Geography Reference
In-Depth Information
Box J.1 (Projection heights in geometry space: minimal distance mapping).
Geodesics in geometry space:
d
d
t
x
k
x
2
+
y
2
+
z
2
=0
.
(J.1)
Minimal distance mapping:
x
(
u,v
)]
2
+[
Y
y
(
u,v
)]
2
+[
Z
Z
(
u,v
)]
2
.
S
:= [
X
−
−
−
(J.2)
Reference surface:
(i)
2
(plane)
,
(ii)
x
(
u,v
)
∈
P
2
x
(
u,v
)
∈
S
r
(sphere)
,
(J.3)
(iii)
2
A
1
,A
2
x
(
u,v
)
∈
E
(ellipsoid-of-revolution)
,
(iv)
A
1
,A
2
,A
3
x
(
u,v
)
(triaxial ellipsoid),
Heights in geometry space:
h
:=
∈
E
X
−
x
(
u, v
)
.
(J.4)
Box J.2 (Projection heights in geometry space: minimal distance mapping, formulae and
relations).
Stationary functional:
(i)
(
x
1
,x
2
,x
3
,x
4
):=
1
2
+
Λ
(
a
1
x
+
a
2
y
+
a
3
z
+
a
4
) =
L
2
X
−
x
(J.5)
=
1
x
1
)
2
+(
Y
x
2
)
2
+(
Z
x
3
)
2
]+
x
4
(
a
1
x
1
+
a
2
x
2
+
a
3
x
3
+
a
4
)
,
2
[(
X
−
−
−
(ii)
(
x
1
,x
2
,x
3
,x
4
):=
2
+
Λ
(
x
2
+
y
2
+
z
2
r
2
) =
L
X
−
x
−
(J.6)
=(
X − x
1
)
2
+(
Y − x
2
)
2
+(
Z − x
3
)
2
+
x
4
[(
x
1
)
2
+(
x
2
)
2
+(
x
3
)
2
− r
2
]
,
(iii)
2
+
Λ
(
x
1
)
2
+(
x
2
)
2
A
1
1
+
(
x
3
)
2
(
x
1
,x
2
,x
3
,x
4
); =
L
X
−
x
A
2
−
(J.7)
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