tailieunhanh - Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 9

Tham khảo tài liệu 'effective computational geometry for curves & surfaces - boissonnat & teillaud part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 192 . Boissonnat D. Cohen-Steiner B. Mourrain G. Rote G. Vegter Global parameterizability of a curve in the parameter x means that the solution consists of a sequence of curves which can be written in the parameterized form y C x over sequence of disjoint intervals for the parameter x. Similarly global parameterizability of a surface in the parameters x and y means that the solution consists of parameterized surface patches z S x y over a set of disjoint domains for x y . Suppose that a curve in a two-dimensional rectangle X is globally parameterizable in x. The curve has at most one intersection with the left and right edge and an arbitrary number of intersections with the bottom and top edge. Let x1 x2 x3 . denote the sequence of intersections sorted from left to right see Fig. . Between the first two successive intersections x1 and x2 there can either be no solution inside X or the solution can be an x-monotone curve in X . These two possibilities can be distinguished by intersecting the curve with a vertical line segment I half-way between x1 and x2. More precisely we just need to compute the signs of f at the endpoints of I. Given this information we can draw polygonal connections between the points xi which are a topologically correct representation of the curve pieces inside X as in Fig. . To connect two points xi and xi i on the same edge we can for example draw two 45 segments. Points on different edges can be connected by straight lines. b Fig. . Finding a correct mesh for a curve in a square a The following lemma summarizes this procedure and it also formulates the three-dimensional version. Lemma 2. 1. If a curve f x y 0 is globally parameterizable in x in a two-dimensional box X and if one can find the zeros of f on the edges of the box then one can construct a topologically correct mesh for the curve inside X . 5 Meshing of Surfaces 193 2. If a surface f x y z 0 is globally parameterizable in x and y in a three-dimensional box X and if