By Jean-Marie Morvan

ISBN-10: 354073791X

ISBN-13: 9783540737919

ISBN-10: 3540737928

ISBN-13: 9783540737926

The purpose of this ebook is to set the fashionable foundations of the idea of generalized curvature measures. This topic has a protracted historical past, starting with J. Steiner (1850), H. Weyl (1939), H. Federer (1959), P. Wintgen (1982), and maintains this present day with younger and impressive mathematicians. within the final a long time, a renewal of curiosity in arithmetic in addition to machine technological know-how has arisen (finding new functions in special effects, scientific imaging, computational geometry, visualization …). Following a old and didactic procedure, the booklet introduces the mathematical historical past of the topic, starting with curves and surfaces, occurring with convex subsets, delicate submanifolds, subsets of optimistic achieve, polyhedra and triangulations, and finishing with floor reconstruction. We specialise in the speculation of ordinary cycle, which permits to compute and approximate curvature measures of a big type of delicate or discrete items of the Euclidean area. We supply specific computations whilst the thing is a 2 or three dimensional polyhedron. This publication 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 (Geometry and Computing)**

**Sample text**

It may be tempting to get a better approximation by using an inscribed conic instead of segments. 4 Curves in E 2 23 p1 p2 α p Fig. 11 The pointwise curvature of a smooth curve approximated by the angle • • • • Consider four points q, r, s,t close to p. Construct the conic C on p, q, r, s,t. Compute the exact value of the curvature κ p of C at p. Approximate the curvature k p of γ at p by κ p . These steps are not valid in general, as the following example3 shows. A counterexample. 8) where g:R→R is a smooth function such that g(x) = 1 2n 1 1 on the segment [ 2n+1 , 2n ].

Let A be a subset of EN and a ∈ A. We denote by reach(A, a) the supremum of real numbers r such that every point of the ball of radius r and center a has a unique projection onto A (realizing the distance dA (m) from m to A): reach(A, a) = sup{r ∈ R for which {m : d(m, a) < r} ⊂ UA }. 2. The reach of A is the real number reach(A) = inf reach(A, a). a∈A In other words, reach(A) is the supremum of real numbers r ≥ 0 such that the orthogonal projection onto A is defined in the tubular neighborhood of radius r.

Theorem 6. The rotation number of a simple closed polygon equals ±1 (the sign depending on the orientation). The reader can consult [11, 40] for details. 5 Signed Curvature and Topology It is important to remark that the rotation number I of a closed plane curve is invariant under isotopy. Consequently, the integral of the signed curvature κ of smooth or polygonal closed plane curves is a topological invariant. On the other hand, the unsigned curvature k is not a topological invariant: an isotopy of a closed plane curve changes its total curvature (Fig.

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