Geography Reference
In-Depth Information
1-5 One Example: Orthogonal Map Projection
One example of deformation analysis (Euler-Lagrange deformation tensor, its eigenspace,
ellipses and hyperbolae of distortion), orthogonal map projection, Hammer equiareal modified
azimuthal projection.
The general eigenspace analysis of the Euler-Lagrange deformation tensor analysis visualized by
ellipses and hyperbolae of distortion is close to the heart of any map projection. It is for this reason
that we present to you as Example 1.7 the orthogonal projection of the northern hemisphere onto
the equatorial plane. We recommend to go through all details with “paper and pencil”.
Example 1.7 (Orthogonal projection of points of the sphere onto the equatorial plane through
the origin).
Let us assume that we make an orthogonal projection of points of the northern hemisphere onto
the equatorial plane
2
O
2
R + . For an illustration of such a
map projection let us refer to Fig. 1.10 . The direct mapping and inverse mapping equations are
given by
P
through the origin
O
of the plane
S
x = X = R cos Φ cos Λ,
Λ ( x, y ) = arctan ( y/x ) ,
versus cos Φ ( x, y )= x 2 + y 2
R
(1.121)
y = Y = R cos Φ sin Λ,
,
α = Λ, r = X 2 + Y 2 = R cos Φ,
Λ = α, cos Φ = r/R.
End of Example.
2 .We
pose two problems. (i) Derive the right Euler-Lagrange deformation tensor. (ii) Solve the right
general eigenvalue-eigenvector problem.
We take advantage of Cartesian coordinates
{
x, y
}
and polar coordinates
{
α,r
}
to cover
R
Solution (the first and the second problem).
We solve the two problems side-by-side in Box 1.18 in Cartesian coordinates and in Box 1.19 in
polar coordinates. Given the right Euler-Lagrange matrix, by means of Box 1.20 ,wearegiving
the transform to the left Euler-Lagrange matrix.
First, we transform the right Cauchy-Green matrix from Boxes 1.13 to 1.18 . Second, we take
advantage of Corollary 1.9 in order to compute the right Euler-Lagrange matrix 2E r =I 2
C r
in Cartesian coordinates as well as its right eigenvalues 2 κ i = λ i
1 from given right eigenvalues
λ i . In particular, we find κ 1
=0and κ 2 = 0. Third, we represent the right Euler-Lagrange
tensor in the Cartesian base e 1
e 1 , e 1
e 2 ,and e 2
e 2 ,where
denotes the symmetric product.
Fourth, in contrast, we transfer the right Cauchy-Green matrix from Boxes 1.14 to 1.19 . Fifth,
we again use Corollary 1.9 in order to compute the right Euler-Lagrange matrix 2E r =G r
C r
(G r =diag( r 2 , 1)) in polar coordinates as well as its right eigenvalues 2 κ i = λ i
1. Again, we find
κ 1
=0and κ 2 = 0. Sixth, we represent the right Euler-Lagrange tensor in the polar base g 2
g 2 .
Seventh, Box 1.20 reviews the transformations of the right Euler-Lagrange tensor E r to the
left Euler-Lagrange tensor E l by means of the left Jacobi matrix J l transferred from Box 1.14
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