 299–319 | Cite as

On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

  • Bohumír Bastl
  • Bert Jüttler
  • Miroslav Lávička
  • Tino Schulz
  • Zbyněk Šír


Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one-parameter family of spheres. Consequently, canal surfaces can be described by a spine curve and a radius function. We present parameterization algorithms for rational ringed surfaces and rational canal surfaces. It is shown that these algorithms may generate any rational parameterization of a ringed (or canal) surface with the property that one family of parameter lines consists of circles. These algorithms are used to obtain rational parameterizations for Darboux cyclides and to construct blends between pairs of canal surfaces and pairs of ringed surfaces.


Ringed surface Canal surface Rational parameterization Darboux cyclide Dupin cyclide Blending 

Mathematics Subject Classification

65D17 68U05 53A05 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Bohumír Bastl
    • 1
  • Bert Jüttler
    • 2
  • Miroslav Lávička
    • 1
  • Tino Schulz
    • 3
  • Zbyněk Šír
    • 4
  1. 1.University of West BohemiaPlzeňCzech Republic
  2. 2.Johannes Kepler University of LinzLinzAustria
  3. 3.INRIA Méditerranée, Sophia-AntipolisValbonneFrance
  4. 4.Charles University in PraguePragueCzech Republic

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