Polyhedral design with blended $n$-sided interpolants

TL;DR

A new parametric surface representation is proposed that interpolates the vertices of a given closed mesh of arbitrary topology by blending local, multi-sided quadratic interpolants.

Abstract

A new parametric surface representation is proposed that interpolates the vertices of a given closed mesh of arbitrary topology. Smoothly connecting quadrilateral patches are created by blending local, multi-sided quadratic interpolants. In the non-four-sided case, this requires a special parameterization technique involving rational curves. Appropriate handling of triangular subpatches and alternative subpatch representations are also discussed.

Figures (7)

• Figure 1: Overlapping quadratic nets around a quad.
• Figure 2: Blending the local interpolants. Dots indicate a point of evaluation in each; the last image shows the resulting patch.
• Figure 3: Quad $\rightarrow$ interpolant mappings.
• Figure 4: Trebol model (left: cage, center: isophotes, right: mean curvature).
• Figure 5: Torus model (left: cage, right: isophote map).