262 9 “Ellipsoid-of-revolution to sphere and from sphere to plane”
The curvature tensors of the ellipsoid-of-revolution and the sphere, namely the Gauss curvature scalar
and the trace as the alternative curvature scalar, are presented in Box 9.4. Note that N and M are the
radii of principal type of the ellipsoid-of-revolution and that r is the curvature radius of the sphere.
Box 9.4 (Curvature tensors, Gauss’s curvature tensors ).
Curvature tensor
(ellipsoid-of-revolution):
Curvature tensor
(sphere):
Grad G 3 =
g r a d g 3 =
= −HG
−1
"
G 1
G 2
#
= K
"
G 1
G 2
#
.
= −hg
−1
"
g 1
g 2
#
= k
"
g 1
g 2
#
.
(9.26)
Gauss’s curvature tensor
(ellipsoid-of-revolution):
Gauss’s curvature tensor
(sphere):
K := −HG
−1 ,
k := hg
−1 ,
K := −HG
−1 =
"
1/N
0
0 1/M
#
,
k := −hg
−1 =
"
1/r 0
0 1/r
#
.
N :=
A 1
(1 − E 2 sin
2 Φ) 1/2 ,
M :=
A 1 (1 − E
2 )
(1 − E 2 sin
2 Φ) 3/2 .
(9.27)
Eigenvalues of the
curvature tensor:
Eigenvalues of the
curvature tensor:
K 1 :=
1
N
, K 2 :=
1
M
,
κ 1 = κ 2 =
1
r
,
Grad G 3 =
G r a d g 3 =
= K
"
G 1
G 2
#
=
"
K 1 G 1
K 2 G 2
#
.
= k
"
g 1
g 2
#
=
"
κ 1 g 1
κ 2 g 2
#
.
(9.28)
Tangent space:
Tangent space:
G 1 :=
∂X
∂Λ
, G 2 :=
∂X
∂Φ
.
g 1 :=
∂x
∂λ
, g 2 :=
∂x
∂φ
.
(9.29)