### Abstract

We introduce a novel method to define Dupin cyclide blends between quadric primitives. Dupin cyclides are non-spherical algebraic surfaces discovered by French mathematician Pierre-Charles Dupin at the beginning of the 19th century. As a Dupin cyclide can be fully characterized by its principal circles, we have focussed our study on how to determine principal circles tangent to both quadrics being blended. This ensures that the Dupin cyclide we are constructing constitutes a G^{1} blend. We use the Rational Quadratic Bélier Curve (RQBC) representation of circular arcs to model the principal circles, so the construction of each circle is reduced to the determination of the three control points of the RQBC representing the circle. In this work, we regard the blending of two quadric primitives A and B as two complementary blending operations: primitive A-cylinder and cylinder-primitive B; two Dupin cyclides and a cylinder are then defined for each blending operation. In general the cylinder is not useful and may be reduced to a simple circle. A complete shape design example is presented to illustrate the modeling of Eurographics'04 Hugo using a limited number of quadrics combined using Dupin cyclide blends.

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

Pages (from-to) | 321-330 |

Number of pages | 10 |

Journal | Computer Graphics Forum |

Volume | 23 |

Issue number | 3 SPEC. ISS. |

DOIs | |

State | Published - Jan 1 2004 |

### ASJC Scopus subject areas

- Computer Networks and Communications
- Computer Graphics and Computer-Aided Design

### Cite this

*Computer Graphics Forum*,

*23*(3 SPEC. ISS.), 321-330. https://doi.org/10.1111/j.1467-8659.2004.00763.x

**Dupin cyclide blends between quadric surfaces for shape modeling.** / Foufou, Sebti; Garnier, Lionel.

Research output: Contribution to journal › Article

*Computer Graphics Forum*, vol. 23, no. 3 SPEC. ISS., pp. 321-330. https://doi.org/10.1111/j.1467-8659.2004.00763.x

}

TY - JOUR

T1 - Dupin cyclide blends between quadric surfaces for shape modeling

AU - Foufou, Sebti

AU - Garnier, Lionel

PY - 2004/1/1

Y1 - 2004/1/1

N2 - We introduce a novel method to define Dupin cyclide blends between quadric primitives. Dupin cyclides are non-spherical algebraic surfaces discovered by French mathematician Pierre-Charles Dupin at the beginning of the 19th century. As a Dupin cyclide can be fully characterized by its principal circles, we have focussed our study on how to determine principal circles tangent to both quadrics being blended. This ensures that the Dupin cyclide we are constructing constitutes a G1 blend. We use the Rational Quadratic Bélier Curve (RQBC) representation of circular arcs to model the principal circles, so the construction of each circle is reduced to the determination of the three control points of the RQBC representing the circle. In this work, we regard the blending of two quadric primitives A and B as two complementary blending operations: primitive A-cylinder and cylinder-primitive B; two Dupin cyclides and a cylinder are then defined for each blending operation. In general the cylinder is not useful and may be reduced to a simple circle. A complete shape design example is presented to illustrate the modeling of Eurographics'04 Hugo using a limited number of quadrics combined using Dupin cyclide blends.

AB - We introduce a novel method to define Dupin cyclide blends between quadric primitives. Dupin cyclides are non-spherical algebraic surfaces discovered by French mathematician Pierre-Charles Dupin at the beginning of the 19th century. As a Dupin cyclide can be fully characterized by its principal circles, we have focussed our study on how to determine principal circles tangent to both quadrics being blended. This ensures that the Dupin cyclide we are constructing constitutes a G1 blend. We use the Rational Quadratic Bélier Curve (RQBC) representation of circular arcs to model the principal circles, so the construction of each circle is reduced to the determination of the three control points of the RQBC representing the circle. In this work, we regard the blending of two quadric primitives A and B as two complementary blending operations: primitive A-cylinder and cylinder-primitive B; two Dupin cyclides and a cylinder are then defined for each blending operation. In general the cylinder is not useful and may be reduced to a simple circle. A complete shape design example is presented to illustrate the modeling of Eurographics'04 Hugo using a limited number of quadrics combined using Dupin cyclide blends.

UR - http://www.scopus.com/inward/record.url?scp=4644291263&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4644291263&partnerID=8YFLogxK

U2 - 10.1111/j.1467-8659.2004.00763.x

DO - 10.1111/j.1467-8659.2004.00763.x

M3 - Article

VL - 23

SP - 321

EP - 330

JO - Computer Graphics Forum

JF - Computer Graphics Forum

SN - 0167-7055

IS - 3 SPEC. ISS.

ER -