From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 39

by J l =J − r . Eighth, we have computed the left eigenvalues of the left Euler-Lagrange defor-
mation tensor. The degenerate distortion ellipse/hyperbola of the right Euler-Lagrange matrix
is finally illustrated by Fig. 1.17 .

End of Solution (the first and the second problem).

2 2

O

Box 1.18 (Orthogonal projection S

R + onto

P , Cartesian coordinates, the first problem and

the second problem).

Right Cauchy-Green matrix in Cartesian coordinates:

R 2

x 2 .

1

y 2

− xy

C r = (1.122)

R 2 xy −

R 2 − ( x 2 + y 2 )

Right Euler-Lagrange matrix in Cartesian coordinates:

x 2 xy

xy y 2 .

1
2

1

2E r =I 2 − C r , E r = − (1.123)

R 2 − ( x 2 + y 2 )

Right eigenvalues:
2 κ i = λ i − 1 ∀i ∈{ 1 , 2 },

R 2 x 2 + y 2

( x 2 + y 2 ) ,λ 2 =1 ,κ 1 = 1

λ 1 =

( x 2 + y 2 ) ,κ 2 =0 .

(1.124)

R 2 − 2 R 2 −

Right Euler-Lagrange tensor:
E r =

x 2 xy

1
2 e 1

1
2 ( e 2

= − ⊗ e 1

( x 2 + y 2 ) −

⊗ e 2 + e 2 ⊗ e 1 )

R 2 − R 2 − ( x 2 + y 2 )

1
2 e 2 ⊗ e 2

y 2

−

( x 2 + y 2 ) = R 2 −

x 2

1
2 e 1 ∨

xy

= − e 1

( x 2 + y 2 ) −

e 1 ∨ e 2

R 2 − R 2 − ( x 2 + y 2 )

y 2

1
2 e 2

− ∨ e 2 (1.125)

R 2 − ( x 2 + y 2 )

subject to

e ν := 1

e μ ∨

2 ( e μ

⊗ e ν ⊗ e ν ⊗ e μ ) .

2 2

O

Box 1.19 (Orthogonal projection S

R + onto

P , polar coordinates, the first and the second

problem).

Right Cauchy-Green matrix in polar coordinates:
r 2 = x 2 + y 2 = X 2 + Y 2 = R 2 cos 2 Φ,