Geography Reference
In-Depth Information
S
r
0
,
κ
0
=1
/r
0
,κ
0
=0
Fig. 23.35.
Minimal distance mapping of a point close to the circle
where
a
:=
tan α
0
,b
:=
y
0
− x
0
tan α
0
.The
necessary condition
∂x
x, y,λ
=0
∂L
aλ
= 0
↔−
(
X
−
x
)
−
(23.158)
x, y,λ
=0
↔−
(
Y − y
)
−
∂L
∂x
λ
= 0
(23.159)
x, y,λ
=0
∂L
∂λ
↔−
ax
+
y
=
b
(23.160)
leads to linear normal equations which are solved as following
x
+
ay
=
X
+
aY
−
λ
=
Y
−
y
→
x
+
a
−
1
y
=
a
−
1
b
a
y
+
a
−
1
y
=
X
+
aY
+
a
−
1
b, a
+
a
−
1
=(1+
a
2
)
/a
y
=
a
X
+
aY
+
a
−
1
b
a
2
a
b
1+
a
2
→
=
1+
a
2
X
+
1+
a
2
Y
+
1+
a
2
a
2
x
=
X
+
aY − ay
=
X
+
aY −
1+
a
2
(
X
+
aY
+
a
−
1
b
)
X
a
ab
1+
a
2
→
x
=
1+
a
2
+
1+
a
2
Y
−
x
y
=
1
a
aa
2
X
+
−
ab
b
1
1+
a
2
1
1+
a
2
Y
The
su
cient condition
H
2
:=
∂
2
L
∂
2
L
∂x∂y
∂x
2
>
0
∂
2
L
∂y∂x
∂
2
L
∂y
2
(
x,y
)
or
H
2
=2
I
2
>
0
(23.161)
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