Contents XV
18 “Sphere to cone”: pseudo-conic projections ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 395
Mapping the sphere to a cone: pseudo-conic projections
395
18-1 General setup and distortion measures of pseudo-conic projections . . . . . . . . . . . . . . 395
18-2 Special pseudo-conic projections based upon the sphere . . . . . . . . . . . . . . . . . . . . . 398
18-21 Stab–Werner mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
18-22 Bonne mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
19 “Ellipsoid-of-revolution to cone”: polar aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 405
Mapping the ellipsoid to a cone: polar aspect
405
19-1 General mapping equations of the ellipsoid-of-revolution to the cone . . . . . . . . . . . . . 405
19-2 Special conic projections based upon the ellipsoid-of-revolution . . . . . . . . . . . . . . . . 406
19-21 Special conic projections of type equidistant on the set of parallel circles . . . . . . 406
19-22 Special conic projections of type conformal . . . . . . . . . . . . . . . . . . . . . . . . 407
19-23 Special conic projections of type equal area . . . . . . . . . . . . . . . . . . . . . . . . 410
20 Geodetic mapping ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 415
Riemann, Soldner, and Fermi coordinates on the ellipsoid-of-revolution, initial values, boundary values 415
20-1 Geodesic, geodesic circle, Darboux frame, Riemann coordinates . . . . . . . . . . . . . . . . 417
20-2 Lagrange portrait, Hamilton portrait, Lie series, Clairaut constant . . . . . . . . . . . . . . 426
20-21 Lagrange portrait of a geodesic: Legendre series, initial/boundary values . . . . . . 426
20-22 Hamilton portrait of a geodesic: Hamilton equations, initial/boundary values . . . 428
20-3 Soldner coordinates: geodetic parallel coordinates . . . . . . . . . . . . . . . . . . . . . . . . 433
20-31 First problem of Soldner coordinates: input {L 0 , B 0 , x c , y c }, output {L, B, γ} . . . . 434
20-32 Second problem of Soldner coordinates: input {L, B, L 0 , B 0 }, output {x c , y c } . . . . 438
20-4 Fermi coordinates: oblique geodetic parallel coordinates . . . . . . . . . . . . . . . . . . . . . 438
20-5 Deformation analysis: Riemann, Soldner, Gauss–Krueger coordinates . . . . . . . . . . . . . 440
21 Datum problems ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 453
Analysis versus synthesis, Cartesian approach versus curvilinear approach
453
21-1 Analysis of a datum problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
21-2 Synthesis of a datum problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
21-3 Error propagation in analysis and synthesis of a datum problem . . . . . . . . . . . . . . . . 467
21-4 Gauss–Krueger/UTM coordinates: from a local to a global datum . . . . . . . . . . . . . . 469
21-41 Direct transformation of local conformal into global conformal coordinates . . . . . 470
21-42 Inverse transformation of global conformal into local conformal coordinates . . . . . 481
21-43 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
21-5 Mercator coordinates: from a global to a local datum . . . . . . . . . . . . . . . . . . . . . . 490
21-51 Datum transformation extended by form parameters of the UMP . . . . . . . . . . . 490
21-52 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
A Law and order ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 497
Relation preserving maps
497
A-1 Law and order: Cartesian product, power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
A-2 Law and order: Fibering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
B The inverse of a multivariate homogeneous polynomial ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 505
Univariate, bivariate, and multivariate polynomials and their inversion formulae
505
B-1 Inversion of a univariate homogeneous polynomial of degree n . . . . . . . . . . . . . . . . . 505
B-2 Inversion of a bivariate homogeneous polynomial of degree n . . . . . . . . . . . . . . . . . . 509
B-3 Inversion of a multivariate homogeneous polynomial of degree n . . . . . . . . . . . . . . . . 516
18 “Sphere to cone”: pseudo-conic projections ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 395
Mapping the sphere to a cone: pseudo-conic projections
395
18-1 General setup and distortion measures of pseudo-conic projections . . . . . . . . . . . . . . 395
18-2 Special pseudo-conic projections based upon the sphere . . . . . . . . . . . . . . . . . . . . . 398
18-21 Stab–Werner mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
18-22 Bonne mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
19 “Ellipsoid-of-revolution to cone”: polar aspect ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 405
Mapping the ellipsoid to a cone: polar aspect
405
19-1 General mapping equations of the ellipsoid-of-revolution to the cone . . . . . . . . . . . . . 405
19-2 Special conic projections based upon the ellipsoid-of-revolution . . . . . . . . . . . . . . . . 406
19-21 Special conic projections of type equidistant on the set of parallel circles . . . . . . 406
19-22 Special conic projections of type conformal . . . . . . . . . . . . . . . . . . . . . . . . 407
19-23 Special conic projections of type equal area . . . . . . . . . . . . . . . . . . . . . . . . 410
20 Geodetic mapping ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 415
Riemann, Soldner, and Fermi coordinates on the ellipsoid-of-revolution, initial values, boundary values 415
20-1 Geodesic, geodesic circle, Darboux frame, Riemann coordinates . . . . . . . . . . . . . . . . 417
20-2 Lagrange portrait, Hamilton portrait, Lie series, Clairaut constant . . . . . . . . . . . . . . 426
20-21 Lagrange portrait of a geodesic: Legendre series, initial/boundary values . . . . . . 426
20-22 Hamilton portrait of a geodesic: Hamilton equations, initial/boundary values . . . 428
20-3 Soldner coordinates: geodetic parallel coordinates . . . . . . . . . . . . . . . . . . . . . . . . 433
20-31 First problem of Soldner coordinates: input {L 0 , B 0 , x c , y c }, output {L, B, γ} . . . . 434
20-32 Second problem of Soldner coordinates: input {L, B, L 0 , B 0 }, output {x c , y c } . . . . 438
20-4 Fermi coordinates: oblique geodetic parallel coordinates . . . . . . . . . . . . . . . . . . . . . 438
20-5 Deformation analysis: Riemann, Soldner, Gauss–Krueger coordinates . . . . . . . . . . . . . 440
21 Datum problems ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 453
Analysis versus synthesis, Cartesian approach versus curvilinear approach
453
21-1 Analysis of a datum problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
21-2 Synthesis of a datum problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
21-3 Error propagation in analysis and synthesis of a datum problem . . . . . . . . . . . . . . . . 467
21-4 Gauss–Krueger/UTM coordinates: from a local to a global datum . . . . . . . . . . . . . . 469
21-41 Direct transformation of local conformal into global conformal coordinates . . . . . 470
21-42 Inverse transformation of global conformal into local conformal coordinates . . . . . 481
21-43 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
21-5 Mercator coordinates: from a global to a local datum . . . . . . . . . . . . . . . . . . . . . . 490
21-51 Datum transformation extended by form parameters of the UMP . . . . . . . . . . . 490
21-52 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
A Law and order ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 497
Relation preserving maps
497
A-1 Law and order: Cartesian product, power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
A-2 Law and order: Fibering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
B The inverse of a multivariate homogeneous polynomial ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 505
Univariate, bivariate, and multivariate polynomials and their inversion formulae
505
B-1 Inversion of a univariate homogeneous polynomial of degree n . . . . . . . . . . . . . . . . . 505
B-2 Inversion of a bivariate homogeneous polynomial of degree n . . . . . . . . . . . . . . . . . . 509
B-3 Inversion of a multivariate homogeneous polynomial of degree n . . . . . . . . . . . . . . . . 516