Map Projections

by Grafarend, Erik W.; Krumm, Friedrich W.

• ISBN13: 9783540367017
• ISBN10: 3540367012
• Format: Hardcover
• Copyright: 2006-12-04
• Publisher: Springer Verlag

Summary

In the context of Geographical Information Systems (GIS) the book offers a timely review of map projections (sphere, ellipsoidal, rotational surfaces) and geodetic datum transformations. For the needs of photogrammetry computer vision and remote sensing space projective mappings are reviewed. A large variety of world map projections can be downloaded from the attached CD-ROM.

Author Biography

Erik W. Grafarend honorary professor of the Wuhan University in China and of Tehran University in Iran Friedrich W. Krumm has been a member of the permanent staff of the Institute of Geodesy, Universitat Stuttgart

Table of Contents

Preface	p. V
From Riemann manifolds to Riemann manifolds	p. 1
Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Riemann manifold	p. 1
Cauchy-Green deformation tensor	p. 5
Stretch or length distortion	p. 11
Two examples: pseudo-cylindrical and orthogonal map projections	p. 19
Euler-Lagrange deformation tensor	p. 29
One example: orthogonal map projection	p. 33
Review: the deformation measures	p. 37
Angular shear	p. 37
Relative angular shear	p. 40
Equivalence theorem of conformal mapping	p. 43
Two examples: Mercator Projection and Stereographic Projection	p. 53
Areal distortion	p. 74
Equivalence theorem of equiareal mapping	p. 76
One example: mapping from an ellipsoid-of-revolution to the sphere	p. 76
Review: the canonical criteria	p. 80
Isometry	p. 80
Equidistant mapping of submanifolds	p. 82
Canonical criteria	p. 83
Optimal map projections	p. 85
Maximal angular distortion	p. 86
Exercise: the Armadillo double projection	p. 92
From Riemann manifolds to Euclidean manifolds	p. 97
Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Euclidean manifold	p. 97
Eigenspace analysis, Cauchy-Green deformation tensor	p. 97
Eigenspace analysis, Euler-Lagrange deformation tensor	p. 99
The equivalence theorem for conformal mappings	p. 101
Conformeomorphism	p. 101
Higher-dimensional conformal mapping	p. 102
The equivalence theorem for equiareal mappings	p. 106
Canonical criteria for conformal, equiareal, and other mappings	p. 111
Polar decomposition and simultaneous diagonalization of three matrices	p. 111
Coordinates	p. 113
Coordinates (direct, transverse, oblique aspects)	p. 113
Coordinates relating to manifolds	p. 113
Killing vectors of symmetry	p. 122
The oblique frame of reference of the sphere	p. 126
A first design of an oblique frame of reference of the sphere	p. 126
A second design of an oblique frame of reference of the sphere	p. 132
The transverse frame of reference of the sphere: part one	p. 136
The transverse frame of reference of the sphere: part two	p. 138
Transformations between oblique frames of reference: first design, second design	p. 139
Numerical Examples	p. 142
The oblique frame of reference of the ellipsoid-of-revolution	p. 143
The direct and inverse transformations of the normal frame to the oblique frame	p. 143
The intersection of the ellipsoid-of-revolution and the central oblique plane	p. 144
The oblique quasi-spherical coordinates	p. 144
The arc length of the oblique equator in oblique quasi-spherical coordinates	p. 146
Direct transformation of oblique quasi-spherical longitude/latitude	p. 148
Inverse transformation of oblique quasi-spherical longitude/latitude	p. 151
Surfaces of Gaussian curvature zero	p. 153
Classification of surfaces of Gaussian curvature zero in a two-dimensional Euclidean space	p. 153
Ruled surfaces	p. 153
Developable surfaces	p. 156
"Sphere to tangential plane": polar (normal) aspect	p. 161
Mapping the sphere to a tangential plane: polar (normal) aspect	p. 161
General mapping equations	p. 163
Special mapping equations	p. 166
Equidistant mapping (Postel projection)	p. 166
Conformal mapping (stereographic projection, UPS)	p. 168
Equiareal mapping (Lambert projection)	p. 171
Normal perspective mappings	p. 174
What are the best polar azimuthal projections of "sphere to plane"?	p. 197
The pseudo-azimuthal projection	p. 202
The Wiechel polar pseudo-azimuthal projection	p. 205
"Sphere to tangential plane": transverse aspect	p. 209
Mapping the sphere to a tangential plane: meta-azimuthal projections in the transverse aspect	p. 209
General mapping equations	p. 209
Special mapping equations	p. 210
Equidistant mapping (transverse Postel projection)	p. 210
Conformal mapping (transverse stereographic projection, transverse UPS)	p. 211
Equal area mapping (transverse Lambert projection)	p. 213
"Sphere to tangential plane": oblique aspect	p. 215
Mapping the sphere to a tangential plane: meta-azimuthal projections in the oblique aspect	p. 215
General mapping equations	p. 215
Special mapping equations	p. 216
Equidistant mapping (oblique Postel projection)	p. 216
Conformal mapping (oblique stereographic projection, oblique UPS)	p. 217
Equal area mapping (oblique Lambert projection)	p. 218
"Ellipsoid-of-revolution to tangential plane"	p. 221
Mapping the ellipsoid to a tangential plane (azimuthal projections in the normal aspect)	p. 221
General mapping equations	p. 223
Special mapping equations	p. 225
Equidistant mapping	p. 225
Conformal mapping	p. 232
Equiareal mapping	p. 238
Perspective mapping equations	p. 240
The first derivation	p. 245
The special case "sphere to tangential plane"	p. 250
An alternative approach for a topographic point	p. 251
"Ellipsoid-of-revolution to sphere and from sphere to plane"	p. 257
Mapping the ellipsoid to sphere and from sphere to plane (double projection, "authalic" projection)	p. 257
General mapping equations "ellipsoid-of-revolution to plane"	p. 257
The setup of the mapping equations "ellipsoid-of-revolution to plane"	p. 257
The metric tensor of the ellipsoid-of-revolution, the first differential form	p. 258
The curvature tensor of the ellipsoid-of-revolution, the second differential form	p. 258
The metric tensor of the sphere, the first differential form	p. 260
The curvature tensor of the sphere, the second differential form	p. 260
Deformation of the first kind	p. 261
Deformation of the second kind	p. 263
The conformal mappings "ellipsoid-of-revolution to plane"	p. 264
The equal area mappings "ellipsoid-of-revolution to plane"	p. 269
"Sphere to cylinder": polar aspect	p. 273
Mapping the sphere to a cylinder: polar aspect	p. 273
General mapping equations	p. 274
Special mapping equations	p. 276
Equidistant mapping (Plate Carree projection)	p. 276
Conformal mapping (Mercator projection)	p. 277
Equal area mapping (Lambert projection)	p. 278
Optimal cylinder projections	p. 279
"Sphere to cylinder": transverse aspect	p. 285
Mapping the sphere to a cylinder: meta-cylindrical projections in the transverse aspect	p. 285
General mapping equations	p. 286
Special mapping equations	p. 286
Equidistant mapping (transverse Plate Carree projection)	p. 286
Conformal mapping (transverse Mercator projection)	p. 287
Equal area mapping (transverse Lambert projection)	p. 287
"Sphere to cylinder": oblique aspect	p. 289
Mapping the sphere to a cylinder: meta-cylindrical projections in the oblique aspect	p. 289
General mapping equations	p. 290
Special mapping equations	p. 290
Equidistant mapping (oblique Plate Carree projection)	p. 290
Conformal mapping (oblique Mercator projection)	p. 291
Equal area mapping (oblique Lambert projection)	p. 291
"Sphere to cylinder": pseudo-cylindrical projections	p. 293
Mapping the sphere to a cylinder: pseudo-cylindrical projections	p. 293
General mapping equations	p. 293
Special mapping equations	p. 294
Sinusoidal pseudo-cylindrical mapping (J. Cossin, N. Sanson, J. Flamsteed)	p. 295
Elliptic pseudo-cylindrical mapping (C. B. Mollweide)	p. 296
Parabolic pseudo-cylindrical mapping (J. E. E. Craster)	p. 298
Rectilinear pseudo-cylindrical mapping (Eckert II)	p. 299
"Ellipsoid-of-revolution to cylinder": polar aspect	p. 301
Mapping the ellipsoid to a cylinder (polar aspect, generalization for rotational-symmetric surfaces)	p. 301
General mapping equations	p. 301
Special mapping equations	p. 302
Special normal cylindric mapping (equidistant: parallel circles, conformal: equator)	p. 302
Special normal cylindric mapping (normal conformal, equidistant: equator)	p. 304
Special normal cylindric mapping (normal equiareal, equidistant: equator)	p. 304
Summary (cylindric mapping equations)	p. 306
General cylindric mappings (equidistant, rotational-symmetric figure)	p. 307
Special normal cylindric mapping (equidistant: equator, set of parallel circles)	p. 308
Special normal conformal cylindric mapping (equidistant: equator)	p. 308
Special normal equiareal cylindric mapping (equidistant + conformal: equator)	p. 309
An example (mapping the torus)	p. 309
"Ellipsoid-of-revolution to cylinder": transverse aspect	p. 313
Mapping the ellipsoid to a cylinder (transverse Mercator and Gauss-Krueger mappings)	p. 313
The equations governing conformal mapping	p. 316
A fundamental solution for the Korn-Lichtenstein equations	p. 319
Constraints to the Korn-Lichtenstein equations (Gauss-Krueger/UTM mappings)	p. 325
Principal distortions and various optimal designs (UTM mappings)	p. 330
Examples (Gauss-Krueger/UTM coordinates)	p. 334
Strip transformation of conformal coordinates (Gauss-Krueger/UTM mappings)	p. 346
Two-step-approach to strip transformations	p. 347
Two examples of strip transformations	p. 354
"Ellipsoid-of-revolution to cylinder": oblique aspect	p. 359
Mapping the ellipsoid to a cylinder (oblique Mercator and rectified skew orthomorphic projections)	p. 359
The equations governing conformal mapping	p. 361
The oblique reference frame	p. 364
The equations of the oblique Mercator projection	p. 369
"Sphere to cone": polar aspect	p. 379
Mapping the sphere to a cone: polar aspect	p. 379
General mapping equations	p. 381
Special mapping equations	p. 382
Equidistant mapping (de L'Isle projection)	p. 382
Conformal mapping (Lambert projection)	p. 386
Equal area mapping (Albers projection)	p. 389
"Sphere to cone": pseudo-conic projections	p. 395
Mapping the sphere to a cone: pseudo-conic projections	p. 395
General setup and distortion measures of pseudo-conic projections	p. 395
Special pseudo-conic projections based upon the sphere	p. 398
Stab-Werner mapping	p. 398
Bonne mapping	p. 400
"Ellipsoid-of-revolution to cone": polar aspect	p. 405
Mapping the ellipsoid to a cone: polar aspect	p. 405
General mapping equations of the ellipsoid-of-revolution to the cone	p. 405
Special conic projections based upon the ellipsoid-of-revolution	p. 406
Special conic projections of type equidistant on the set of parallel circles	p. 406
Special conic projections of type conformal	p. 407
Special conic projections of type equal area	p. 410
Geodetic mapping	p. 415
Riemann, Soldner, and Fermi coordinates on the ellipsoid-of-revolution, initial values, boundary values	p. 415
Geodesic, geodesic circle, Darboux frame, Riemann coordinates	p. 417
Lagrange portrait, Hamilton portrait, Lie series, Clairaut constant	p. 426
Lagrange portrait of a geodesic: Legendre series, initial/boundary values	p. 426
Hamilton portrait of a geodesic: Hamilton equations, initial/boundary values	p. 428
Soldner coordinates: geodetic parallel coordinates	p. 433
First problem of Soldner coordinates: input {L[subscript O], B[subscript O], x[subscript c], y[subscript c]}, output {L, b, [Gamma]}	p. 434
Second problem of Soldner coordinates: input {L, B, L[subscript O], B[subscript O]}, output {x[subscript c], y[subscript c]}	p. 438
Fermi coordinates: oblique geodetic parallel coordinates	p. 438
Deformation analysis: Riemann, Soldner, Gauss-Krueger coordinates	p. 440
Datum problems	p. 453
Analysis versus synthesis, Cartesian approach versus curvilinear approach	p. 453
Analysis of a datum problem	p. 454
Synthesis of a datum problem	p. 463
Error propagation in analysis and synthesis of a datum problem	p. 467
Gauss-Krueger/UTM coordinates: from a local to a global datum	p. 469
Direct transformation of local conformal into global conformal coordinates	p. 470
Inverse transformation of global conformal into local conformal coordinates	p. 481
Numerical results	p. 484
Mercator coordinates: from a global to a local datum	p. 490
Datum transformation extended by form parameters of the UMP	p. 490
Numerical results	p. 492
Law and order	p. 497
Relation preserving maps	p. 497
Law and order: Cartesian product, power sets	p. 497
Law and order: Fibering	p. 502
The inverse of a multivariate homogeneous polynomial	p. 505
Univariate, bivariate, and multivariate polynomials and their inversion formulae	p. 505
Inversion of a univariate homogeneous polynomial of degree n	p. 505
Inversion of a bivariate homogeneous polynomial of degree n	p. 509
Inversion of a multivariate homogeneous polynomial of degree n	p. 516
Elliptic integrals	p. 519
Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals	p. 519
Introductory example	p. 519
Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals	p. 519
Korn-Lichtenstein and d'Alembert-Euler equations	p. 527
Conformal mapping, Korn-Lichtenstein equations and d'Alembert-Euler (Cauchy-Riemann) equations	p. 527
Korn-Lichtenstein equations	p. 527
D'Alembert-Euler (Cauchy-Riemann) equations	p. 529
Geodesics	p. 543
Geodetic curvature and geodetic torsion, the Newton form of a geodesic in Maupertuis gauge	p. 543
Geodetic curvature, geodetic torsion, and normal curvature	p. 543
The differential equations of third order of a geodesic circle	p. 545
The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution)	p. 546
The Lagrange portrait and the Hamilton portrait of a geodesic	p. 546
The Maupertuis gauge and the Newton portrait of a geodesic	p. 550
A geodesic as a submanifold of the sphere (conformal coordinates)	p. 551
A geodesic as a submanifold of the ellispoid-of-revolution (conformal coordinates)	p. 557
Maupertuis gauged geodesics (normal coordinates, local tangent plane)	p. 563
Maupertuis gauged geodesics (Lie series, Hamilton portrait)	p. 565
Mixed cylindric map projections	p. 569
Mixed cylindric map projections of the ellipsoid-of-revolution, Lambert/Sanson-Flamsteed projections	p. 569
Pseudo-cylindrical mapping: biaxial ellipsoid onto plane	p. 570
Mixed equiareal cylindric mapping: biaxial ellipsoid onto plane	p. 572
Deformation analysis of vertically/horizontally averaged equiareal cylindric mappings	p. 579
Generalized Mollweide projection	p. 589
Generalized Mollweide projection of the ellipsoid-of-revolution	p. 589
The pseudo-cylindrical mapping of the biaxial ellipsoid onto the plane	p. 589
The generalized Mollweide projections for the biaxial ellipsoid	p. 593
Examples	p. 597
Generalized Hammer projection	p. 601
Generalized Hammer projection of the ellipsoid-of-revolution: azimuthal, transverse, resolved equiareal	p. 601
The transverse equiareal projection of the biaxial ellipsoid	p. 602
The transverse reference frame	p. 602
The equiareal mapping of the biaxial ellipsoid onto a transverse tangent plane	p. 605
The equiareal mapping in terms of ellipsoidal longitude, ellipsoidal latitude	p. 607
The ellipsoidal Hammer projection	p. 609
The equiareal mapping from a left biaxial ellipsoid to a right biaxial ellipsoid	p. 610
The explicit form of the mapping equations generating an equiareal map	p. 611
An integration formula	p. 617
The transformation of the radial function r(A*, B*) into r([Delta]*, [Phi]*)	p. 618
The inverse of a special univariate homogeneous polynomial	p. 619
Mercator projection and polycylindric projection	p. 623
Optimal Mercator projection and optimal polycylindric projection of conformal type	p. 623
The optimal Mercator projection (UM)	p. 624
The optimal polycylindric projection of conformal type (UPC)	p. 630
Gauss surface normal coordinates in geometry and gravity space	p. 637
Three-dimensional geodesy, minimal distance mapping, geometric heights	p. 637
Projective heights in geometry space: from planar/spherical to ellipsoidal mapping	p. 638
Gauss surface normal coordinates: case study ellipsoid-of-revolution	p. 643
Review of surface normal coordinates for the ellipsoid-of-revolution	p. 644
Buchberger algorithm of forming a constraint minimum distance mapping	p. 648
Gauss surface normal coordinates: case study triaxial ellipsoid	p. 652
Review of surface normal coordinates for the triaxial ellipsoid	p. 652
Position, orientation, form parameters: case study Earth	p. 653
Form parameters of a surface normal triaxial ellipsoid	p. 655
Bibliography	p. 657
Index	p. 709
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