By Jean-Marie Morvan
The cause of this e-book is to set the trendy foundations of the speculation of generalized curvature measures. This topic has an extended background, starting with J. Steiner (1850), H. Weyl (1939), H. Federer (1959), P. Wintgen (1982), and maintains this present day with younger and excellent mathematicians. within the final a long time, a renewal of curiosity in arithmetic in addition to laptop technology has arisen (finding new purposes in special effects, scientific imaging, computational geometry, visualization …).
Following a ancient and didactic process, the e-book introduces the mathematical history of the topic, starting with curves and surfaces, occurring with convex subsets, soft submanifolds, subsets of confident achieve, polyhedra and triangulations, and finishing with floor reconstruction. We concentrate on the speculation of ordinary cycle, which permits to compute and approximate curvature measures of a big category of delicate or discrete gadgets of the Euclidean area. We supply particular computations while the article is a 2 or three dimensional polyhedron.
This ebook can function a textbook to any mathematician or desktop scientist, engineer or researcher who's drawn to the speculation of curvature measures.
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Additional info for Generalized Curvatures
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 .
Generalized Curvatures by Jean-Marie Morvan