Contents
Preface ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ V
1
From Riemann manifolds to Riemann manifolds ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1
Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Riemann manifold
1
1-1 Cauchy–Green deformation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1-2 Stretch or length distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1-3 Two examples: pseudo-cylindrical and orthogonal map projections . . . . . . . . . . . . . . 19
1-4 Euler–Lagrange deformation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1-5 One example: orthogonal map projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1-6 Review: the deformation measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1-7 Angular shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1-8 Relative angular shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1-9 Equivalence theorem of conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1-10 Two examples: Mercator Projection and Stereographic Projection . . . . . . . . . . . . . . . 53
1-11 Areal distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
1-12 Equivalence theorem of equiareal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
1-13 One example: mapping from an ellipsoid-of-revolution to the sphere . . . . . . . . . . . . . 76
1-14 Review: the canonical criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
1-141 Isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
1-142 Equidistant mapping of submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1-143 Canonical criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1-144 Optimal map projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
1-145 Maximal angular distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
1-15 Exercise: the Armadillo double projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2
From Riemann manifolds to Euclidean manifolds ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 97
Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Euclidean manifold 97
2-1 Eigenspace analysis, Cauchy–Green deformation tensor . . . . . . . . . . . . . . . . . . . . . 97
2-2 Eigenspace analysis, Euler–Lagrange deformation tensor . . . . . . . . . . . . . . . . . . . . 99
2-3 The equivalence theorem for conformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . 101
2-31 Conformeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2-32 Higher-dimensional conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2-4 The equivalence theorem for equiareal mappings . . . . . . . . . . . . . . . . . . . . . . . . . 106
2-5 Canonical criteria for conformal, equiareal, and other mappings . . . . . . . . . . . . . . . . 111
2-6 Polar decomposition and simultaneous diagonalization of three matrices . . . . . . . . . . . 111
3
Coordinates ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 113
Coordinates (direct, transverse, oblique aspects)
113
3-1 Coordinates relating to manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3-2 Killing vectors of symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Preface ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ V
1
From Riemann manifolds to Riemann manifolds ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1
Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Riemann manifold
1
1-1 Cauchy–Green deformation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1-2 Stretch or length distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1-3 Two examples: pseudo-cylindrical and orthogonal map projections . . . . . . . . . . . . . . 19
1-4 Euler–Lagrange deformation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1-5 One example: orthogonal map projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1-6 Review: the deformation measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1-7 Angular shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1-8 Relative angular shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1-9 Equivalence theorem of conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1-10 Two examples: Mercator Projection and Stereographic Projection . . . . . . . . . . . . . . . 53
1-11 Areal distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
1-12 Equivalence theorem of equiareal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
1-13 One example: mapping from an ellipsoid-of-revolution to the sphere . . . . . . . . . . . . . 76
1-14 Review: the canonical criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
1-141 Isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
1-142 Equidistant mapping of submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1-143 Canonical criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1-144 Optimal map projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
1-145 Maximal angular distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
1-15 Exercise: the Armadillo double projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2
From Riemann manifolds to Euclidean manifolds ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 97
Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Euclidean manifold 97
2-1 Eigenspace analysis, Cauchy–Green deformation tensor . . . . . . . . . . . . . . . . . . . . . 97
2-2 Eigenspace analysis, Euler–Lagrange deformation tensor . . . . . . . . . . . . . . . . . . . . 99
2-3 The equivalence theorem for conformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . 101
2-31 Conformeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2-32 Higher-dimensional conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2-4 The equivalence theorem for equiareal mappings . . . . . . . . . . . . . . . . . . . . . . . . . 106
2-5 Canonical criteria for conformal, equiareal, and other mappings . . . . . . . . . . . . . . . . 111
2-6 Polar decomposition and simultaneous diagonalization of three matrices . . . . . . . . . . . 111
3
Coordinates ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 113
Coordinates (direct, transverse, oblique aspects)
113
3-1 Coordinates relating to manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3-2 Killing vectors of symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122