XIV Contents
13 “Sphere to cylinder”: pseudo-cylindrical projections ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 293
Mapping the sphere to a cylinder: pseudo-cylindrical projections
293
13-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
13-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
13-21 Sinusoidal pseudo-cylindrical mapping (J. Cossin, N. Sanson, J. Flamsteed) . . . . . 295
13-22 Elliptic pseudo-cylindrical mapping (C. B. Mollweide) . . . . . . . . . . . . . . . . . . 296
13-23 Parabolic pseudo-cylindrical mapping (J. E. E. Craster) . . . . . . . . . . . . . . . . . 298
13-24 Rectilinear pseudo-cylindrical mapping (Eckert II) . . . . . . . . . . . . . . . . . . . . 299
14 “Ellipsoid-of-revolution to cylinder”: polar aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 301
Mapping the ellipsoid to a cylinder (polar aspect, generalization for rotational-symmetric surfaces)
301
14-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
14-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
14-21 Special normal cylindric mapping (equidistant: parallel circles, conformal: equator) 302
14-22 Special normal cylindric mapping (normal conformal, equidistant: equator) . . . . . 304
14-23 Special normal cylindric mapping (normal equiareal, equidistant: equator) . . . . . 304
14-24 Summary (cylindric mapping equations) . . . . . . . . . . . . . . . . . . . . . . . . . . 306
14-3 General cylindric mappings (equidistant, rotational-symmetric figure) . . . . . . . . . . . . 307
14-31 Special normal cylindric mapping (equidistant: equator, set of parallel circles) . . . 308
14-32 Special normal conformal cylindric mapping (equidistant: equator) . . . . . . . . . . 308
14-33 Special normal equiareal cylindric mapping (equidistant + conformal: equator) . . 309
14-34 An example (mapping the torus) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
15 “Ellipsoid-of-revolution to cylinder”: transverse aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 313
Mapping the ellipsoid to a cylinder (transverse Mercator and Gauss–Krueger mappings)
313
15-1 The equations governing conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
15-2 A fundamental solution for the Korn–Lichtenstein equations . . . . . . . . . . . . . . . . . . 319
15-3 Constraints to the Korn–Lichtenstein equations (Gauss–Krueger/UTM mappings) . . . . . 325
15-4 Principal distortions and various optimal designs (UTM mappings) . . . . . . . . . . . . . . 330
15-5 Examples (Gauss–Krueger/UTM coordinates) . . . . . . . . . . . . . . . . . . . . . . . . . . 334
15-6 Strip transformation of conformal coordinates (Gauss–Krueger/UTM mappings) . . . . . . 346
15-61 Two-step-approach to strip transformations . . . . . . . . . . . . . . . . . . . . . . . . 347
15-62 Two examples of strip transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
16 “Ellipsoid-of-revolution to cylinder”: oblique aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 359
Mapping the ellipsoid to a cylinder (oblique Mercator and rectified skew orthomorphic projections)
359
16-1 The equations governing conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
16-2 The oblique reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
16-3 The equations of the oblique Mercator projection . . . . . . . . . . . . . . . . . . . . . . . . 369
17 “Sphere to cone”: polar aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 379
Mapping the sphere to a cone: polar aspect
379
17-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
17-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
17-21 Equidistant mapping (de L’Isle projection) . . . . . . . . . . . . . . . . . . . . . . . . 382
17-22 Conformal mapping (Lambert projection) . . . . . . . . . . . . . . . . . . . . . . . . . 386
17-23 Equal area mapping (Albers projection) . . . . . . . . . . . . . . . . . . . . . . . . . . 389
13 “Sphere to cylinder”: pseudo-cylindrical projections ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 293
Mapping the sphere to a cylinder: pseudo-cylindrical projections
293
13-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
13-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
13-21 Sinusoidal pseudo-cylindrical mapping (J. Cossin, N. Sanson, J. Flamsteed) . . . . . 295
13-22 Elliptic pseudo-cylindrical mapping (C. B. Mollweide) . . . . . . . . . . . . . . . . . . 296
13-23 Parabolic pseudo-cylindrical mapping (J. E. E. Craster) . . . . . . . . . . . . . . . . . 298
13-24 Rectilinear pseudo-cylindrical mapping (Eckert II) . . . . . . . . . . . . . . . . . . . . 299
14 “Ellipsoid-of-revolution to cylinder”: polar aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 301
Mapping the ellipsoid to a cylinder (polar aspect, generalization for rotational-symmetric surfaces)
301
14-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
14-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
14-21 Special normal cylindric mapping (equidistant: parallel circles, conformal: equator) 302
14-22 Special normal cylindric mapping (normal conformal, equidistant: equator) . . . . . 304
14-23 Special normal cylindric mapping (normal equiareal, equidistant: equator) . . . . . 304
14-24 Summary (cylindric mapping equations) . . . . . . . . . . . . . . . . . . . . . . . . . . 306
14-3 General cylindric mappings (equidistant, rotational-symmetric figure) . . . . . . . . . . . . 307
14-31 Special normal cylindric mapping (equidistant: equator, set of parallel circles) . . . 308
14-32 Special normal conformal cylindric mapping (equidistant: equator) . . . . . . . . . . 308
14-33 Special normal equiareal cylindric mapping (equidistant + conformal: equator) . . 309
14-34 An example (mapping the torus) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
15 “Ellipsoid-of-revolution to cylinder”: transverse aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 313
Mapping the ellipsoid to a cylinder (transverse Mercator and Gauss–Krueger mappings)
313
15-1 The equations governing conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
15-2 A fundamental solution for the Korn–Lichtenstein equations . . . . . . . . . . . . . . . . . . 319
15-3 Constraints to the Korn–Lichtenstein equations (Gauss–Krueger/UTM mappings) . . . . . 325
15-4 Principal distortions and various optimal designs (UTM mappings) . . . . . . . . . . . . . . 330
15-5 Examples (Gauss–Krueger/UTM coordinates) . . . . . . . . . . . . . . . . . . . . . . . . . . 334
15-6 Strip transformation of conformal coordinates (Gauss–Krueger/UTM mappings) . . . . . . 346
15-61 Two-step-approach to strip transformations . . . . . . . . . . . . . . . . . . . . . . . . 347
15-62 Two examples of strip transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
16 “Ellipsoid-of-revolution to cylinder”: oblique aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 359
Mapping the ellipsoid to a cylinder (oblique Mercator and rectified skew orthomorphic projections)
359
16-1 The equations governing conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
16-2 The oblique reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
16-3 The equations of the oblique Mercator projection . . . . . . . . . . . . . . . . . . . . . . . . 369
17 “Sphere to cone”: polar aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 379
Mapping the sphere to a cone: polar aspect
379
17-1 General mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
17-2 Special mapping equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
17-21 Equidistant mapping (de L’Isle projection) . . . . . . . . . . . . . . . . . . . . . . . . 382
17-22 Conformal mapping (Lambert projection) . . . . . . . . . . . . . . . . . . . . . . . . . 386
17-23 Equal area mapping (Albers projection) . . . . . . . . . . . . . . . . . . . . . . . . . . 389