Contents

Motivations

1 Motivation -Curves

1.1 The length of a curve

1.2 The curvature of a curve

1.3 The Gauss map of a curve

1.4 Curves in E2

2 Motivation -Surfaces

2.1 The area of a surface

2.2 The pointwise Gauss curvature

2.3 The Gauss map of a surface

2.4 The global Gauss curvature

2.5 ... and the volume...

3 Distance and Projection

3.1 The distance function

3.2 The projection map

3.3 The reach of a subset

3.4 The Voronoi diagrams

3.5 The medial axis of a subset

4 Elements of Measure Theory

4.1 Outer measures and measures

4.2 Measurable functions and their integrals

4.3 The standard Lebesgue measure on EN

4.4 Hausdorff measures

4.5 Area and co-area formula

4.6 Radon measures

4.7 Convergence of measures

5 Polyhedra

5.1 Definitions and properties of polyhedra

5.2 Euler characteristic

5.3 Gauss curvature of a polyhedron

6 Convex Subsets

6.1 Convex subsets

6.2 Differential properties of the boundary

6.3 The volume of the boundary of a convex body

6.4 The transversal integral and the Hadwiger theorem

7 Differential Forms and Densities on EN

7.1 Differential forms and their integrals

7.2 Densities

8 Measures on Manifolds

8.1 Integration of differential forms

8.2 Density and measure on a manifold

8.3 The Fubini theorem on a fiber bundle

9 Background on Riemannian Geometry

9.1 Riemannian metric and Levi-Civita connexion

9.2 Properties of the curvature tensor

9.3 Connexion forms and curvature forms

9.4 The volume form

9.5 The Gauss-Bonnet theorem

9.6 Spheres and balls

9.7 The Grassmann manifolds

10 Riemannian Submanifolds

10.1 Some generalities on (smooth) submanifolds

10.2Thevolumeofasubmanifold

10.3 Hypersurfaces in EN

10.4 Submanifolds in EN of any codimension

10.5TheGaussmapofasubmanifold..... 140

11 Currents

11.1 Basic definitions and properties on currents

11.2 Rectifiable currents

11.3Three theorems

12 Approximation of the Volume

12.1 Thegeneralframework

12.2 A general evaluation theorem for the volume

12.3 An approximation result

12.4 Aconvergence theorem for the volume

13 Approximation of the Length of Curves

13.1 A general approximation result

13.2 An approximation by a polygonal line

14 Approximation of the Area of Surfaces

14.1 A general approximation of the area

14.2 Triangulations

14.3 Relative height of a triangulation inscribed in a surface

14.4 A bound on the deviation angle

14.5 Approximation of the area of a smooth surface by the

area of a triangulation

15 The Steiner Formula for Convex Subsets

15.1 The Steiner formula for convex bodies (1840)

15.2 Examples:segments,discsandballs

15.3 Convex bodies in EN whose boundary is a polyhedron

15.4 Convex bodies with smooth boundary

15.5 Evaluation of the Quermassintegrale by means of transversal integrals

15.6 Continuity of the k

15.7 Anadditivity formula

16 Tubes Formula

16.1 The Lipschitz-Killingcurvatures

16.2 The tubes formulaofH.Weyl(1939)

16.3 The Eule rcharacteristic

16.4 Partial continuity of the k

16.5 Transversal integrals

16.6 On the differentiability of the immersions

17 Subsets of Positive Reach

17.1 Subsets of positive reach (H. Federer, 1958)

17.2 The Steiner formula

17.3 Curvature measures

17.4 The Euler characteristic

17.5 The problem of continuity of the k

17.6 Thetransversalintegralses

18 Invariant Forms

18.1 Invariant forms on EN × EN

18.2 Invariant differential forms on EN × SN-1

18.3 Examplesinlow dimensions

19 The Normal Cycle

19.1 The notion of a normal cycle

19.2 Existence and uniqueness of the normal cycle

19.3 A convergence theorem

19.4 Approximation of normal cycles

20 Curvature Measures of Geometric Sets

20.1 Definition of curvatures

20.2 Continuity of the Mk

20.3 Curvature measures of geometric sets

20.4 Convergence and approximation theorems

21 Second Fundamental Measure

21.1 A vector valued invariant form

21.2 Second fundamental measure associated to a geometric set

21.3 The case of a smooth hypersurface

21.4 The case of a polyhedron

21.5 Convergence and approximation

21.6 An example of application

22 Curvature Measures in E2

22.1 Invariant forms of E2 × S1

22.2 Bounded domains in E2

22.3 Plane curves

22.4 The length of plane curves

22.5 The curvature of plane curves

23 Curvature Measures in E3

23.1 Invariant forms of E3 × S2

23.2 Space curves and polygons

23.3 Surfaces and bounded domains in E3

23.4 Second fundamental measure for surfaces

24 Approximation of the Curvature of Curves

24.1 Curves in E2

24.2 Curves in E3

25 Approximation of the Curvatures of Surfaces

25.1 The general approximation result

25.2 Approximation by a triangulation

26 On Restricted Delaunay Triangulations

26.1 Delaunay triangulation

26.2 Approximation using a Delaunay triangulation