XVI Contents
C Elliptic integrals ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 519
Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals
519
C-1 Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
C-2 Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals . . . . . . . . . . . . . . 519
D Korn–Lichtenstein and d’Alembert–Euler equations ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 527
Conformal mapping, Korn–Lichtenstein equations and d’Alembert–Euler (Cauchy–Riemann) equations 527
D-1 Korn–Lichtenstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
D-2 D’Alembert–Euler (Cauchy–Riemann) equations . . . . . . . . . . . . . . . . . . . . . . . . . 529
E Geodesics ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 543
Geodetic curvature and geodetic torsion, the Newton form of a geodesic in Maupertuis gauge
543
E-1 Geodetic curvature, geodetic torsion, and normal curvature . . . . . . . . . . . . . . . . . . 543
E-2 The differential equations of third order of a geodesic circle . . . . . . . . . . . . . . . . . . 545
E-3 The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution) . . . 546
E-31 The Lagrange portrait and the Hamilton portrait of a geodesic . . . . . . . . . . . . 546
E-32 The Maupertuis gauge and the Newton portrait of a geodesic . . . . . . . . . . . . . 550
E-33 A geodesic as a submanifold of the sphere (conformal coordinates) . . . . . . . . . . 551
E-34 A geodesic as a submanifold of the ellispoid-of-revolution (conformal coordinates) . 557
E-35 Maupertuis gauged geodesics (normal coordinates, local tangent plane) . . . . . . . 563
E-36 Maupertuis gauged geodesics (Lie series, Hamilton portrait) . . . . . . . . . . . . . . 565
F
Mixed cylindric map projections ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 569
Mixed cylindric map projections of the ellipsoid-of-revolution, Lambert/Sanson–Flamsteed projections 569
F-1 Pseudo-cylindrical mapping: biaxial ellipsoid onto plane . . . . . . . . . . . . . . . . . . . . . 570
F-2 Mixed equiareal cylindric mapping: biaxial ellipsoid onto plane . . . . . . . . . . . . . . . . 572
F-3 Deformation analysis of vertically/horizontally averaged equiareal cylindric mappings . . . 579
G Generalized Mollweide projection ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 589
Generalized Mollweide projection of the ellipsoid-of-revolution
589
G-1 The pseudo-cylindrical mapping of the biaxial ellipsoid onto the plane . . . . . . . . . . . . 589
G-2 The generalized Mollweide projections for the biaxial ellipsoid . . . . . . . . . . . . . . . . . 593
G-3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
H Generalized Hammer projection ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 601
Generalized Hammer projection of the ellipsoid-of-revolution: azimuthal, transverse, resolved equiareal 601
H-1 The transverse equiareal projection of the biaxial ellipsoid . . . . . . . . . . . . . . . . . . . 602
H-11 The transverse reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
H-12 The equiareal mapping of the biaxial ellipsoid onto a transverse tangent plane . . 605
H-13 The equiareal mapping in terms of ellipsoidal longitude, ellipsoidal latitude . . . . . 607
H-2 The ellipsoidal Hammer projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
H-21 The equiareal mapping from a left biaxial ellipsoid to a right biaxial ellipsoid . . . 610
H-22 The explicit form of the mapping equations generating an equiareal map . . . . . . 611
H-3 An integration formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
H-4 The transformation of the radial function r(A
∗ , B
∗ ) into r(Λ
∗ , Φ
∗ ) . . . . . . . . . . . . . . 618
H-5 The inverse of a special univariate homogeneous polynomial . . . . . . . . . . . . . . . . . . 619
C Elliptic integrals ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 519
Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals
519
C-1 Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
C-2 Elliptic kernel, elliptic modulus, elliptic functions, elliptic integrals . . . . . . . . . . . . . . 519
D Korn–Lichtenstein and d’Alembert–Euler equations ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 527
Conformal mapping, Korn–Lichtenstein equations and d’Alembert–Euler (Cauchy–Riemann) equations 527
D-1 Korn–Lichtenstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
D-2 D’Alembert–Euler (Cauchy–Riemann) equations . . . . . . . . . . . . . . . . . . . . . . . . . 529
E Geodesics ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 543
Geodetic curvature and geodetic torsion, the Newton form of a geodesic in Maupertuis gauge
543
E-1 Geodetic curvature, geodetic torsion, and normal curvature . . . . . . . . . . . . . . . . . . 543
E-2 The differential equations of third order of a geodesic circle . . . . . . . . . . . . . . . . . . 545
E-3 The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution) . . . 546
E-31 The Lagrange portrait and the Hamilton portrait of a geodesic . . . . . . . . . . . . 546
E-32 The Maupertuis gauge and the Newton portrait of a geodesic . . . . . . . . . . . . . 550
E-33 A geodesic as a submanifold of the sphere (conformal coordinates) . . . . . . . . . . 551
E-34 A geodesic as a submanifold of the ellispoid-of-revolution (conformal coordinates) . 557
E-35 Maupertuis gauged geodesics (normal coordinates, local tangent plane) . . . . . . . 563
E-36 Maupertuis gauged geodesics (Lie series, Hamilton portrait) . . . . . . . . . . . . . . 565
F
Mixed cylindric map projections ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 569
Mixed cylindric map projections of the ellipsoid-of-revolution, Lambert/Sanson–Flamsteed projections 569
F-1 Pseudo-cylindrical mapping: biaxial ellipsoid onto plane . . . . . . . . . . . . . . . . . . . . . 570
F-2 Mixed equiareal cylindric mapping: biaxial ellipsoid onto plane . . . . . . . . . . . . . . . . 572
F-3 Deformation analysis of vertically/horizontally averaged equiareal cylindric mappings . . . 579
G Generalized Mollweide projection ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 589
Generalized Mollweide projection of the ellipsoid-of-revolution
589
G-1 The pseudo-cylindrical mapping of the biaxial ellipsoid onto the plane . . . . . . . . . . . . 589
G-2 The generalized Mollweide projections for the biaxial ellipsoid . . . . . . . . . . . . . . . . . 593
G-3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
H Generalized Hammer projection ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 601
Generalized Hammer projection of the ellipsoid-of-revolution: azimuthal, transverse, resolved equiareal 601
H-1 The transverse equiareal projection of the biaxial ellipsoid . . . . . . . . . . . . . . . . . . . 602
H-11 The transverse reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
H-12 The equiareal mapping of the biaxial ellipsoid onto a transverse tangent plane . . 605
H-13 The equiareal mapping in terms of ellipsoidal longitude, ellipsoidal latitude . . . . . 607
H-2 The ellipsoidal Hammer projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
H-21 The equiareal mapping from a left biaxial ellipsoid to a right biaxial ellipsoid . . . 610
H-22 The explicit form of the mapping equations generating an equiareal map . . . . . . 611
H-3 An integration formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
H-4 The transformation of the radial function r(A
∗ , B
∗ ) into r(Λ
∗ , Φ
∗ ) . . . . . . . . . . . . . . 618
H-5 The inverse of a special univariate homogeneous polynomial . . . . . . . . . . . . . . . . . . 619