Geodesics - Map Projections: Cartographic Information Systems - page 720

= ⇒

D 1 = ( κ g + κ n ) D 1 +( κ g − κ n τ g ) D 2 +( κ g τ g + κ n ) D 3 ,

D 1 = G K U K ,
D 1 = G K,L U K U L + G K U K ,

−

(E.9)

D 1 = G K,LM U K U L U M + G K,L ( U K U L + U K U L )+ G K,L U K U L + G K U K ,
G K,LM =

= N G N + N

G N,M + H KL,M G 3 + H KL G 3 ,M ,

KL KL

,M

G K,LM =

(E.10)

H KL H MM 2 G M 2 K 2 G K 2 +

= −

+ N 1

M 2
N 1 M

G M 2 + N 1 H N 1 ,M G 3 + N ,M G N + H KL,M G 3 ,

KL KL KL

D 1 =

= U M +(2 U K U L + U K U L ) M

+ U K U L U M 2

KL

M
KL

,M 2 + M 1 M

M 1 M 2

− (E.11)

KL

H KL H M 1 M 2 G M 1 M )] G M +[(2 U K U L + U K U L ) H KL +

−

+ U K U L U M 2 M 1

H M 1 M 2 + H KL,M 2 G 3 .

KL

M
KL

= M , “symmetry” of Christoffel symbols leads to

LK

D 1 =

= U M +3 U K U L M

+ U K U L U N

KL

M
KL

,N + Q M

NQ

H KL H NQ G QM G M + − (E.12)

KL

+ 3 U K U L H KL + U K U L U N Q

H NQ + H KL,N D 3 .

KL

τ g then is obtained as
κ g τ g + κ n =3 U K U L H KL + U K U L U N ( Q

H NQ + H KL,N ) ,

KL

κ n = H KL,N U K U L U N +2 H KL U K U L
⇒
κ g τ g = H KL U K U L + Q

H NQ U K U L U N

(E.13)

KL

⇒