Buch, Englisch, 266 Seiten, Previously published in hardcover, Format (B × H): 155 mm x 235 mm, Gewicht: 429 g
Reihe: Geometry and Computing
Buch, Englisch, 266 Seiten, Previously published in hardcover, Format (B × H): 155 mm x 235 mm, Gewicht: 429 g
Reihe: Geometry and Computing
ISBN: 978-3-642-09300-5
Verlag: Springer
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.
Zielgruppe
Research
Autoren/Hrsg.
Fachgebiete
- Mathematik | Informatik EDV | Informatik Informatik Mathematik für Informatiker
- Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
- Mathematik | Informatik Mathematik Geometrie Euklidische Geometrie
- Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
Weitere Infos & Material
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.
