Geodetic Mapping - Map Projections: Cartographic Information Systems - page 469

A 1 ,A 2

Box 20.1 (Surface geometry of E ).

Matrices
(Frobenius matrix F (elements a, b, c, d ) , Gauss matrix G = J T J ,
(elements e, f, g ) ,
Hesse matrix H = { X , KL , G 3 } (elements l, m, n ) , curvature matrix K ,
Jacobi matrix J = ∂X J /∂U K { } ):

⎡ ⎤

( √ 1 − E 2 sin 2 B )
A 1 cos B

0

⎣ ⎦ , F

F 1

2 × 2 ,

F= {

KL }

= ∈ R (20.13)

(1 − E 2 sin 2 B ) 3 / 2
A 1 (1 −E 2 )

0

= A 2 cos 2 B

, G

0

1 −E 2 sin 2 B

G 1

2 × 2 , G=J T J ,

G= {

KL }

∈ R (20.14)

(1 −E 2 ) 2
(1 −E 2 sin 2 B ) 3

A 1

0

⎡
⎣ −

⎤

√ 1 −E 2 sin 2 B

A 1 cos 2 B

0

⎦ , H

H 1

2 × 2 ,

H= {

KL }

= ∈ R (20.15)

E 2 )
(1 −E 2 sin 2 B ) 3 / 2

A 1 (1 −

0

⎡ ⎤

√ 1 − E 2 sin 2 B
A 1

0

⎣ ⎦ =

K 1 HG − 1 , K

2 × 2 ,

K= {

KL }

= − ∈ R (20.16)

(1 − E 2 sin 2 B ) 3 / 2
A 1 (1 −E 2 )

0

1

E 2 sin 2 B (2 E 2 (1 + sin 2 B )) tr[ K ]

2

− −

h = − = − , (20.17)

2 A 1 (1 − E 2 )

E 2 sin 2 B ) 2
A 1 (1

k =det[ K ]= (1

−

, (20.18)

− E 2 )

⎡ ⎤

A 1 (1 − E 2 )sin B cos L
(1 −E 2 sin 2 B ) 3 / 2

√ 1 −E 2 sin 2 B −

A 1 cos B sin L

−

⎣ ⎦

= ∂ ( X,Y )

A 1 (1 − E 2 )sin B sin L
(1 −E 2 sin 2 B ) 3 / 2

J 1

√ 1 −E 2 sin 2 B

A 1 cos B cos L 3 × 2 .

J= {

KL }

∂ ( L, B ) =

− , J ∈ R (20.19)

+ A 1 (1 − E 2 )cos B 0

(1 −E 2 sin 2 B ) 3 / 2

Eigenvalues:
1st eigenvalue of K: κ 1 = 1 − E 2 sin 2 B/A 1 ;
κ − 1 =: N ( B )= A 1 / 1

E 2 sin 2 B (1st curvature radius);
2nd eigenvalue of K: κ 2 =(1

−

E 2 sin 2 B ) 3 / 2 /A 1 (1 E 2 );

− −