TY  - BOOK
AU  - Morvan,Jean-Marie
ED  - SpringerLink (Online service)
TI  - Generalized Curvatures
T2  - Geometry and Computing,
SN  - 9783540737926
AV  - QA641-670 
U1  - 516.36 23
PY  - 2008///
CY  - Berlin, Heidelberg
PB  - Springer Berlin Heidelberg
KW  - Mathematics
KW  - Computer graphics
KW  - Computer mathematics
KW  - Differential geometry
KW  - Differential Geometry
KW  - Computational Mathematics and Numerical Analysis
KW  - Computer Imaging, Vision, Pattern Recognition and Graphics
N1  - 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
N2  - 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
UR  - http://dx.doi.org/10.1007/978-3-540-73792-6
ER  -