A subdivision scheme for Poisson curves and surfaces

• Published in: Computer aided geometric design Vol. 17; no. 9; pp. 813 - 833
• Main Authors: Morin, Géraldine, Goldman, Ron
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.10.2000; Elsevier
• Subjects: Analytic function; Applied sciences; Artificial intelligence; Bézier curve; Computer aided design; Computer science; control theory; systems; de Casteljau algorithm; Exact sciences and technology; Pattern recognition. Digital image processing. Computational geometry; Poisson distribution; Software; Stationary subdivision; Analytic function; Stationary subdivision; Bézier curve; de Casteljau algorithm; Poisson distribution; Polyhedron; Geometrical shape; Blending; Tensor product; Analytical function; Control point; Stationary condition
• ISSN: 0167-8396; 1879-2332
• DOI: 10.1016/S0167-8396(00)00028-5

The de Casteljau evaluation algorithm applied to a finite sequence of control points defines a Bézier curve. This evaluation procedure also generates a subdivision algorithm and the limit of the subdivision process is this same Bézier curve. Extending the de Casteljau subdivision algorithm to an infinite sequence of control points defines a new family of curves. Here, limits of this stationary non-uniform subdivision process are shown to be equivalent to curves whose control points are the original data points and whose blending functions are given by the Poisson distribution. Thus this approach generalizes standard subdivision techniques from polynomials to arbitrary analytic functions. Extensions of this new subdivision scheme from curves to tensor product surfaces are also discussed.