Geodetic Mapping - Map Projections: Cartographic Information Systems - page 470

κ − 2 =: M ( B )= A 1 (1 E 2 sin 2 B ) 3 / 2 (2nd curvature radius) .

Christoffel symbols M

E 2 ) / (1

− −

:

KL

1
11

( L, B )= 1 ( L, B )= 2 ( L, B )=0 , 1

12

( L, B )

22 12

− −

tan B (1 − E 2 )

= , (20.20)

E 2 sin 2 B

1 −

2
11

( L, B )= sin B cos B (1

, 2

( L, B )

E 2 sin 2 B ) −

22

1 − E 2

=3 E 2 sin B cos B (1 E 2 sin 2 B ) .

−

Box 20.2 (Surface geometry of E

A 1 ,A 2 ).

Matrices
(Frobenius matrix F (elements a, b, c, d ) , Gauss matrixG = J T J ,
(elements e, f, g ) ,

Hesse matrix H = { (elements l, m, n ) , curvature matrix K ,

Jacobi matrixJ =

X ,KL , G 3 }

∂X J /∂U K { } ):

E 2 sin 2 U cos 2 V ) 1 / 2

KL } = (1

−

F= {F 2

sin 2 U cos 2 V ) 1 / 2 A 1 (1 −

− sin U sin V
cos V

,

E 2 sin 2 U cos 2 V )
(1 −E 2 )cos V

cos U (1 −

(20.21)

− sin U sin V (1 − E 2 sin 2 U cos 2 V )
(1 −E 2 )

− cos U

G 2

G= {

KL }

=

= G 11 G 12

,

G 21 G 22

⎧
⎨

G 11 = A 1 cos 2 V (1 − 2 E 2 (1 − sin 2 U sin 2 V )+ E 4 (1 − sin 2 U sin 2 V (1+sin 2 U cos 2 V ))) ,

(1 −E 2 sin 2 U cos 2 V ) 3

G 12 = G 21 = A 1 E 2 cos U sin U cos V sin V (2 − E 2 (1+cos 2 V sin 2 U )) , , (20.22)

E 2 sin 2 U cos 2 V ) 3

⎩

(1 −

G 22 = A 1 (1 − 2 E 2 sin 2 U + E 4 sin 2 U (1 − cos 2 V cos 2 U )) ,

(1 −E 2 sin 2 cos 2 V ) 3

= − A 1 cos 2 V (1 − E 2 (1 − sin 2 U sin 2 V))

,

− A 1 E 2 sin U cos U sin V cos V
(1 −E 2 sin 2 U cos 2 V ) 3 / 2 (1 −E 2 sin 2 U cos 2 V ) 3 / 2

H 2 H= {

KL }

(20.23)

E 2 sin 2 U )
(1 −E 2 sin 2 U cos 2 V ) 3 / 2

− A 1 (1 −

symmetric