108 2 From Riemann manifolds to Euclidean manifolds
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
−1
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
−1
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 . We have 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
−1
l
= 1. By means of implicit differentation of the generalized Kepler equation, we compute
(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
−1
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
−1
l
= 1,
accounts for computing the first left invariant tr
C l G
−1
l
. Indeed, a simple form of such an invariant
is not available. Accordingly, we left tr
C l G
−1
l
with a formula for (t
)
2 and 1/G 11 G 22 , respectively.
End of Solution (the second problem).
Fig. 2.1. Mollweide projection of an ellipsoid-of-revolution, E. Grafarend and A. Heidenreich (1995).
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
−1
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
−1
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 . We have 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
−1
l
= 1. By means of implicit differentation of the generalized Kepler equation, we compute
(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
−1
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
−1
l
= 1,
accounts for computing the first left invariant tr
C l G
−1
l
. Indeed, a simple form of such an invariant
is not available. Accordingly, we left tr
C l G
−1
l
with a formula for (t
)
2 and 1/G 11 G 22 , respectively.
End of Solution (the second problem).
Fig. 2.1. Mollweide projection of an ellipsoid-of-revolution, E. Grafarend and A. Heidenreich (1995).