1-3 Two examples: pseudo-cylindrical and orthogonal map projections 27
Box 1.15 (Orthogonal projection S
2
R + onto P
2
O , Cartesian coordinates, the second problem).
G r =
»
1 0
0 1
–
,
C r according to Box 1.13.
(1.99)
Right general eigenvalue problem:
˛
˛ C r − λ
2 G r
˛
˛ = 0 ⇔ λ
2
1,2 = λ
2
+,− =
1
2
„
tr
ˆ
C r G
−1
r
˜ ±
q `
tr
ˆ
C r G
−1
r
˜´ 2 − 4det
ˆ
C r G
−1
r
˜
«
,
C r =
1
R 2 − (x 2 + y 2 )
»
R
2 − y
2
xy
xy
R
2 − x
2
–
, G r = I 2 ,
tr
ˆ
C r G
−1
r
˜
= tr [C r ] =
2R
2 − (x
2 + y
2 )
R 2 − (x 2 + y 2 )
,
det
ˆ
C r G
−1
r
˜
= det [C r ] =
`
R
2 − x
2
´`
R
2 − y
2
´ − x
2 y
2
ˆ
R 2 − (x 2 + y 2 )
˜ 2
=
R
2
R 2 − (x 2 + y 2 )
,
`
tr
ˆ
C r G
−1
r
˜´ 2 − 4det
ˆ
C r G
−1
r
˜
=
(x
2 + y
2 )
2
ˆ
R 2 − (x 2 + y 2 )
˜ 2 ,
λ
2
1 = λ
2
+ =
R
2
R 2 − (x 2 + y 2 )
, λ 1 = λ + = +
R
p
R 2 − (x 2 + y 2 )
, λ
2
2 = λ
2
− = 1 , λ 2 = λ − = +1 .
(1.100)
Right eigencolumns:
(i)
√ :=
p
g 11 (C 22 − λ
2
1 g 22 ) 2 − 2g 12 (C 12 − λ
2
1 g 12 )(C 22 − λ
2
1 g 22 ) + g 22 (C 12 − λ
2
1 g 12 ) 2 =
=
q
(C 22 − λ
2
1 ) 2 + C
2
12 = x
p
x 2 + y 2
R 2 − (x 2 + y 2 )
,
"
f 11
f 21
#
=
1
√
"
C 22 − λ
2
1
−C 12
#
= −
1
p
x 2 + y 2
" x
y
#
;
(1.101)
(ii)
√ :=
p
g 11 (C 12 − λ
2
2 g 12 ) 2 − 2g 12 (C 11 − λ
2
2 g 11 )(C 12 − λ
2
2 g 12 ) + g 22 (C 11 − λ
2
2 g 11 ) 2 =
=
q
(C 11 − λ
2
2 ) 2 + C
2
12 = x
p
x 2 + y 2
R 2 − (x 2 + y 2 )
,
"
f 12
f 22
#
=
1
√
" −C 12
C 11 − λ
2
2
#
= +
1
p
x 2 + y 2
" −y
x
#
.
(1.102)
Right eigenvectors:
1st eigenvector: f 1 := e 1 f 11 + e 2 f 21 , f 1 (x, y) = −e 1
x
p
x 2 + y 2
− e 2
y
p
x 2 + y 2
;
2nd eigenvector: f 2 := e 1 f 12 + e 2 f 22 , f 2 (x, y) = −e 1
y
p
x 2 + y 2
+ e 2
x
p
x 2 + y 2
.
(1.103)
Notes:
˙
f 1 f 2
¸
= 0 ⇒
`
f 1 , f 2
´
= π/2 ,
‚
‚ f 1
‚
‚
2
=
‚
‚ f 2
‚
‚
2
= 1 .
(1.104)