Generalized Curvatures

di Jean-Marie Morvan (Autore)

Springer-Verlag Berlin and Heidelberg GmbH & Co. KG, 2008

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.

• Autore: Jean-Marie Morvan
• Editore: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
• Collana: Geometry and Computing
• Anno: 2008
• Rilegatura: Hardback
• Pagine: 266 p.
• Dimensioni: 235 x 155 mm
• EAN: 9783540737919

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