From Riemann Manifolds to Euclidean Manifolds - Map Projections: Cartographic Information Systems - page 122

Here, we only have specialized Theorem 1.14 to M

r := { R 2 ,δ μν } =: E 2 . One of the most popular

equiareal mappings E

2 ,δ μν } is the Mollweide projection of the ellipsoid-of-revolution
to the plane, which is presented in Example 2.2 and is illustrated in Fig. 2.1 .

A 1 ,A 1 ,A 2 →{ R

Example 2.2 (Mollweide projection of the ellipsoid-of-revolution, with reference to Grafarend
et al. ( 1995a )).

Let us assume that we have found a solution of the right characteristic equation, which gen-
erates an equiareal mapping of the ellipsoid-of-revolution

2
A 1 ,A 1 ,A 2

E parameterized by the two

coordinates )as

outlined in Box 2.3 , also called generalized Mollweide projection . Such a generalized Mollweide
projection is classified as “pseudo-cylindric” and equiareal, mapping the circular equator equidis-
tantly. Its mapping equations x ( Λ, Φ )and y ( Φ ), where

{ Λ, Φ } (called { Gauss surface normal longitude,Gauss surface normal latitude }

{ x, y } are Cartesian coordinates that

2 , depend on cos t ( Φ )andsin t ( Φ ). The auxiliary function t ( Φ )isasolution
of the generalized Kepler equation since for relative eccentricity E 2 =( A 1 −

2 ,δ μν

cover { R } = E

A 2 ) /A 1 → 0 the gen-

eralized Kepler equations reduces to the Kepler equation. Such a Kepler equation is known from
the classical Mollweide projection of the sphere or from solving the Kepler two-body problem in
mechanics.

End of Example.

We pose two problems. (i) Prove that the generalized Mollweide projection of the ellipsoid-of-
revolution is equiareal. For this purpose, observe the postulate det [ C l G − l ] = 1. (ii) Determine the
left principal stretches Λ 1 and Λ 2 by setting up the characteristic equations of the left eigenvalue
problem that is presented in Box 2.4 .

Solution (the first problem).

Here, we set up the test of an equiareal mapping to be based upon the postulate det [C l G − l ]=1.
First, by means of Box 2.5 , we compute the left Jacobi matrix substituted by D Λ x, D Φ x, D Λ y ,and
D Φ y . Second, we set up the left Cauchy-Green matrix C l =J ∗ G r J l subject to G r =I 2 .Wehave
to emphasize that C l is not a diagonal matrix. Third, we adopt the left matrix of the metric G l .
Fourth, given the left Cauchy-Green matrix, C l , and the left matrix of the metric, G l , we derive
the determinantal identity det [C l G − l ] = 1. By means of implicit differentiation of the generalized
Kepler equation ,wecompute( t ) , ( t ) 2 , ( t ) 2 cos 4 t, a 2 b 2 and 1 /G 11 G 22 in step five. Sixth, taking
all individual terms into one, we have proven det[C l G − l ]=1.

End of Solution (the first problem).

Solution (the second problem).

First, we set up the characteristic equations of the left general eigenvalue problem of Box 2.4 in
order to compute the left principal stretches Λ 1 and Λ 2 , respectively. Second, the solution of the left
characteristic equation subject to the condition of an equiareal mapping, namely det [C l G − l ]=1,
accounts for computing the first left invariant tr [C l G − l ]. Indeed, a simple form of such an invariant
is not available. Accordingly, we left tr[C l G − l ] with a formula for ( t ) 2 and 1 /G 11 G 22 , respectively.

End of Solution (the second problem).