Construction and smoothing of triangular Coons patches with geodesic boundary curves

Given three regular space curves r1(t), r2(t) r3(t), t ∈ [ 0, 1 ] that define a curvilinear triangle, we consider the problem of constructing a C 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 [6] 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.