The theory of generalized curvature measures has a long history, beginning with J. Steiner (1850), H. Weyl (1939), H. Federer (1959), P. Wintgen (1982), and continues today with young and brilliant mathematicians. In the last decades, a renewal of interest in mathematics as well as computer science has arisen (finding new applications in computer graphics, medical imaging, computationalgeometry, visualization …). Following a historical and didactic approach, thebook introduces the mathematical background of the subject, beginning with curves and surfaces, going on with convex subsets, smooth submanifolds, subsets of positive reach, polyhedra and triangulations, and ending with surface reconstruction. We focus on the theory of normal cycle, which allows to compute andapproximate curvature measures of a large class of smooth or discrete objectsof the Euclidean space. We give explicit computations when the object is a 2 or 3 dimensional polyhedron. First coherent and complete account of this subject in book form INDICE: 1 Motivation - Curves 2 Motivation - Surfaces 3 Distance and Projection 4 Elements of Measure Theory 5 Polyhedra 6 Convex Subsets 7 DifferentialForms and Densities on EN8 Measures on Manifolds 9 Background on Riemannian Geometry 10 Riemannian Submanifolds 11 Currents 12 Approximation of the Volume 13 Approximation of the Length of Curves 14 Approximation of the Area of Surfaces 15 The Steiner Formula for Convex Subsets 16 Tubes Formula of the immersions 17 Subsets of Positive Reach 18 Invariant Forms 19 The Normal Cycle 20 Curvature Measures of Geometric Sets 21 Second Fundamental Measure 22 Curvature Measures in E223 Curvature Measures in E3 24 Approximation of the Curvature of Curves25 Approximation of the Curvatures of Surfaces 26 On Restricted Delaunay Triangulations.
- ISBN: 978-3-540-73791-9
- Editorial: Springer
- Encuadernacion: Cartoné
- Páginas: 300
- Fecha Publicación: 01/04/2008
- Nº Volúmenes: 1
- Idioma: Inglés