tailieunhanh - Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 10

Tham khảo tài liệu 'effective computational geometry for curves & surfaces - boissonnat & teillaud part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 5 Meshing of Surfaces 217 the topology of the surface does not change . The points at which we cut will be called slab points. These points include all x-critical points of the polar variety as well as all points where the projection of the polar variety on the x-y-plane intersects itself. The system of equations that characterize the x-critical points has been given in for two general directions d and d1. In our case d is the z-direction and d is the x-direction. Thus the critical points are given by the system fz fyz - fy fzz x y z 0 fz x y z 0 f x y z 0. This includes the x-critical points of the surface itself i. e. the points where x has a local extremum these points have a tangent plane perpendicular to the x-axis and a fortiori a vertical tangent line and therefore they lie on the silhouette. There are cases when the system does not have a zerodimensional solution set and therefore it cannot be used to define slab points. The example of Fig. below is an instance of this. In these cases one must modify the system to obtain a finite set of slab points as described in 263 327 . The points where the vertical projection of the polar variety onto the x-y-plane crosses itself are the points x y for which has more than one solution z. For a polynomial f these points can be found by computing the resultant of the polynomials in see Chap. 3 for details. A slab point x y of this type will be called a multiple slab point if more than two curves of the polar variety pass through the vertical line at x y without going through the same point in space. We make the following important nondegeneracy assumption There is a finite set of slab points there are no multiple slab points and no two slab points have the same x-coordinate. This assumption excludes for example a surface which consists of two equal spheres vertically above each other. The two silhouettes equators would coincide in the projection. It also excludes a torus with a horizontal axis or