From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 32

2 2

O

Box 1.16 (Orthogonal projection S

R + onto

P , polar coordinates, the second problem).

G r = r 2 0

,

C r according to Box 1.14 . (1.105)

01

Right general eigenvalue problem :

2 tr[C r G − r ]

λ 1 , 2 = λ 2 + ,− = 1

λ 2 G r | C r − | =0 ⇔

4det[C r G − r ] ,

(tr[C r G − r ]) 2

± −

G − r = r 2 0

, C r G − r = 10 ,

R 2
R 2 −r 2

01

0

tr[C r G − r ]= 2 R 2

r 2 R 2

R 2

−

− r 2 , det[C r G − r ]=

− r 2 ,

(1.106)

R 2

(tr[C r G − r ]) 2

r 2

− 4det[C r G − r ]=

r 2 ,

R 2 −

R 2
R 2

R

λ 1 = λ 2 + =

r 2 , 2 = λ 2

√ R 2

r 2 ,

1 = λ + =+ =1 ,

−

−

−

λ 2 = λ − =+1 .

Alternative solution, right general eigenvalue problem:

C r = diag r 2 ,

r 2 , G r =diag( r 2 , 1) ,

R 2

λ 2 G r | | C r − =0

R 2 −

⇔

λ 2

=

r 2 (1 λ 2 ) − 0

C 11 − λ 2 g 11 0

=0

R 2

0 C 22 − λ 2 g 22

0

R 2 −r 2 −

⇔ (1.107)

R 2
R 2

λ 2 g 11 = r 2 (1 λ 2 )=0 , λ 2 g 22 = λ 2 =0 C 11 − − 22 −

− r 2 −

⇒

R 2
R 2

R

λ 1 = λ 2 + =

r 2 , 2 = λ 2

√ R 2

r 2 , 1 = λ + =+

=1 ,

−

−

−

λ 2 = λ − =+1 .

Right eigencolumns:

f 11
f 21

= 1

0

= 1
r

1
0

, f 12 = 0

1

= 0 .

1
√ g 11

1
√ g 22

(1.108)

f 22 1