1-5 One example: orthogonal map projection 35
e1
e2
√ κ 1
√ κ 2
r
y
x
O
p
Fig. 1.17. Orthogonal projection S
2
R + onto P
2
O , degenerate Euler–Lagrange ellipse/hyperbola.
Box 1.19 (Orthogonal projection S
2
R + onto P
2
O , 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 ,
C r (r) =
2
4
r
2
0
0
R
2
R 2 − r 2
3
5 , G r (r) =
»
r
2
0
0 1
–
.
(1.126)
Right Euler–Lagrange matrix in polar coordinates:
2E r = G r − C r , E r =
1
2
2
4
0
0
0 −
r
2
R 2 − r 2
3
5 .
(1.127)
Right eigenvalues:
2κ i = λ
2
i − 1 ∀ i ∈ {1, 2} , λ
2
1 =
R
2
R 2 − r 2 , λ
2
2 = 1 , κ 1 =
1
2
r
2
R 2 − r 2 > 0 , κ 2 = 0 .
(1.128)
Right Euler–Lagrange tensor:
E r = −
1
2
g 2 ⊗ g 2
r
2
R 2 − r 2 .
(1.129)
Box 1.20 (Orthogonal projection S
2
R + onto P
2
O , polar coordinates, the transformations from the right Euler–
Lagrange matrix to the left Euler–Lagrange matrix).
E r → E l :
E l = J
∗
l E r J l , r
2 = R
2 cos
2 Φ ,
J l =
» 1
0
0 −
p
R 2 − r 2
–
, E l =
»
0
0
0 −r
2 /2
–
= −
1
2
R
2
»
0
0
0 cos
2 Φ
–
.
(1.130)
Left eigenvalues:
2K 1 = Λ
2
1 − 1 =
1
λ
2
1
− 1 , 2K 2 = Λ
2
2 − 1 =
1
λ
2
2
− 1 , K 1 = −
1
2
cos
2 Φ , K 2 = 0 .
(1.131)