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.
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Notes
- 1.
K here is normalised to sum to 1: the kernel is thus 4K. This factor of 4 is analogous to the factor of 2 in the expression for the univariate scheme mentioned above.
- 2.
In the case of schemes having multiple kinds of e- and f-vertices (for example, that of [9] or of higher arity) each kind will need individual calculations of this type.
- 3.
For example, to get better continuity when the set of support points needs to change because of changes in the set of local neighbours.
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Acknowledgements
My thanks go to a very diligent referee who bothered to construct a counterexample disproving a conjecture in my first draft of this paper and also pointed out many places where the original text was not clear. I also thank colleagues: Cedric Gerot for helping me understand that counterexample, Leif Kobbelt and Ulrich Reif for prompt replies to my emails requesting clarifications, and to Ioannis Ivrissimtzis for bringing the moving least squares ideas to my attention.
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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 laterFootnote 3, 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
If the regular coefficients are \(\bar{a}_i\), then we can set up the system
and solve for the \(\delta _j\).
The actual coefficients can then be determined from these \(\delta _j\) values.
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
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.
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Sabin, M. (2017). Towards Subdivision Surfaces C2 Everywhere. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2016. Lecture Notes in Computer Science(), vol 10521. Springer, Cham. https://doi.org/10.1007/978-3-319-67885-6_11
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