Geography Reference
In-Depth Information
r
0
,bymeansof(
23.155
), (
23.156
)wemayuse(
0
x,
0
y
) as a zero order approximation of
(
x, y
)
∈
C
.With(
0
x,
0
y
) as input we start immediately the Newton iteration up to a fixed point.
For easy reference we collect the result in
(
x, y
)
∈
S
Lemma 23.3 (minimal distance mapping of a point close to the clothoid).
2
Be a point close to the clothoid
C
. The point (
x, y
)
∈
C
is at minimal Euclidean
distance to
X
−
x
2
Let (
X,Y
)
∈
R
to (
X,Y
)
∈
R
2
if
(
x, y
)
∈
X
−
x
2
2
,
(
x, y
)
∈
C
=min
|
(
X,Y
)
∈
R
(23.164)
The Lagrangean 2
L
(
x, y
(
x
)) := (
X − x
)
2
+(
Y − y
)
2
subject to
y=
J
x
0
)
j
j
=0
c
j
(
x
−
(23.165)
with the clothoidal coecients
c
j
given by Table
23.6
is minimal, if
(
i
)
the necessary condition
L
x
:=
∂L
∂x
(
x, y
(
x
)) = 0 or
x
0
)
2
+
f
3
(
x
x
0
)
3
+
f
4
(
x
x
0
)
4
f
(
x
−
x
0
)=
f
0
+
f
1
(
x
−
x
0
)+
f
2
(
x
−
−
−
x
0
)
5
+
f
6
(
x
x
0
)
6
+
f
7
(
x
x
0
)
7
+
+
f
5
(
x
−
−
−
(23.166)
O
(
x
x
0
)
8
=0
−
with coe
cients
f
n
givenuptoorder8inTable
23.7
and
(
ii
)
the su
ciency condition
L
xx
:=
∂
2
L
∂x
2
(
x, y
(
x
))
>
0or
x
0
)
2
+4
f
4
(
x
x
0
)
3
+5
f
5
(
x
x
0
)
4
+
f
(
x
−
x
0
)=
f
1
+2
f
2
(
x
−
x
0
)+3
f
3
(
x
−
−
−
6
f
6
(
x − x
0
)
5
+7
f
7
(
x − x
0
)
6
+
O
(
x − x
0
)
7
>
0
(23.167)
are fulfilled. Since the curvature
κ
C
(
x, y
(
x
))
>
0for
all points
(
x, y
)
∈
C
of the clothoid is
positive,
namely
C
is a
convex curve,
there is always
one
minimal distance mapping (
X,Y
)
→
(
x, y
)
∈
C
.
End of Lemma.
Corollary 23.8 (minimal distance mapping of a point close to the clothoid, Newton iteration).
Let (
0
x,
0
y
) be a zero order approximation of the minimal distance mapping
R
2
(
X,Y
)
→
(
x, y
)
∈
C
. Then a standard Newton iteration is
f
(
0
x
)
f
(
0
x
)
1
x
=
0
x
−
(23.168)
...
f
(
n
−
1
x
)
f
(
n−
1
x
)
n
x
=
n−
1
x
−
(23.169)
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