The Importance of Polynomial Reproduction in Piecewise-Uniform Subdivision

Adi Levin

September 2005


We survey a number of related methods, which have been published by the author and collaborators, in the field of subdivision schemes for curves and surfaces. The theory presented in these works relies mainly on the notion of polynomial reproduction, i.e. the ability of a scheme to reproduce all polynomials up to a certain degree as limit functions. We demonstrate that the study of polynomial reproduction is central to smoothness analysis and to approximation. In particular, we show how to exploit polynomial reproduction in the context of piecewise-uniform stationary subdivision. The applications include boundary treatments for subdivision surfaces, interpolation of curves by surfaces, subdivision stencils around extraordinary vertices (construction of C^2 schemes), as well as schemes that involve different kinds of grids (triangular / quadrilateral).

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BiBTeX entry

author = "Adi Levin",
title = "The Importance of Polynomial Reproduction in Piecewise-Uniform Subdivision",
year = "2005",
booktitle = "Mathematics of Surfaces XI",
editor = "Ralph Martin and Helmut Bez and Malcolm Sabin",
publisher = "Springer",
pages = "272-307")