A Class of C 2 Point-Normal Subdivision Surfaces and Their Exact Evaluation at Arbitrary Parameters

Abstract

This paper proposes techniques to construct and evaluate C 2 subdivision surfaces under control of quadrilateral meshes with unit normal vectors at the vertices. An improved C 2 modification of the Catmull-Clark subdivision scheme is first given by tuning the eigenvalues of subdivision matrices. Just like the existing C 2 modified Catmull-Clark subdivision, the improved scheme can also generate C 2 subdivision surfaces with vanishing curvature at the extraordinary points, but the flat regions near the extraordinary points have been significantly reduced. By independently applying the new modified subdivision scheme to control points and control normals of the control meshes, a class of C 2 point-normal (PN) subdivision surfaces is defined by updating the limit positions of subdivided points along the limit vectors of subdivided normals. Unlike their linear counterparts, PN subdivision surfaces can approximate control points with even higher-order accuracy while spheres and circular cylinders can be reproduced exactly. The curvatures at the extraordinary points of C 2 PN subdivision surfaces are generally non-vanishing, unless the surfaces reduce to linear subdivision surfaces. Like the classical Catmull-Clark subdivision surfaces, the C 2 modified Catmull-Clark subdivision surfaces and the C 2 PN Catmull-Clark subdivision surfaces can be evaluated exactly at arbitrary parameters. Numerical examples also demonstrate the effectiveness of the proposed techniques for modeling and evaluating curvature-continuous subdivision surfaces.