Improved shape for refinable surfaces with singularly parameterized irregularities

Abstract

To date, singularly-parameterized surface constructions suffer from poor highlight line distributions, ruling them out as a surface representation of choice for primary design surfaces. This paper explores graded, many-piece, everywhere C1 singularly-parameterized surface caps that mimic the shape of a high-quality guide surface. The approach illustrates the trade-off between polynomial degree and surface quality. For bi-degree 5, minor flaws in the highlight line distribution are still visible when zooming in on the singularity, but the distribution is good at the macroscopic level. Constructions of degree bi-4 or bi-3 may require one or more steps of guided subdivision to reach the same macroscopic quality. Akin to subdivision surfaces, singularly-parameterized functions on the surfaces are straightforward to refine.

G1 – Smooth biquintic approximation of Catmull-Clark subdivision surfaces

“…• , δ even , δ even ] T = 1 10 −M 0,0 + 5M 1,0 + 10M 3,0 − 5M 4,0 + M 5,0 . Checking property (21) for the masks M 0,0 , M 1,0 and M even 2,0 , we notice that Y T M 0,0 = 0, Y T M 1,0 = 0 but Y T M even 2,0 = 0. Thus, we need to compute a new mask M 2,0 = [δ • , δ, δ ] T ensuring (21) in order to have a solution for system (17).…”

“…Differently from that work, we provide a complete and explicit construction for the surface without solving any global system. In [21] a guided subdivision approach is developed to construct a bi-3, bi-4 or bi-5 C 1 Bézier surface. This approach, in which a face split is carried out, is intended to obtain a final surface with regular isophotes across extraordinary vertices; to do that, free parameters are introduced and fixed using minimization of a functional to obtain a better result, whereas we provide explicit mask formulas for the construction of the patches.…”

Splines for Meshes with Irregularities

“…However subdivision schemes governed by simple stencils do not meet the "class A" requirements. Detailed analysis as well as test cases in [58,50] have exposed major shape deficiencies. Figure 4.6 shows three types of flaws: (A) saddles arising in the limit from convex input nets for valence above n = 4, (B) unwanted flatness due to neighboring extraordinary points and (C) pinched highlight line distributions for higher-order saddles.…”

Multisided generalisations of Gregory patches

“…Either the domain is lled up by separate parametric triangular or quadrilateral patches (see e.g. [22,11,15]), or a single patch is used to cover the whole domain. Our approach uses the single-patch strategy.…”