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We generalize the transfinite triangular interpolant of (Nielson, 1987) inorder to generate visually smooth (not necessarily polynomial) localinterpolating quasi-optimal triangular spline surfaces. Given as input atriangular mesh stored in a half-edge data structure, at first we produce alocal interpolating network of curves by optimizing quadratic energyfunctionals described along the arcs as weighted combinations of squared lengthvariations of first and higher order derivatives, then by optimizing weightedcombinations of first and higher order quadratic thin-plate-spline-likeenergies we locally interpolate each curvilinear face of the previous curvenetwork with triangular patches that are usually only $C^0$ continuous alongtheir common boundaries. In a following step, these local interpolating optimaltriangular surface patches are used to construct quasi-optimal continuousvector fields of averaged unit normals along the joints, and finally we extendthe $G^1$ continuous transfinite triangular interpolation scheme of (Nielson,1987) by imposing further optimality constraints concerning the isoparametriclines of those groups of three side-vertex interpolants that have to beconvexly blended in order to generate the final visually smooth localinterpolating quasi-optimal triangular spline surface. While we describe theproblem in a general context, we present examples in special polynomial,trigonometric, hyperbolic and algebraic-trigonometric vector spaces offunctions that may be useful both in computer-aided geometric design and incomputer graphics.

Nielson-type transfinite triangular interpolants by means of quadratic energy functional optimizations