XII
Contents
3-3 The oblique frame of reference of the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3-31 A first design of an oblique frame of reference of the sphere . . . . . . . . . . . . . . 126
3-32 A second design of an oblique frame of reference of the sphere . . . . . . . . . . . . . 132
3-33 The transverse frame of reference of the sphere: part one . . . . . . . . . . . . . . . . 136
3-34 The transverse frame of reference of the sphere: part two . . . . . . . . . . . . . . . . 138
3-35 Transformations between oblique frames of reference: first design, second design . . 139
3-36 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3-4 The oblique frame of reference of the ellipsoid-of-revolution . . . . . . . . . . . . . . . . . . 143
3-41 The direct and inverse transformations of the normal frame to the oblique frame . 143
3-42 The intersection of the ellipsoid-of-revolution and the central oblique plane . . . . . 144
3-43 The oblique quasi-spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3-44 The arc length of the oblique equator in oblique quasi-spherical coordinates . . . . 146
3-45 Direct transformation of oblique quasi-spherical longitude/latitude . . . . . . . . . . 148
3-46 Inverse transformation of oblique quasi-spherical longitude/latitude . . . . . . . . . 151
4
Surfaces of Gaussian curvature zero ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 153
Classification of surfaces of Gaussian curvature zero in a two-dimensional Euclidean space
153
4-1 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4-2 Developable surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5
“Sphere to tangential plane”: polar (normal) aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 161
Mapping the sphere to a tangential plane: polar (normal) aspect
161
5-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5-21 Equidistant mapping (Postel projection) . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5-22 Conformal mapping (stereographic projection, UPS) . . . . . . . . . . . . . . . . . . 168
5-23 Equiareal mapping (Lambert projection) . . . . . . . . . . . . . . . . . . . . . . . . . 171
5-24 Normal perspective mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5-25 What are the best polar azimuthal projections of “sphere to plane”? . . . . . . . . . 197
5-3 The pseudo-azimuthal projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5-4 The Wiechel polar pseudo-azimuthal projection . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6
“Sphere to tangential plane”: transverse aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 209
Mapping the sphere to a tangential plane: meta-azimuthal projections in the transverse aspect
209
6-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6-21 Equidistant mapping (transverse Postel projection) . . . . . . . . . . . . . . . . . . . 210
6-22 Conformal mapping (transverse stereographic projection, transverse UPS) . . . . . 211
6-23 Equal area mapping (transverse Lambert projection) . . . . . . . . . . . . . . . . . . 213
7
“Sphere to tangential plane”: oblique aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 215
Mapping the sphere to a tangential plane: meta-azimuthal projections in the oblique aspect
215
7-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7-21 Equidistant mapping (oblique Postel projection) . . . . . . . . . . . . . . . . . . . . . 216
7-22 Conformal mapping (oblique stereographic projection, oblique UPS) . . . . . . . . . 217
7-23 Equal area mapping (oblique Lambert projection) . . . . . . . . . . . . . . . . . . . . 218
Contents
3-3 The oblique frame of reference of the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3-31 A first design of an oblique frame of reference of the sphere . . . . . . . . . . . . . . 126
3-32 A second design of an oblique frame of reference of the sphere . . . . . . . . . . . . . 132
3-33 The transverse frame of reference of the sphere: part one . . . . . . . . . . . . . . . . 136
3-34 The transverse frame of reference of the sphere: part two . . . . . . . . . . . . . . . . 138
3-35 Transformations between oblique frames of reference: first design, second design . . 139
3-36 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3-4 The oblique frame of reference of the ellipsoid-of-revolution . . . . . . . . . . . . . . . . . . 143
3-41 The direct and inverse transformations of the normal frame to the oblique frame . 143
3-42 The intersection of the ellipsoid-of-revolution and the central oblique plane . . . . . 144
3-43 The oblique quasi-spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3-44 The arc length of the oblique equator in oblique quasi-spherical coordinates . . . . 146
3-45 Direct transformation of oblique quasi-spherical longitude/latitude . . . . . . . . . . 148
3-46 Inverse transformation of oblique quasi-spherical longitude/latitude . . . . . . . . . 151
4
Surfaces of Gaussian curvature zero ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 153
Classification of surfaces of Gaussian curvature zero in a two-dimensional Euclidean space
153
4-1 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4-2 Developable surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5
“Sphere to tangential plane”: polar (normal) aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 161
Mapping the sphere to a tangential plane: polar (normal) aspect
161
5-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5-21 Equidistant mapping (Postel projection) . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5-22 Conformal mapping (stereographic projection, UPS) . . . . . . . . . . . . . . . . . . 168
5-23 Equiareal mapping (Lambert projection) . . . . . . . . . . . . . . . . . . . . . . . . . 171
5-24 Normal perspective mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5-25 What are the best polar azimuthal projections of “sphere to plane”? . . . . . . . . . 197
5-3 The pseudo-azimuthal projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5-4 The Wiechel polar pseudo-azimuthal projection . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6
“Sphere to tangential plane”: transverse aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 209
Mapping the sphere to a tangential plane: meta-azimuthal projections in the transverse aspect
209
6-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6-21 Equidistant mapping (transverse Postel projection) . . . . . . . . . . . . . . . . . . . 210
6-22 Conformal mapping (transverse stereographic projection, transverse UPS) . . . . . 211
6-23 Equal area mapping (transverse Lambert projection) . . . . . . . . . . . . . . . . . . 213
7
“Sphere to tangential plane”: oblique aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 215
Mapping the sphere to a tangential plane: meta-azimuthal projections in the oblique aspect
215
7-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7-21 Equidistant mapping (oblique Postel projection) . . . . . . . . . . . . . . . . . . . . . 216
7-22 Conformal mapping (oblique stereographic projection, oblique UPS) . . . . . . . . . 217
7-23 Equal area mapping (oblique Lambert projection) . . . . . . . . . . . . . . . . . . . . 218