# Generalized Curvatures

• Jean-Marie Morvan Book

Part of the Geometry and Computing book series (GC, volume 2)

1. Front Matter
Pages i-xi
2. ### Motivations

1. Pages 1-10
2. Pages 13-28
3. Pages 29-44
3. ### Background: Metrics and Measures

1. Pages 47-56
2. Pages 57-68
4. ### Background: Polyhedra and Convex Subsets

1. Pages 71-76
2. Pages 77-88
5. ### Background: Classical Tools in Differential Geometry

1. Pages 91-95
2. Pages 97-99
3. Pages 101-107
4. Pages 109-119
5. Pages 121-125
6. ### On Volume

1. Pages 129-137
2. Pages 139-141
3. Pages 143-150
7. ### The Steiner Formula

1. Pages 153-164
2. Pages 165-175
3. Pages 177-186
8. ### The Theory of Normal Cycles

1. Pages 189-192
2. Pages 193-203
3. Pages 206-211
4. Pages 213-218
9. ### Applications to Curves and Surfaces

1. Pages 221-229
2. Pages 231-239
3. Pages 241-247
4. Pages 249-252
5. Pages 253-259
10. Back Matter
Pages 261-266

### Introduction

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.

### Keywords

Gaussian curvature Riemannian geometry Riemannian manifold computational geometry computer graphics curvature curvature measure differential geometry discrete geometry manifold submanifold triangulation visualization

#### Authors and affiliations

• Jean-Marie Morvan
• 1
1. 1.Institut Camille JordanUniversité Claude Bernard Lyon 169622France