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International audience; Given three regular space curves r1(t), r2(t), r3(t) for t in [0,1] that define a curvilinear triangle, we consider the problem of constructing a triangular surface patch R(u1,u2,u3) bounded by these three curves, such that they are geodesics of the constructed surface. Results from a prior study (Farouki et al., 2009a) concerned with tensor-product patches are adapted to identify constraints on the given curves for the existence of such geodesic-bounded triangular surface patches. For curves satisfying these conditions, the patch is constructed by means of a cubically-blended triangular Coons interpolation scheme. A formulation of thin-plate spline energy in terms of barycentric coordinates with respect to a general domain triangle is also derived, and used to optimize the smoothness of the geodesic-bounded triangular surface patches.

International audience; 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.