Jörg Peters
Abstract
Given a polyhedron, construct a new polyhedron by connecting every edge-midpoint to its four neighboring edge-midpoints. This refinement rule yields a C1 surface and the surface has a piecewise quadratic parametrozation except at a finite number of isolated points. We analyze and improve the construction.
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Index Terms
- The simplest subdivision scheme for smoothing polyhedra
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Reviews
Nickolas S. Sapidis
This short note deals with the smoothness of the surface produced when the following subdivision scheme is applied to a polyhedron: “connect every edge-midpoint to the four midpoints of the edges sharing a vertex and a face with the current edge.” Section 2 focuses on a regular mesh, defined to have the standard valence and number of edges. It is proven that, on a regular mesh, midedge subdivision converges to a surface parameterized by shifts of the 4-direction box spline. Unfortunately, the authors choose to offer no details on this particular surface, which is not well known even among surfacing specialists. Section 3 establishes smoothness of the limit surface at extraordinary points, using properties of the local subdivision matrix that maps points surrounding the extraordinary point to layers of new points. Indeed, the eigenanalysis of this matrix is satisfactorily detailed along with a proof of the main result: “the characteristic map is regular and injective.” A convergence analysis is presented in section 4, where a modified subdivision mask is discussed that preserves important properties and, at the same time, significantly improves convergence. Finally, examples demonstrating the advantages of the proposed subdivision technique are presented. I wish the authors had presented more details on the principal mathematical tools used here to further the usefulness of this short paper.
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