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

TL;DR

This paper extends Nielson's transfinite triangular interpolants to generate smooth, quasi-optimal triangular spline surfaces using energy functional optimization, applicable in geometric design and graphics.

Contribution

It introduces a generalized method for constructing smooth triangular spline surfaces through energy optimization, including polynomial, trigonometric, and hyperbolic function spaces.

Findings

Produces visually smooth surfaces with $C^0$ continuity along patches.

Uses quadratic energy functionals for local curve and surface optimization.

Extends Nielson's scheme to include optimality constraints for better smoothness.

Abstract

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