A general 4th-order PDE method to generate Bézier surfaces from the boundary. (English) Zbl 1083.65014

Summary: We present a method for generating Bézier surfaces from the boundary information based on a general 4th-order partial differential equations (PDE). This is a generalisation of our previous work on harmonic and biharmonic Bézier surfaces whereby we studied the Bézier solutions for Laplace and the standard biharmonic equation, respectively.
Here we study the Bézier solutions of the Euler-Lagrange equation associated with the most general quadratic functional. We show that there is a large class of fourth-order operators for which Bézier solutions exist and hence we show that such operators can be utilised to generate Bézier surfaces from the boundary information. As part of this work we present a general method for generating these Bézier surfaces. Furthermore, we show that some of the existing techniques for boundary based surface design, such as Coons patches and Bloor-Wilson PDE method, are indeed particular cases of the generalised framework we present here.


65D17 Computer-aided design (modeling of curves and surfaces)
Full Text: DOI


[1] Arnal, A., 2006. Generation of triangular Bézier surfaces. Doctoral thesis, Univ. Jaume I, Castelló, Spain, in preparation
[2] Bloor, M.I.G.; Wilson, M.J., An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch, Computer aided geometric design, 22, 3, 203-219, (2005) · Zbl 1205.65069
[3] Botsch, M.; Kobbelt, L., An intuitive framework for real-time freeform modeling, ACM trans. graph., 23, 3, 630-634, (2004)
[4] Evans, D.J., A recursive algorithm for determining the eigenvalues of a quindiagonal matrix, Computer J., 18, 1, 70-73, (1973) · Zbl 0296.65015
[5] Farin, G.; Hansford, D., Discrete coons patches, Computer aided geometric design, 16, 691-700, (1999) · Zbl 0997.65033
[6] Kang, H.; Kak, A., Deforming virtual objects interactively in accordance with an elastic model, Computer-aided design, 28, 4, 251-262, (1996)
[7] Light, R.; Gossard, D., Modification of geometric models through variational geometry, Computer-aided design, 14, 4, 209-214, (1982)
[8] Monterde, J., Bézier surfaces of minimal area: the Dirichlet approach, Computer aided geometric design, 21, 117-136, (2004) · Zbl 1069.65559
[9] Monterde, J.; Ugail, H., On harmonic and biharmonic Bézier surfaces, Computer aided geometric design, 21, 697-715, (2004) · Zbl 1069.65526
[10] Olver, P., Applications of Lie groups to differential equations, Graduate texts in mathematics, vol. 107, (1986), Springer-Verlag New York
[11] Sederberg, T.W.; Cardon, D.L.; Finnigan, G.T.; North, N.S.; Zheng, J.; Lyche, T., T-spline simplification and local refinement, ACM trans. graph., 23, 3, 276-283, (2004)
[12] Schneider, R.; Kobbelt, L., Geometric fairing of irregular meshes for free-form surface design, Computer aided geometric design, 18, 4, 359-379, (2001) · Zbl 0969.68154
[13] Sweet, R.A., A recursive relation for the determinant of a pentadiagonal matrix, Comm. ACM, 12, 6, 330-332, (1969) · Zbl 0182.48903
[14] Welch, W.; Witkin, A., Variational surface modeling, Computer graphics, 26, 157-166, (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.