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. Let G be the group of rigid motions of E . We say that a 0 quantity Q(S) associated to S is geometric with respect to G if the corresponding 0 quantity Q[g(S)] associated to g(S) equals Q(S), for all g?G . For instance, the 0 diameter of S and the area of the convex hull of S are quantities geometric with respect to G . But the distance from the origin O to the closest point of S is not, 0 since it is not invariant under translations of S. It is important to point out that the property of being geometric depends on the chosen group. For instance, if G is the 1 N group of projective transformations of E , then the property of S being a circle is geometric for G but not for G , while the property of being a conic or a straight 0 1 line is geometric for both G and G . This point of view may be generalized to any 0 1 subset S of any vector space E endowed with a group G acting on it.
Mục lục
Motivations.- Motivation: Curves.- Motivation: Surfaces.- Background: Metrics and Measures.- Distance and Projection.- Elements of Measure Theory.- Background: Polyhedra and Convex Subsets.- Polyhedra.- Convex Subsets.- Background: Classical Tools in Differential Geometry.- Differential Forms and Densities on EN.- Measures on Manifolds.- Background on Riemannian Geometry.- Riemannian Submanifolds.- Currents.- On Volume.- Approximation of the Volume.- Approximation of the Length of Curves.- Approximation of the Area of Surfaces.- The Steiner Formula.- The Steiner Formula for Convex Subsets.- Tubes Formula.- Subsets of Positive Reach.- The Theory of Normal Cycles.- Invariant Forms.- The Normal Cycle.- Curvature Measures of Geometric Sets.- Second Fundamental Measure.- Applications to Curves and Surfaces.- Curvature Measures in E2.- Curvature Measures in E3.- Approximation of the Curvature of Curves.- Approximation of the Curvatures of Surfaces.- On Restricted Delaunay Triangulations.
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