Geography Reference
In-Depth Information
Box 9.4 (Curvature tensors, Gauss's curvature tensors).
Curvature tensor
Curvature tensor
(ellipsoid-of-revolution):
(sphere):
(9.26)
Grad G 3
grad g 3
= HG 1 G 1
=K G 1
.
= hg 1 g 1
=k g 1
.
G 2
G 2
g 2
g 2
Gauss's curvature tensor
Gauss's curvature tensor
(ellipsoid-of-revolution):
(sphere):
(9.27)
HG 1
hg 1
K:=
k:=
HG 1 = 1 /N
,
hg 1 = 1 /r
.
0
1 /M
0
1 /r
K:=
k:=
0
0
A 1
N :=
(1 −E 2 sin 2 Φ ) 1 / 2 ,
A 1 (1 −E 2 )
M :=
(1 −E 2 sin 2 Φ ) 3 / 2 .
Eigenvalues of the
Eigenvalues of the
curvature tensor:
curvature tensor:
K 1 := N ,K 2 := M ,
κ 1 = κ 2 = r ,
Grad G 3 =
Grad g 3 =
=K G 1
= K 1 G 1
.
=k g 1
= κ 1 g 1
.
(9.28)
G 2
K 2 G 2
g 2
κ 2 g 2
Tangent space:
Tangent space:
G 1 := ∂Λ ,G 2 := ∂Φ .
g 1 := ∂x
∂λ ,g 2 := ∂x
∂φ ..
(9.29)
 
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