Geography Reference
In-Depth Information
or
L
(
x
1
,x
2
,x
3
,x
4
):=(
X − x
1
)
2
+(
Y − x
2
)
2
+(
Z − x
3
)
2
+
x
4
(
A
2
[(
x
1
)
2
+(
x
2
)
2
]+
A
1
(
x
3
)
2
A
1
A
2
)
,
−
(iv)
2
+
Λ
(
x
1
)
2
A
1
1
+
(
x
2
)
2
A
2
+
(
x
3
)
2
(
x
1
,x
2
,x
3
,x
4
):=
L
X
−
x
A
3
−
(J.8)
(
x
1
,x
2
,x
3
,x
4
):=
2
+
x
4
[
A
2
A
3
x
2
+
A
1
A
3
y
2
+
A
1
A
2
z
2
L
X
−
x
A
1
A
2
A
3
]
.
First variation:
−
∂L
∂x
μ
(
x
ν
)=0
∀ μ, ν ∈{
1
,
2
,
3
,
4
}
(J.9)
⇐⇒
(i) plane
2
:
P
x
1
)+
a
1
x
4
=0
,
x
2
)+
a
2
x
4
=0
,
x
3
)+
a
3
x
4
=0
,
−
(
X
−
−
(
Y
−
−
(
Z
−
(J.10)
a
1
x
1
+
a
2
x
2
+
a
3
x
3
+
a
4
=0;
(ii) sphere
S
2
r
:
−
(
X − x
1
)+
x
1
x
4
=0
, −
(
Y − x
2
)+
x
2
x
4
=0
, −
(
Z − x
3
)+
x
3
x
4
=0
,
(J.11)
r
2
=0;
(iii) ellipsoid-of-revolution
(
x
1
)
2
+(
x
2
)
2
+(
x
3
)
2
−
2
A
1
,A
2
E
:
x
1) +
A
2
x
1
x
4
=0
,
x
2
)+
A
2
x
2
x
4
=0
,
x
3
)
−
(
X
−
−
(
Y
−
−
(
Z
−
+
A
1
x
3
x
4
=0
,
(J.12)
A
2
[(
x
1
)
2
+(
x
2
)
2
]+
A
1
(
x
3
)
2
A
1
A
2
=0;
−
2
(iv) triaxial ellipsoid
E
A
1
,A
2
,A
2
:
x
1
)+
A
2
A
3
x
1
x
4
=0
,
x
2
)+
A
1
A
3
x
2
x
4
=0
,
x
3
)
−
(
X
−
−
(
Y
−
−
(
Z
−
+
A
1
A
2
x
3
x
4
=0
,
(J.13)
A
2
A
3
(
x
1
)
2
+
A
1
A
3
(
x
2
)
2
+
A
1
A
2
(
x
3
)
2
− A
1
A
2
A
3
=0
.
Second variation.
The second variation decides about the solution of type “minimum” or
“maximum” or “turning point”.
In our case
∂
2
1
2
L
∂x
k
∂x
l
(
x
γ
)
>
0
(J.14)
⇐⇒
⎡
⎤
+1 0 0
0+10
00+1
⎣
⎦
>
0;
2
:
(i) plane
P
(J.15)
Search WWH ::
Custom Search