Generalized Curvatures PDF

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.