Geography Reference
In-Depth Information
Table 23.5
Initial value data for a numerical analysis of the clothoid
Case 1
Case 2
Case 3
κ
0
=1
/
500
κ
0
=1
/
500
κ
0
=1
/
500
κ
0
=0
κ
0
=0
κ
0
=10
−
6
α
0
=10
◦
α
0
=10
◦
α
0
=10
◦
x
0
=
y
0
=
s
0
=0
x
0
=
y
0
=
s
0
=0
x
0
=
y
0
=
s
0
=0
(Figure
23.30
)
(Figure
23.30
)
(Figure 23.33)
x
0
=
y
0
= 200
x
0
=
y
0
= 200
κ
0
=1
/
500
κ
0
=10
−
6
s
0
=0
s
0
=0
α
0
=45
◦
(Figure
23.31
)
(Figure
23.31
)
x
0
=
y
0
= 200
x
0
=
y
0
= 200
x
0
=
y
0
=0
s
0
=50
s
0
=50
s
0
=0
(Figure
23.32
)
(Figure
23.32
)
(Figure 23.34)
23-6 Minimum Distance Mapping of a Point Close to the Clothoid:
Sensitive Control of High-Speed-Railway Track
High-speed-railways/bullet trains/Trans rapids
need
extreme reliable control systems
to avoid catastrophic
events. Here we have concentrated on a focal sensitive problem, the
local high resolution
of the track design. It
is assumed that the track design in terms of
clothoid circle or straight
line in
UTM map matching coordinates
is given. We separate measurement errors from track dissolution.
If a railway track has been surveyed from points close to the clothoid, the circle or the straight
line we are left with the problem to
relate a measurement point
X
2
close to
1
r
0
∈
R
C
(or
S
or
1
)
we shall construct an operational procedure for such an optimal relation.
First
we go through
two examples (circle
1
). By means of the minimal distance mapping of (
X, Y
)
2
to (
x, y
)
(or
S
r
0
or
L
∈
R
∈
C
L
1
) in order to make the acquaintance with the minimal
distance mapping.
Second
we develop the general minimal distance mapping for the clothoid, in
particular by a lemma and two corollaries.
Third
we illustrate the minimal distance mapping by
a numerical example of a number of points which are
outside
and
inside
a specific clothoid.
1
r
0
S
, straight line
L
Two examples: The minimal distance mapping of points close to the straight line
or to the circle
In order to gain some insight into the minimal distance mapping of “real points” to the straight
line or the circle we begin examples.
Example 23.3 (minimal distance mapping of a point close to the circle
S
r
0
κ
0
=1
/r
0
,κ
0
=0).
According to Fig.
23.35
we minimize the Euclidean distance
d
(
x
,
X
)=
X-x
of a given point
X
∈
R
2
and an unknown point
x
∈
S
r
0
which is an element of the circle
x
0
)
2
+(
y
y
0
)
2
=
r
0
}
1
r
0
2
S
:=
{
x
∈
R
|
(
x
−
−
of radius
r
0
around the point (
x
0
,y
0
). Indeed by means of the
Lagrange multiplier λ
the constrained
optimal location of
x
∼
(
x, y
) is generated by
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