# COONS TRIANGULAR BÉZIER SURFACES

### A. Arnal, A. Lluch

#### Abstract

In this paper we give some different surface generation methods starting out from prescribed boundary curves. If the boundary control points are known it is natural to think of Coons patches, a popular solution of the problem of finding a surface given its boundary curves. We have developed three methods to generate triangular patches given the boundary curves. First we give a discrete version of the triangular Coons patch. A second method lets us to find the extremals of a functional as a solution of a linear system of the control points. That functional is the one that minimizes the Coons patch. The third method makes it possible to build a Bézier triangle by means of a mask deduced from the characterization of cubical extremals of the functional.

#### References

- Arnal, A., Lluch, A., and Monterde, J. (2003). Triangular bézier surfaces of minimal area. In ICCSA (3), volume 2669 of Lecture Notes in Computer Science, pages 366-375. Springer.
- Arnal, A., Lluch, A., and Monterde, J. (2008). Triangular bézier approximations to constant mean curvature surfaces. In ICCS (2), volume 5102 of Lecture Notes in Computer Science, pages 96-105. Springer.
- Coons, S. A. (1967). Surfaces for computer aided design of space forms. Technical Report MAC-TR-41, MIT.
- Farin, G. and Hansford, D. (1999). Discrete Coons patches. Comput. Aided Geom. Design, 16(7):691-700. Dedicated to Paul de Faget de Casteljau.
- Monterde, J. (2003). The plateau-bézier problem. In IMA Conference on the Mathematics of Surfaces, volume 2768 of Lecture Notes in Computer Science, pages 262-273. Springer.
- Monterde, J. (2004). Bézier surfaces of minimal area: the Dirichlet approach. Comput. Aided Geom. Design, 21(2):117-136.
- Monterde, J. and Ugail, H. (2004). On harmonic and biharmonic Bézier surfaces. Comput. Aided Geom. Design, 21(7):697-715.
- Monterde, J. and Ugail, H. (2006). A general 4th-order pde method to generate bézier surfaces from the boundary. Computer Aided Geometric Design, 23(2):208-225.
- Nielson, G. M., Thomas, D., and Wixom, J. (1978). Boundary data interpolation on triangular domains. Technical Report GMR-2834, General Motors Research Laboratories.

#### Paper Citation

#### in Harvard Style

Arnal A. and Lluch A. (2010). **COONS TRIANGULAR BÉZIER SURFACES** . In *Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)* ISBN 978-989-674-026-9, pages 148-153. DOI: 10.5220/0002848601480153

#### in Bibtex Style

@conference{grapp10,

author={A. Arnal and A. Lluch},

title={COONS TRIANGULAR BÉZIER SURFACES},

booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)},

year={2010},

pages={148-153},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002848601480153},

isbn={978-989-674-026-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)

TI - COONS TRIANGULAR BÉZIER SURFACES

SN - 978-989-674-026-9

AU - Arnal A.

AU - Lluch A.

PY - 2010

SP - 148

EP - 153

DO - 10.5220/0002848601480153