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where
f
(
x
)
,f
(
x
) is taken from (
23.166
), (
23.167
) and Table
23.7
. The reproducing point
m
x
=
m
1
x
generated the fixed point within “machine arithmetic”.
−
End of Corollary.
For the proof of Lemma
23.3
we start from the Lagrangean
2
X
−
x
2
=
1
L
(
x, y
(
x
)) :=
1
2
(
X − x
)
2
+
1
2
(
Y − y
(
x
))
2
=min
(23.170)
z
subject to the clothoid
C
parameterized by the means of
x
0
)
J
=
J
x
0
)
2
+
...
+
c
J
(
x
x
0
)
j
y(x)=
c
0
+
c
1
(
x
−
x
0
)+
c
2
(
x
−
−
j
=0
c
j
(
x
−
The function
L
(
x
,
y
(
x
)) is minimal if the following two equivalent conditions are fulfilled.
1st condition (“necessary”)
1
st version: 1
st
derivative of the Lagrangean
dL
dx
(
x
)=0
2
nd version: orthogonality
X
−
x
(
x
)
|
dx
(
x
)
dx
−
(23.171)
3
rd
version: normal equations
dy
dx
(
x
)=0
−
(
X
−
x
)(
Y
−
y
(
x
))
·
x
0
)
J−
1
=
K
∂y
∂x
=
c
1
+2
c
2
(
x
x
0
)
k−
1
−
x
0
)+
...
+
Jc
J
(
x
−
k
=1
kc
k
(
x
−
(23.172)
j
=0
c
j
(
x − x
0
)
j
J
K
k
=1
kc
k
(
x − x
0
)
k−
1
=0
↔
x − X −
Y −
Y
c
1
+2
c
2
(
x
x
0
)
K−
1
x
0
)
2
+
...
+
Kc
K
(
x
x
−
x
0
−
X
+
x
0
−
−
x
0
)+3
c
3
(
x
−
−
+
c
0
+
c
1
(
x
x
0
)
J
x
0
)
2
+
...
+
c
J
(
x
−
x
0
)+
c
2
(
x
−
−
x
0
)
2
+
...
+
Kc
K
(
x
x
0
)
K−
1
]=0
∗
[
c
1
+2
c
2
(
x
−
x
0
)+3
c
2
(
x
−
−
x
0
)
1
2
Yc
2
+2
c
0
c
2
+
c
1
+(
x
x
0
)
2
↔−
X
+
x
0
−
Yc
1
+
c
0
c
1
+(
x
−
−
−
x
0
)
3
−
4
Yc
4
+4
c
0
c
4
+4
c
1
c
3
+2
c
2
+
...
= 0
(
−
3
Yc
3
+3
c
0
c
3
+3
c
1
c
2
)+(
x
−
(23.173)
In Table
23.6
we have collected all the coecients of the above quoted polynomial
x
0
)
2
+
...
+
f
7
(
x
x
0
)
7
+
f
(
x
−
x
0
)=
f
0
+
f
1
(
x
−
x
0
)+
f
2
(
x
−
−
···
(23.174)
The second version of the normal equations can be interpreted as an
orthogonality condition
:
the
displacement vector
X
−
x
(
x
)andthe
tangent vector d
x
/
d
x
(
x
)atthepoint
x
are
orthogonal.
X
f
2
(
x
) is in the direction of the
clothoid unit normal vector
f
2
(
x
)at
x
point to the exterior. There are two cases to be discussed:
−
x
(
x
)=
f
2
|
X
−
x
(
x
)
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