1-10 Two examples: Mercator Projection and Stereographic Projection 55
Example 1.11 (Conformal mapping of an ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
to a sphere S
2
r : the Universal
Mercator Projection (UMP) of type left E
2
A 1 ,A 1 ,A 2
and right S
2
r , the special Korn–Lichtenstein equations,
and the Cauchy–Riemann equations (d’Alembert–Euler equations)).
Let us assume that we have found a solution of the left Korn–Lichtenstein equations of the ellipsoidof-revolution E
2
A 1 ,A 1 ,A 2
parameterized by the two coordinates {Λ, Φ} which conventionally are called
{Gauss surface normal longitude, Gauss surface normal latitude}. Similarly, let us depart from a solution of the right Korn–Lichtenstein equations of the sphere S
2
r parameterized by the two coordinates
{λ, φ} which are called {spherical longitude, spherical latitude}. Here, we follow the commutative diagram of Fig. 1.25 and identify the left conformal coordinates {P, Q} with the Universal Mercator
Projection (UMP) of E
2
A 1 ,A 1 ,A 2
, and the right conformal coordinates {p, q} with the Universal Mercator Projection (UMP) of S
2
r , which is outlined in Box 1.24. The ratios Q/A 1 and q/r are also called
{ellipsoidal isometric latitude, spherical isometric latitude} or ellipsoidal spherical Lambert functions
Q = A 1 lamΦ and q = r lamφ, respectively. In addition, we adopt the left and right matrices of the
metric {G l , G r } of Example 1.3.
End of Example.
We pose four problems. (i) Do the left and right conformal maps that are parameterized by
{P (Λ), Q(Φ)} and {p(λ), q(φ)} as “UMP left” and “UMP right” fulfil the Korn–Lichtenstein equations, the integrability conditions (vector-valued Laplace–Beltrami equations of harmonicity), and the
condition “orientation preserving conformeomorphism”? (ii) Derive the left and right factors of conformality, Λ
2 = Λ
2
1 = Λ
2
2 and λ
2 = λ
2
1 = λ
2
2 . Do the factors of conformality fulfill a special Helmholtz
equation? (iii) Prove that under “UMP left” as well as “UMP right” both the equators of E
2
A 1 ,A 1 ,A 2
and
S
2
r are mapped equidistantly. Interpret this result as a boundary condition of the Korn–Lichtenstein
equations. (iv) Derive a “simple conformal mapping” E
2
A 1 ,A 1 ,A 2
→ S
2
r .
Fig. 1.25. Universal Mercator Projection (UMP) of the sphere S
2
r with shorelines and Tissot ellipses of
distortion. Graticule: 30
◦ in longitude, 15
◦ in latitude. Domain: {−180
◦ < Λ ≤ +180
◦ , −80
◦ < Φ < +80
◦ }.
Example 1.11 (Conformal mapping of an ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
to a sphere S
2
r : the Universal
Mercator Projection (UMP) of type left E
2
A 1 ,A 1 ,A 2
and right S
2
r , the special Korn–Lichtenstein equations,
and the Cauchy–Riemann equations (d’Alembert–Euler equations)).
Let us assume that we have found a solution of the left Korn–Lichtenstein equations of the ellipsoidof-revolution E
2
A 1 ,A 1 ,A 2
parameterized by the two coordinates {Λ, Φ} which conventionally are called
{Gauss surface normal longitude, Gauss surface normal latitude}. Similarly, let us depart from a solution of the right Korn–Lichtenstein equations of the sphere S
2
r parameterized by the two coordinates
{λ, φ} which are called {spherical longitude, spherical latitude}. Here, we follow the commutative diagram of Fig. 1.25 and identify the left conformal coordinates {P, Q} with the Universal Mercator
Projection (UMP) of E
2
A 1 ,A 1 ,A 2
, and the right conformal coordinates {p, q} with the Universal Mercator Projection (UMP) of S
2
r , which is outlined in Box 1.24. The ratios Q/A 1 and q/r are also called
{ellipsoidal isometric latitude, spherical isometric latitude} or ellipsoidal spherical Lambert functions
Q = A 1 lamΦ and q = r lamφ, respectively. In addition, we adopt the left and right matrices of the
metric {G l , G r } of Example 1.3.
End of Example.
We pose four problems. (i) Do the left and right conformal maps that are parameterized by
{P (Λ), Q(Φ)} and {p(λ), q(φ)} as “UMP left” and “UMP right” fulfil the Korn–Lichtenstein equations, the integrability conditions (vector-valued Laplace–Beltrami equations of harmonicity), and the
condition “orientation preserving conformeomorphism”? (ii) Derive the left and right factors of conformality, Λ
2 = Λ
2
1 = Λ
2
2 and λ
2 = λ
2
1 = λ
2
2 . Do the factors of conformality fulfill a special Helmholtz
equation? (iii) Prove that under “UMP left” as well as “UMP right” both the equators of E
2
A 1 ,A 1 ,A 2
and
S
2
r are mapped equidistantly. Interpret this result as a boundary condition of the Korn–Lichtenstein
equations. (iv) Derive a “simple conformal mapping” E
2
A 1 ,A 1 ,A 2
→ S
2
r .
Fig. 1.25. Universal Mercator Projection (UMP) of the sphere S
2
r with shorelines and Tissot ellipses of
distortion. Graticule: 30
◦ in longitude, 15
◦ in latitude. Domain: {−180
◦ < Λ ≤ +180
◦ , −80
◦ < Φ < +80
◦ }.