Contents XIII
8
“Ellipsoid-of-revolution to tangential plane” ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 221
Mapping the ellipsoid to a tangential plane (azimuthal projections in the normal aspect)
221
8-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8-21 Equidistant mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8-22 Conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8-23 Equiareal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8-3 Perspective mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8-31 The first derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8-32 The special case “sphere to tangential plane” . . . . . . . . . . . . . . . . . . . . . . . 250
8-33 An alternative approach for a topographic point . . . . . . . . . . . . . . . . . . . . . 251
9
“Ellipsoid-of-revolution to sphere and from sphere to plane” ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 257
Mapping the ellipsoid to sphere and from sphere to plane (double projection, “authalic” projection)
257
9-1 General mapping equations “ellipsoid-of-revolution to plane” . . . . . . . . . . . . . . . . . . 257
9-11 The setup of the mapping equations “ellipsoid-of-revolution to plane” . . . . . . . . 257
9-12 The metric tensor of the ellipsoid-of-revolution, the first differential form . . . . . . 258
9-13 The curvature tensor of the ellipsoid-of-revolution, the second differential form . . 258
9-14 The metric tensor of the sphere, the first differential form . . . . . . . . . . . . . . . 260
9-15 The curvature tensor of the sphere, the second differential form . . . . . . . . . . . . 260
9-16 Deformation of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9-17 Deformation of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9-2 The conformal mappings “ellipsoid-of-revolution to plane” . . . . . . . . . . . . . . . . . . . 264
9-3 The equal area mappings “ellipsoid-of-revolution to plane” . . . . . . . . . . . . . . . . . . . 269
10 “Sphere to cylinder”: polar aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 273
Mapping the sphere to a cylinder: polar aspect
273
10-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
10-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
10-21 Equidistant mapping (Plate Carr´ ee projection) . . . . . . . . . . . . . . . . . . . . . . 276
10-22 Conformal mapping (Mercator projection) . . . . . . . . . . . . . . . . . . . . . . . . 277
10-23 Equal area mapping (Lambert projection) . . . . . . . . . . . . . . . . . . . . . . . . . 278
10-3 Optimal cylinder projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
11 “Sphere to cylinder”: transverse aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 285
Mapping the sphere to a cylinder: meta-cylindrical projections in the transverse aspect
285
11-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
11-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
11-21 Equidistant mapping (transverse Plate Carr´ ee projection) . . . . . . . . . . . . . . . 286
11-22 Conformal mapping (transverse Mercator projection) . . . . . . . . . . . . . . . . . . 287
11-23 Equal area mapping (transverse Lambert projection) . . . . . . . . . . . . . . . . . . 287
12 “Sphere to cylinder”: oblique aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 289
Mapping the sphere to a cylinder: meta-cylindrical projections in the oblique aspect
289
12-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
12-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
12-21 Equidistant mapping (oblique Plate Carr´ ee projection) . . . . . . . . . . . . . . . . . 290
12-22 Conformal mapping (oblique Mercator projection) . . . . . . . . . . . . . . . . . . . . 291
12-23 Equal area mapping (oblique Lambert projection) . . . . . . . . . . . . . . . . . . . . 291
8
“Ellipsoid-of-revolution to tangential plane” ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 221
Mapping the ellipsoid to a tangential plane (azimuthal projections in the normal aspect)
221
8-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8-21 Equidistant mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8-22 Conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8-23 Equiareal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8-3 Perspective mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8-31 The first derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8-32 The special case “sphere to tangential plane” . . . . . . . . . . . . . . . . . . . . . . . 250
8-33 An alternative approach for a topographic point . . . . . . . . . . . . . . . . . . . . . 251
9
“Ellipsoid-of-revolution to sphere and from sphere to plane” ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 257
Mapping the ellipsoid to sphere and from sphere to plane (double projection, “authalic” projection)
257
9-1 General mapping equations “ellipsoid-of-revolution to plane” . . . . . . . . . . . . . . . . . . 257
9-11 The setup of the mapping equations “ellipsoid-of-revolution to plane” . . . . . . . . 257
9-12 The metric tensor of the ellipsoid-of-revolution, the first differential form . . . . . . 258
9-13 The curvature tensor of the ellipsoid-of-revolution, the second differential form . . 258
9-14 The metric tensor of the sphere, the first differential form . . . . . . . . . . . . . . . 260
9-15 The curvature tensor of the sphere, the second differential form . . . . . . . . . . . . 260
9-16 Deformation of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9-17 Deformation of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9-2 The conformal mappings “ellipsoid-of-revolution to plane” . . . . . . . . . . . . . . . . . . . 264
9-3 The equal area mappings “ellipsoid-of-revolution to plane” . . . . . . . . . . . . . . . . . . . 269
10 “Sphere to cylinder”: polar aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 273
Mapping the sphere to a cylinder: polar aspect
273
10-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
10-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
10-21 Equidistant mapping (Plate Carr´ ee projection) . . . . . . . . . . . . . . . . . . . . . . 276
10-22 Conformal mapping (Mercator projection) . . . . . . . . . . . . . . . . . . . . . . . . 277
10-23 Equal area mapping (Lambert projection) . . . . . . . . . . . . . . . . . . . . . . . . . 278
10-3 Optimal cylinder projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
11 “Sphere to cylinder”: transverse aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 285
Mapping the sphere to a cylinder: meta-cylindrical projections in the transverse aspect
285
11-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
11-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
11-21 Equidistant mapping (transverse Plate Carr´ ee projection) . . . . . . . . . . . . . . . 286
11-22 Conformal mapping (transverse Mercator projection) . . . . . . . . . . . . . . . . . . 287
11-23 Equal area mapping (transverse Lambert projection) . . . . . . . . . . . . . . . . . . 287
12 “Sphere to cylinder”: oblique aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 289
Mapping the sphere to a cylinder: meta-cylindrical projections in the oblique aspect
289
12-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
12-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
12-21 Equidistant mapping (oblique Plate Carr´ ee projection) . . . . . . . . . . . . . . . . . 290
12-22 Conformal mapping (oblique Mercator projection) . . . . . . . . . . . . . . . . . . . . 291
12-23 Equal area mapping (oblique Lambert projection) . . . . . . . . . . . . . . . . . . . . 291