Construction of Bézier surface patches with Bézier curves as geodesic boundaries

Rida Farouki (1) , Nicolas Szafran (2) , Luc Biard (2)

1 MAE - Department of Mechanical and Aeronautical Engineering
2 MGMI - Modélisation Géométrique & Multirésolution pour l'Image

Given four polynomial or rational Bézier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bézier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four Bézier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.

Mots clés: geodesic curves; surface reconstruction; Hermite/Coons interpolation; Bézier curves and surfaces

Domaines: Géométrie algorithmique [cs.CG]; Modélisation et simulation; Géométrie différentielle [math.DG]

Rida Farouki, Nicolas Szafran, Luc Biard. Construction of Bézier surface patches with Bézier curves as geodesic boundaries. Computer-Aided Design, 2009, 41 (11), pp.772-781. ⟨10.1016/j.cad.2009.02.019⟩. ⟨hal-00381037⟩