tailieunhanh - Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 8

Tham khảo tài liệu 'effective computational geometry for curves & surfaces - boissonnat & teillaud part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 166 D. Cohen-Steiner J-M. Morvan An Approximation Theorem We shall now compare the curvatures of a smooth surface M at a point p with the angular defect of a polyhedron P inscribed in M having this point p as an interior vertex. Our problem is local around p so we consider the set of triangles incident to p which we call the one ring of P . A normal section through p is the intersection of M with a plane containing p and the normal vector to p. Definition 7. Let p be a point of a smooth surface M and letpi i 1 . . n be its neighbors. Point p is called a regular vertex if its neighbors lie in normal sections two consecutive of which form an angle of h n 2n n for all i the distance from p to pi is a constant p. Definition 8. Let vmax and vmin be the principal directions at p. The offset angle a is defined as the angle between the directions vmax and pn pi where n pi is the orthogonal projection of pi onto the tangent plane. In 68 the following is proved Theorem 3. Consider a regular vertex p of valence n. If Yi denotes the angle between directions ppi and ppi 1 There exist two functions A a n and B a n such that 2n - Yi A a n G p B a n km ax km nW o n2 . i 1 The only value of n such that the functions A a n and B a n depend upon a is n 4 and then 2n- 2 Yi 1-2 cos2 a sin2 a G p cos2 a sin2 a km kmnW oW . i 1 If n 4 n En rz 4n 2n t_ff 16sin2n d 2 - cos n - cos n G p I1 1 cos 47 - 3 cos m A ax AM 2 o n2 . 2 n 2 n In particular the only value of n such that B a n 0 is n 6 and then A a 6 Ự3 2 that is 2n - Yi À3G p n2 o n2 . i 1 4 Differential Geometry on Discrete Surfaces 167 Remark that the principal curvatures can be estimated from two different meshes of valences n1 and n2 such that n1 4 n2 4 n1 n2 by solving a system of equations deduced from Theorem 3. Finally the previous result shows that in general at a point p of a smooth surface endowed with a triangulated polyhedron the defect angle G p is not a good approximation of the pointwise Gauss curvature at p. The .