From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 30

D α Λ =1 ,D r Λ =0 , (1.96)

1 1

R =

1
√ R 2

D α Φ =0 ,D r Φ = − 1 −

− r 2 .

− r 2 /R 2

Right Cauchy-Green matrix:
C r := J r G l J r = r 2

,

0

R 2
R 2 −r 2

0

(1.97)

G l = R 2 cos 2 Φ 0

= r 2

0 R 2 .

0

01

Right Cauchy-Green tensor:
x ( α,r )= e 1 r cos α + e 2 r sin α,

g 1 := D α x = − e 1 r sin α + e 2 r cos α, g 2 := D r x =+ e 1 cos α + e 2 sin α,

= r 2 ,g 12 := g 11 := g 1 | g 1 g 1 | g 2 =0 ,g 22 := g 2 | g 2 =1 ,

G r = r 2 0

,

01

(d s ) 2 = r 2 (d α ) 2 +(d r ) 2 ,

(1.98)

2

g 11 g 1 = 1

1 1

g 22 g 2 = g 2 ,

g μ = g μν g ν , g 1 =

r 2 g 1 , g 2 =

ν =1

2

g μ g ν C μν = C r = ⊗

μ,ν =1

g 2 R 2
R 2

R 2
R 2

= g 1 g 1 r 2 + g 2

⊗ ⊗

r 2 = g 1 ⊗

g 1 1+ g 2 ⊗ g 2

r 2 . − −

2 2

O

Box 1.15 (Orthogonal projection S

R + onto

P , Cartesian coordinates, the second problem).

G r = 10

, C r according to Box 1.13 .

(1.99)

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

± −

R 2

x 2 , G r =I 2 ,

1

y 2

− xy

C r =

R 2

xy −

R 2 − ( x 2 + y 2 )