By John L. Morris

ISBN-10: 3540056564

ISBN-13: 9783540056560

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7 We assume here that S has no boundary, but an equivalent result is valid for general oriented surfaces (with or without boundary). 44 3 Motivation: Surfaces Fig. 20 The area of the 2 smooth surface S of E is approximated by the area of the polyhedron P (or Q) S C Q P By analogy with the smooth case, it is then natural to define the (global) Gauss curvature of P by setting K(P) = v ∑ αv . 15) vertices in P This point of view will be generalized to polyhedra of any dimension in Chap. 6. In our context (of approximation), these fundamental results appear as negative ones.

The local feature size lfs(a) at a point a ∈ A is the distance of a from the medial axis of A. 1 There is no uniform definition of these notions in the literature. We give here the simplest one. 56 4 Distance and Projection pr (x) Med(A) A pr (y) x y Fig. 13 Any point on the medial axis has at least two orthogonal projections. One has lfs(x) = |x − pr(x)|, lfs(y) = |y − pr(y)| In other words, using the notion of reach introduced above, one has lfs(a) = reach(A, a). Note that if A is finite, the medial axis of A is nothing but the boundary of the Voronoi regions associated to A (Fig.

15 Here is the intersection of a hyperboloid (of negative Gauss curvature) and an ellipsoid (of positive Gauss curvature). This intersection is the union of two curves. By constructing a sequence of triangulations inscribed in these curves and tending to one of their intersection point, (Tn )n∈N , one sees that any sequence of number ζn can converge to both the Gauss curvature of the hyperboloid and the Gauss curvature of the ellipsoid. A. project team Vegas e2 G(p) TpM ξ p. e 1 ξp M S2 G Fig. 1 The Gauss Map of a Smooth Surface Let S be an oriented (smooth) surface of E3 .

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Conference on Applications of Numerical Analysis by John L. Morris

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