Geometry and Computing

By: Jean-Marie Morvan

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Hardcover | 13 June 2008

At a Glance

Hardcover
280 Pages

Dimensions(cm)
23.5 x 15.88 x 1.27

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The central object of this book is the measure of geometric quantities describing N a subset of the Euclidean space (E ,), endowed with its standard scalar product. Let us state precisely what we mean by a geometric quantity. Consider a subset N S of points of the N-dimensional Euclidean space E , endowed with its standard N scalar product. LetG be the group of rigid motions of E . We say that a 0 quantity Q(S) associated toS is geometric with respect toG if the corresponding 0 quantity Q[g(S)] associated to g(S) equals Q(S), for all g?G . For instance, the 0 diameter ofS and the area of the convex hull ofS are quantities geometric with respect toG . But the distance from the origin O to the closest point ofS is not, 0 since it is not invariant under translations ofS. It is important to point out that the property of being geometric depends on the chosen group. For instance, ifG is the 1 N group of projective transformations of E , then the property ofS being a circle is geometric forG but not forG , while the property of being a conic or a straight 0 1 line is geometric for bothG andG . This point of view may be generalized to any 0 1 subsetS of any vector space E endowed with a groupG acting on it.

Industry Reviews

From the reviews:

"This book is a welcome addition to the literature in differential geometry. The main aim of this book is the measure of geometric quantities describing a subset of the Euclidean space ... endowed with its standard scalar product. ... The book contains 107 figures and the bibliography contains about 89 entries. The book covers an active, interesting and fresh research area. It is very useful for researchers in differential geometry and related subjects." (Kazim Ilarslan, Zentralblatt MATH, Vol. 1149, 2008)

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You Can Find This Book In

Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Computing & I.T.Graphical & Digital Media Applications Digital Animation Calculus & Mathematical Analysis Numerical Analysis Computer Science Artificial Intelligence Computer Vision

Image Processing

Introduction	p. 1
Motivations
Motivation: Curves	p. 13
Motivation: Surfaces	p. 29
Background: Metrics and Measures
Distance and Projection	p. 47
Elements of Measure Theory	p. 57
Background: Polyhedra and Convex Subsets
Polyhedra	p. 71
Convex Subsets	p. 77
Background: Classical Tools in Differential Geometry
Differential Forms and Densities on E[superscript N]	p. 91
Measures on Manifolds	p. 97
Background on Riemannian Geometry	p. 101
Riemannian Submanifolds	p. 109
Currents	p. 121
On Volume
Approximation of the Volume	p. 129
Approximation of the Length of Curves	p. 139
Approximation of the Area of Surfaces	p. 143
The Steiner Formula
The Steiner Formula for Convex Subsets	p. 153
Tubes Formula	p. 165
Subsets of Positive Reach	p. 177
The Theory of Normal Cycles
Invariant Forms	p. 189
The Normal Cycle	p. 193
Curvature Measures of Geometric Sets	p. 205
Second Fundamental Measure	p. 213
Applications to Curves and Surfaces
Curvature Measures in E[superscript 2]	p. 221
Curvature Measures in E[superscript 3]	p. 231
Approximation of the Curvature of Curves	p. 241
Approximation of the Curvatures of Surfaces	p. 249
On Restricted Delaunay Triangulations	p. 253
Bibliography	p. 261
Index	p. 265
Table of Contents provided by Blackwell. All Rights Reserved.

Geometry and Computing

At a Glance

Hardcover

Industry Reviews

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