Trimming analytic functions using right sided Poisson subdivision Ge Âraldine Morin, Ron Goldman * Computer Science Department, Rice University, 6100 Main Street, Houston, TX 77005-1892, USA Accepted 15 January 2001 Abstract Just as the Bernstein basis and Be Âzier points give us tools to manipulate polynomials, the Poisson basis and control points provide a new framework for investigating analytic functions. Using the Poisson representation, two different subdivision algorithms for analytic functions can be derived as extensions of the Be Âzier polynomial subdivision algorithm. The ®rst procedure generates an approximation algorithm, the second a trimming algorithm. Three applications of these two subdivision algorithms are developed here: an evaluation algorithm, a method for extending the convergence domain of the Poisson representation of an analytic function, and an intersection algorithm for analytic curves. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Analytic function; Poisson distribution; Subdivision; Approximation; Trimming; Intersection 1. Introduction The blending functions for Be Âzier curves and surfaces are provided by the binomial distribution. Although the Poisson distribution has long been known in probability theory to be a limiting case of the binomial distribution [1], in geometric modeling the idea of considering limiting cases of the poly- nomial setting only appeared later on [2,5,12]. In approx- imation theory, however, the Poisson distribution has already been used in the Szasz±Mirakian operator [8]. Analytic functions provide a richer collection of model- ing tools than polynomials. In Computer Aided Geometric Design, the Bernstein basis is acknowledged to be more suitable for representing polynomials than the monomial basis. Analogously, we shall ®nd it more natural to use a Poisson series to describe analytic functions, rather than the usual Taylor expansion. The generalization from polyno- mials in Bernstein±Be Âzier form to analytic functions in the Poisson representation is quite natural and extensive [7], but convergence issues speci®c to the Poisson represen- tation need to be addressed. Moreover, to provide useful algorithms, we need to consider ®nite procedures. Based on the Poisson representation of an analytic func- tion, we shall present two subdivision algorithms for analy- tic curves. These algorithms are limiting cases of the Be Âzier subdivision algorithm for polynomial curves [4]. The Pois- son representation and these associated subdivision algo- rithms provide new tools for representing and manipulating analytic functions intuitively and ef®ciently. The ®rst part of this paper establishes the theoretical foun- dations and addresses convergence issues. The second part concentrates on several applications of these tools: repre- senting an analytic function both inside and outside the convergence domain of its Taylor series, evaluating an analytic function or its derivative at a point, and, ®nally, intersecting analytic curves. The Poisson representation has already been exploited in [12], where the Poisson representation for analytic functions is introduced along with a subdivision algorithm that generates a rendering procedure for analytic functions inside the domain of convergence of their Taylor series. Here, we shall begin by recalling this work, although our de®nition of a Poisson function differs slightly. We shall then present a new subdivision procedure, together with corresponding analysis algorithms. In the last part of this paper, we shall develop some applications of these new procedures. 2. The Poisson representation 2.1. The Poisson basis The Poisson basis is an in®nite collection of functions that we shall denote by b k t k$0 . If B n k t denotes the k-th Bern- stein polynomial of degree n, then: b k t lim n!1 B n k t n   ; k $ 0: 1 Computer-Aided Design 33 2001) 813±824 COMPUTER-AIDED DESIGN 0010-4485/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0010-448501)00099-9 www.elsevier.com/locate/cad * Corresponding author. E-mail address: rug@cs.rice.edu R. Goldman).