Smooth polar caps for locally quad-dominant meshes

A general class of $C^1$ smooth rational splines: Application to construction of exact ellipses and ellipsoids

“…The 'shape' basis does not enjoy non-negativity and does not form a partition of unity, and extensions of it to higher smoothness leads to degrees of freedom that control non-intuitive shape parameters. Similar recent constructions for obtaining C 1 polar spline caps can be found in [11]. Curvature continuous polar NURBS surfaces were discussed in [21], and [20] presented a construction of polar caps using periodic B-spline surfaces with G n continuity for arbitrary n. On the CAE side, a standard circular serendipity-type element for IGA was proposed in [12], and C k smooth basis functions over singular parametrizations of triangular domains were constructed in [23].…”

“…In particular, with regard to the last two bullets above, we believe that the explicit, NURBS-compatible C 1 descriptions of ellipses and ellipsoids provided herein will be of use to geometric modellers [11] and computational scientists [24] alike. For instance, the exact C 1 (re)parameterizations at polar points may make the design of algorithms more stable and efficient; it may also avoid the need for special treatment of polar points.…”

Evolving Guide Subdivision

“…A Catmull‐Clark subdivision surface ‘cap’ can therefore be replaced by a EG subdivision cap near extraordinary points. We can therefore interpret regular subnets as uniform bi‐3 splines that smoothly join the surface caps generated by EG subdivision, see [LKP22]. A control‐net modeling session with EG subdivision then looks like a modeling session with Catmull‐Clark.…”

“…8b. Analogous to the construction for a c‐net, in [KP18, Appendix], we construct a map g , consisting of n G 1 ‐connected bi‐5 sectors with a unique quadratic expansion at the common point, as an affine combination of the nodes d and c 0 . Restricting g to the same subdomain as χ T therefore allows sampling the L‐shaped bi‐4 tensor‐borders from g ○ λχ T , see Fig.…”

“…Also in the spirit of [Sab91], [LFS16] prescribe part of the Taylor expansion to prevent shape defects where the knot‐spacing near the extraordinary point is highly unequal. Guided subdivision [KP18, KP19] provides a still more complete geometric Taylor expansion at the extraordinary point by making the subdivision surface follow an initially‐computed, hence static guide surface. The shape of guided subdivision is reportedly very good, but guided subdivision is complex to implement due to its two construction stages: first construct the guide surface and then the subdivision surface, as a series of surface extension operators.…”