Abstract
Given three regular space curves r 1 ( t ) , r 2 ( t ) , r 3 ( t ) for t ∈ [ 0 , 1 ] that define a curvilinear triangle, we consider the problem of constructing a triangular surface patch R ( u 1 , u 2 , u 3 ) 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.
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