Towards Subdivision Surfaces C2 Everywhere

Abstract

The conditions for subdivision surfaces which are piecewise polynomial in the regular region to have continuity higher than C1 were identified by Reif [7]. The conditions are ugly and although schemes have been identified and implemented which satisfy them, those schemes have not proved satisfactory from other points of view. This paper explores what can be created using schemes which are not piecewise polynomial in the regular regions, and the picture looks much rosier. The key ideas are (i) use of quasi-interpolation (ii) local evaluation of coefficients in the irregular context. A new method for determining lower bounds on the Hölder continuity of the limit surface is also proposed.

Appendices

Annex 1: Computation of Stencil Nearest to a Regular Stencil

In the topologically regular but geometrically irregular case, I suggest that the nearest solution to the regular one should be chosen. This gives some measure of continuity with respect to changing layouts in the abscissa plane. The metric for ‘nearest’ might be chosen by more sophisticated arguments later³, but for the moment I just use the euclidean distance in coefficient space.

Call the number of coefficients c and the number of quasi-interpolation conditions n.

The quasi-interpolating conditions define a linear subspace of dimension $c-n$ in the space of dimension $c-1$ of sets of coefficients: nearness to the regular coefficients defines a complementary subspace orthogonal to it, and the solution can be found in that subspace by solving a linear system of size $n\times n$.

Let the set of coefficients be $a_i,\ i\in 1..c$, and let the quasi-interpolation conditions be

$$\begin{aligned} \forall j\in 1..n\quad \varSigma _i a_if_j(v_i) = 0 \end{aligned}$$

If the regular coefficients are $\bar{a}_i$, then we can set up the system

$$\forall i\ a_i = \bar{a}_i + \varSigma _{k\in 1..n}\,\delta _k f_k(v_i)$$

and solve for the $\delta _j$.

$$\begin{aligned} \forall j\in 1..n\quad 0= & {} \varSigma _i a_if_j(v_i) \\= & {} \varSigma _i\, \left( \bar{a}_i + \varSigma _{k}\,\delta _k f_k(v_i)\right) f_j(v_i) \\= & {} \varSigma _i\, \bar{a}_if_j(v_i) + \varSigma _i \,\varSigma _{k}\,\delta _k f_k(v_i)f_j(v_i) \\= & {} \varSigma _i\, \bar{a}_if_j(v_i) + \varSigma _{k}\,\varSigma _i\,\delta _k f_k(v_i)f_j(v_i) \\= & {} \varSigma _i\, \bar{a}_if_j(v_i) + \varSigma _{k}\,\delta _k\varSigma _i\, f_k(v_i)f_j(v_i) \\\left[ \begin{matrix}\ddots &{}&{}\\ &{}\varSigma _i\, f_k(v_i)f_j(v_i)&{}\\ &{}&{}\ddots \end{matrix}\right] \left[ \begin{matrix}\vdots \\ \delta _k\\ \vdots \\ \end{matrix}\right]= & {} -\left[ \begin{matrix}\vdots \\ \varSigma _i\, \bar{a}_if_j(v_i)\\ \vdots \\ \end{matrix}\right] \\\end{aligned}$$

The actual coefficients can then be determined from these $\delta _j$ values.

$$\begin{aligned} \forall i\ a_i:= & {} \bar{a}_i + \varSigma _{k}\,\delta _k f_k(v_i) \end{aligned}$$

The rate of convergence with distance from the EV in an interesting natural configuration can be measured by how $\varSigma _j \delta _j^2$ varies with distance.

Annex 2: Convergence for Conformal Characteristic Map

Table 7. Valency = 5, reproduction degree = 2

Table 8. Valency = 5, reproduction degree = 3

Table 9. Valency = 7, reproduction degree = 2

Table 10. Valency = 7, reproduction degree = 3

These figures in Tables 7, 8, 9 and 10 ($E1=\varSigma _i \delta _i^2$ and $E2=max_i |\delta _i|$) are disappointing in that the convergence is so slow. A conjecture that the rate of convergence would be $d^{-6}$ has been soundly disproved by these calculations.