### Abstract

This paper considers the conversion of the parametric Bézier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.

Original language | English (US) |
---|---|

Title of host publication | Intelligent Computer Vision and Image Processing |

Subtitle of host publication | Innovation, Application, and Design |

Publisher | IGI Global |

Pages | 113-127 |

Number of pages | 15 |

ISBN (Electronic) | 9781466639096 |

ISBN (Print) | 9781466639072 |

DOIs | |

State | Published - Apr 30 2013 |

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### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Intelligent Computer Vision and Image Processing: Innovation, Application, and Design*(pp. 113-127). IGI Global. https://doi.org/10.4018/978-1-4666-3906-5.ch009

**Construction of 3D triangles on Dupin cyclides.** / Belbis, Bertrand; Garnier, Lionel; Foufou, Sebti.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Intelligent Computer Vision and Image Processing: Innovation, Application, and Design.*IGI Global, pp. 113-127. https://doi.org/10.4018/978-1-4666-3906-5.ch009

}

TY - CHAP

T1 - Construction of 3D triangles on Dupin cyclides

AU - Belbis, Bertrand

AU - Garnier, Lionel

AU - Foufou, Sebti

PY - 2013/4/30

Y1 - 2013/4/30

N2 - This paper considers the conversion of the parametric Bézier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.

AB - This paper considers the conversion of the parametric Bézier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.

UR - http://www.scopus.com/inward/record.url?scp=84944111922&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84944111922&partnerID=8YFLogxK

U2 - 10.4018/978-1-4666-3906-5.ch009

DO - 10.4018/978-1-4666-3906-5.ch009

M3 - Chapter

SN - 9781466639072

SP - 113

EP - 127

BT - Intelligent Computer Vision and Image Processing

PB - IGI Global

ER -