Mathematical Aspects of Geometric Modeling by Charles A. Micchelli

By Charles A. Micchelli

This monograph examines intimately yes techniques which are worthwhile for the modeling of curves and surfaces and emphasizes the mathematical concept that underlies those principles. the 2 vital issues of the textual content are using piecewise polynomial illustration (this topic appears to be like in a single shape or one other in each chapter), and iterative refinement, often known as subdivision. the following, uncomplicated iterative geometric algorithms produce, within the restrict, curves with advanced analytic constitution. within the first 3 chapters, the de Casteljau subdivision for Bernstein-Bezier curves is used to introduce matrix subdivision, and the Lane-Riesenfield set of rules for computing cardinal splines is tied into desk bound subdivision. This finally results in the development of prewavelets of compact aid. the rest of the ebook offers with innovations of "visual smoothness" of curves, in addition to the interesting notion of producing delicate multivariate piecewise polynomials as volumes of "slices" of polyhedra.

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2 fc+1 — MATRIX SUBDIVISION FIG. 7. 62). then ti and *2 are both in the interval [d*+i, d^), while if t2 - d1- < 2~ fc ~ 1 then ti and £2 are in [d£+1, d^+j). Now, suppose dj = . e i - - - £ f e for some £ . \ , . . ,  k 6 {0,1} and p = # {r : er = 0, r = 1 , . . , k}. 67) is positive. 62) and establishes that gi is decreasing and g? is increasing. Our interest in the curve g(-|x) : [0,1] —> R 2 , comes from its usefulness in identifying the limit of the following variation of de Casteljau subdivision.

49) persists for all t 6 (0,1] by showing, inductively on r = 1 , 2 , . . , that it holds on the interval [p$r, l] • Suppose we have established it for t in the range p$r < t < 1. Consider values of t in the next interval Par~l < t < pQT. Since in this range p$r < pot, we have by (1-48) and our induction hypothesis that because we chose // so that pp$ — 1. Although it is not important for us here, we remark that it is easy to see when ^ £ (0,1] that there is a positive constant 61 such that for t e [0,1], f ( t ) > 6\t^.

Moreover, x£ has nonnegative components. Proof. 9 to the matrices Ae, e G {0,1}, we conclude that (Ae)a > 0, i — 1,... ,n. Now choose 1 < j < i < n and observe that the inequality implies that (Ao)ij(Ai)ji > 0. In particular, the first column of AQ and the last column of A\ have only positive entries. The remaining conclusions follow from MATRIX SUBDIVISION 45 this fact. To this end, fix any x 6 Kra and e 6 {0,1}, and consider the sequence of vectors xfc := A^x, k = 0 , 1 , . . Set mk := ruin {xf : I < i < n} and Mk := max {xf : 1 < z < n} and observe that, sinceAf is a stochastic matrix.

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