Geography Reference
In-Depth Information
8
Ellipsoid-of-Revolution to Tangential Plane
Mapping the ellipsoid-of-revolution to a tangential plane. Azimuthal projections in the nor-
mal aspect (polar aspect): equidistant, conformal, equiareal, and perspective mapping.
A
1
,A
2
:seeBox
8.1
.As
before, F (with elements
a, b, c, d
)istheFrobeniusmatrix,G=J
∗
J (with elements
e, f, g
)isthe
Gauss matrix, H = [
X
KL
|G
3
] (with elements
l, m, n
) is the Hesse matrix, J = [
∂X
J
/∂U
K
]is
the Jacobi matrix, and K =
−
HG
−
1
is the curvature matrix, finally leading to the mean curvature
h
=
First and foremost, let us consider the ID card of the
ellipsoid-of-revolution
E
−
tr[K]
/
2 and to the Gaussian curvature
k
= det[K].
2
A
1
,A
2
Box 8.1 (ID card of the ellipsoid-of-revolution
E
).
Surface normal ellipsoidal coordinates(1st chart:
Λ, Φ
):
2
/
3
(
X
2
+
Y
2
)
/A
1
+
Z
2
/A
2
{
Λ, Φ
}∈
E
{
Z
=
±
A
2
}
:=
{
X
∈
R
|
=1
,A
1
>A
2
,Z
=
±
A
2
}
,
X
:=
E
1
A
1
cos
Φ
cos
Λ
+
E
2
A
1
cos
Φ
sin
Λ
+
E
3
A
1
(1
−
E
2
)sin
Φ
1
1
1
,
(8.1)
E
2
sin
2
Φ
E
2
sin
2
Φ
E
2
sin
2
Φ
−
−
−
X
+ 180
◦
2
sgn
Y
sgn
X
+1
,
Λ
(
X
) = arctan
Y
1
2
sgn
Y
1
−
−
1
Z
√
X
2
+
Y
2
.
Φ
(
X
) = arctan
1
−
E
2
Matrices F
,
G
,
H
,
J
,
K
,
andI (elements
a, b, c, d
;
e. f, g
;
l, m, n
):
⎡
⎤
√
1
−
E
2
sin
2
Φ
A
1
cos
Φ
⎦
=
=
1
√
G
11
0
1
0
N
cos
Φ
0
0
⎣
F=
1
√
G
22
1
M
0
E
2
sin
2
Φ
)
3
/
2
A
1
(1
−E
2
)
(1
−
0
=
ab
2
×
2
,
∈
R
(8.2)
cd
Search WWH ::
Custom Search