Bicubic polar subdivision

Abstract

We describe and analyze a subdivision scheme that generalizes bicubic spline subdivision to control nets with polar structure. Such control nets appear naturally for surfaces with the combinatorial structure of objects of revolution and at points of high valence in subdivision meshes. The resulting surfaces are C 2 except at a finite number of isolated points where the surface is C 1 and the curvature is bounded.

Pairs of bi-cubic surface constructions supporting polar connectivity

“…The analysis in (Karčiauskas and Peters, 2007) shows that, for n ≥ 6, this radial subdivision results in an everywhere C 2 surface except at the central limit point. At the central point the surface is C 1 with bounded curvature.…”

“…All four Table 1 Bi-cubic surface constructions: (Catmull and Clark, 1978), (Karčiauskas and Peters, 2007), (Peters, 2000) (See also Figure 11). …”

“…Trying to suppress such macroscopic ripples arising in the first subdivision step motivated the global shape optimization (Halstead et al, 1993). In (Karčiauskas et al, 2006), it was argued that both ripple and saddle problems can be more simply resolved by switching to a polar layout of facets ( Figure 1). A neighborhood of a central high-valence vertex ( Figure 2) has polar layout if the vertex is surrounded by one layer of triangles while the remaining mesh consists of quadrilaterals, with always four facets joining at each non-central vertex.…”

“…Polar meshes can also be refined using the more general tool of Quad-tri subdivision (Stam and Loop, 2003;Schaefer and Warren, 2005); but BP subdivision (Karčiauskas and Peters, 2007) is preferable for polar layout because it generates a finite number of bi-cubic patches in the transition from a quad to a triangle facet while Quad-tri subdivision creates infinitely many patches, half of them three-sided. Also for the polar layout, Jet subdivision (Karčiauskas et al, 2006) generates C 2 surfaces with good curvature distribution. But the Jet subdivision is more complex and results in surfaces of degree (6,5); and are therefore outside of our focus on bi-cubic schemes.…”

“…While step (a) preserves the valence, simplifying implementation, applying step (b) a posteriori amounts to knot insertion that does not change the surface. Therefore the analysis of (Karčiauskas and Peters, 2007) applies. By contrast, the equally obvious approach of alternating radial and circular subdivision creates local curvature fluctuations and requires a new analysis.…”