Geography Reference
In-Depth Information
Example 23.5 (minimal distance mapping of a point close to the clothoid, polygon approximation
of the clothoid).
According to Fig.
23.41
we minimize the Euclidean distance
d
(
x
,
X
)=
X
−
x
of a given
2
outside or inside a clothoid to an unknown point (
x, y
)
point
X
C
is approximated by a polygon. For instance, let (
x
1
,y
1
)and(
x
2
,y
2
) be two points of a clothoid,
namely
∈
R
∈
C
where the clothoid
y
1
=
J
x
0
)
j
versus
J
x
0
)
j
=
y
2
.
j
=0
c
j
(
x
1
−
j
=0
c
j
(
x
2
−
1
passing through these two points is oriented according to
A straight line
L
y
2
−
y
1
=tan
α
12
(x
2
−
x
1
)
or
tanα
12
=
y
2
−
y
1
x
1
.
(23.186)
x
2
−
Accordingly the analytical form of
L
1
(x
1
−
x
2
)may be chosen as
y
−
y
1
=
tan
α
12
(
x
2
−
x
1
)
subject to
dy
dx
=
tan α
12
The minimal distance mapping a point (X, Y
∈
R
2
to (
x, y
)
∈
L
1
(x
1
−
x
2
) is materialized by
the Lagrangean
L
(
x, y
(
x
)) =
1
2
(
X − x
)
2
+
1
2
(
Y − y
(
x
))
2
dL
dx
(
x
)=0
y
(
x
))
dy
↔−
(
X
−
x
)
−
(
Y
−
dx
(
x
)=0
↔ x − X − Y
dy
dx
(
x
)+
y
(
x
)
dy
dx
(
x
)=0
↔
x
−
X
−
Y tan α
12
+
tanα
12
(
y
1
−
tan α
12
x
1
+
tan α
12
x
)=0
x
1+
tan
2
α
12
=
X
+
Y tan α
12
y
1
tan α
12
+
x
1
tan
2
α
12
↔
−
x
=
Xcos
2
α
12
+
Ysinα
12
cos
α
12
y
1
sinα
12
cosα
12
+
x
1
sin
2
α
12
↔
−
d
2
L
dx
2
(
x
)
>
0
1
cos
2
α
12
>
0
1+tan
2
α
12
=
↔
End of Example.
The result will be presented in
Corollary 23.9 (minimal distance mapping of a point close to the clothoid, polygon approxima-
tion of the clothoid).
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