Geography Reference
In-Depth Information
The
su
cient condition
(
x,y
)
∂
2
L
∂x
2
∂
2
L
∂x∂y
H
2
:=
>
0
∂
2
L
∂y∂x
∂
2
L
∂y
2
or
H
2
=2
1+
λ
0
>
0
(23.153)
01+
λ
leads to the 2
2
Hesse matrix
of second derivatives with respect to (
x
,
y
)atthepoint(
x, y
)
to be
positive-definite.
Indeed the diagonal form of
H
2
allows positive eigenvalues for 1 +
λ
=
×
+
r
−
0
(X
y
0
)
2
1
/
2
.
x
0
)
2
+(Y
−
−
End of Example.
In summary, we can make the following
Corollary 23.6 (minimal distance mapping of a point close to the circle).
Let (
X,Y
)
∈
R
2
be a point close to the circle
S
r
0
. The point (
x, y
)
∈
S
r
0
is at minimal Euclidean
distance
X
−
x
2
to (
X,Y
)
∈
R
2
if
(
x, y
)
∈{
X
−
x
2
2
,
(
x, y
)
∈
S
1
r
0
=min
|
(
X,Y
)
∈
R
(23.154)
and
r
0
(
X
−
x
0
)
(X
x
=
x
0
+
(23.155)
x
0
)
2
+(Y
y
0
)
2
−
−
r
0
(
Y − y
0
)
y
=
y
0
+
(X
(23.156)
x
0
)
2
+(Y
y
0
)
2
−
−
solves the optimization problem.
End of Corollary.
Example 23.4 (minimal distance mapping of a point close to the straight line
L
1
,
κ
0
=0
,κ
0
=0).
According to Fig.
23.36
we minimize the Euclidean distance
d
(
x,X
)=
X
−
x
of a given
point
X ∈
R
2
and an unknown point
x
∈
L
1
which is an element of the straight line
1
:=
x
∈
R
|y− y
0
=tan
α
0
(
x − x
0
)
2
L
passing the point (
x
0
,y
0
) with
α
0
as its orientation. Indeed by means of the
Lagrange multiplier
λ
the constrained optimal location of
x
∼
(
x
,
y
) is generated by
2
+2
λ
(
y
L
(
x, y, λ
):=
X
−
x
−
ax
−
b
)=min
x,y,λ
(23.157)
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