Cover -- Half Title -- Title -- Copyright -- CONTENTS -- PREFACE -- 1 Cavalieri principle and other prerequisites -- 1.1 The Cavalieri principle -- 1.2 Lebesgue factorization -- 1.3 Haar factorization -- 1.4 Further remarks on measures -- 1.5 Some topological remarks -- 1.6 Parametrization maps -- 1.7 Metrics and convexity -- 1.8 Versions of Crofton's theorem -- 2 Measures invariant with respect to translations -- 2.1 The space G of directed lines on R2 -- 2.2 The space G of (non-directed) lines in R2 -- 2.3 The space E of oriented planes in R3 -- 2.4 The space E of planes in R3
2.5 The space D of directed lines in R3 -- 2.6 The space D of (non-directed) lines in R3 -- 2.7 Measure-representing product models -- 2.8 Factorization of measures on spaces with slits -- 2.9 Dispensing with slits -- 2.10 Roses of directions and roses of hits -- 2.11 Density and curvature -- 2.12 The roses of T3-invariant measures on E -- 2.13 Spaces of segments and flats -- 2.14 Product spaces with slits -- 2.15 Almost sure T-invariance of random measures -- 2.16 Random measures on G -- 2.17 Random measures on E -- 2.18 Random measures on D
3 Measures invariant with respect to Euclidean motions -- 3.1 The group W2 of rotations of R2 -- 3.2 Rotations of R3 -- 3.3 The Haar measure on W3 -- 3.4 Geodesic lines on a sphere -- 3.5 Bi-invariance of Haar measures on Euclidean groups -- 3.6 The invariant measure on G and G -- 3.7 The form of dg in two other parametrizations of lines -- 3.8 Other parametrizations of geodesic lines on a sphere -- 3.9 The invariant measure on D and D
3.10 Other parametrizations of lines in R3 -- 3.11 The invariant measure in the spaces E and E -- 3.12 Other parametrizations of planes in R3 -- 3.13 The kinematic measure -- 3.14 Position-size factorizations -- 3.15 Position-shape factorizations -- 3.16 Position-size-shape factorizations -- 3.17 On measures in shape spaces -- 3.18 The spherical topology of V -- 4 Haar measures on groups of affine transformations -- 4.1 The group Ag and its subgroups -- 4.2 Affine deformations of R2 -- 4.3 The Haar measure on Ag -- 4.4 The Haar measure on A2
4.5 Triads of points in R2 -- 4.6 Another representation of d(r)V -- 4.7 Quadruples of points in R2 -- 4.8 The modified Sylvester problem: four points in R2 -- 4.9 The group Ag and its subgroups -- 4.10 The group of affine deformations of R3 -- 4.11 Haar measures on Ag and A3 -- 4.12 V 3-invariant measure in the space of tetrahedral shapes -- 4.13 Quintuples of points in R3 -- 4.14 Affine shapes of quintuples in R3 -- 4.15 A general theorem -- 4.16 The elliptical plane as a space of affine shapes -- 5 Combinatorial integral geometry -- 5.1 Radon rings in G and G
Summary
This unique book develops the classical subjects of geometric probability and integral geometry
