Geography Reference
In-Depth Information
Fig. 23.39.
Height
h<
0 of a point outside the clothoid
and accordingly to the celebrated representation of the
Hesse scalar
h
c
h
=
1
h
c
hg
−
1
g
=(1+
h
c
κ
)
g,if h
c
>
0
h
2
=
g
−
−
(23.182)
h
=
1+
hg
−
1
g
=(1
h
2
=
g
+
|
h
c
|
|
h
c
|
−|
h
c
|
κ
)
g,if h
c
<
0
(23.183)
we can guarantee
h
2
>
0 immediately
for case one
h
2
>
0. For case two
h
2
>
0
,h
2
<
0, can be fulfilled if the condition
Since
g
c
(
x
)
>
0aswellas
κ
c
(
x
)
>
0 for all points
x
∈
C
1
|h
c
|
h
c
<
0:
κ
c
≤
or r
c
≥|
h
c
|
(23.184)
is met. In words, the radius of curvature of
the clothoid must be larger or equal to the absolute
value of the clothoidal height, the length of the displacement vector
, a condition
which is in general fulfilled in practice. By means of
r
c
≥|h
c
|
for points “
inside
” the clothoid we
have also given a
definition
of the ration “a point (
X,Y
)
∈
R
x
(
x
)
−
X
2
close to the point (
x, y
)
∈
C
”
3rd version: analytical for
d
2
L
dx
2
(
x
)
>
0
y
(
x
)]
d
2
y
1+
dy
dx
(
x
)
dy
↔
dx
(
x
)
−
[
Y
−
dx
2
(
x
)
>
0
Y
x
0
)
j
1+
K
k
=1
L
J
j
=0
c
j
(
x
x
0
)
k
+
l−
2
l
=1
klc
k
c
l
(
x
−
−
−
−
K
k
=1
k
(
k
x
0
)
k−
2
>
0
−
1) (
x
−
f
(
x
−
x
0
)
>
0
↔
f
1
+2
f
2
(
x −
x
0
)+3
f
3
(
x −
x
0
)
2
+
...
+7
f
7
(
x −
x
0
)
6
+
...>
0
(23.185)
Since in practice the clothoid is often approximated by a polygon we summarize by Example
23.5
the related minimal distance mapping.
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