Geography Reference
In-Depth Information
Corollary 23.4 (embedding of
S
r
0
in
R
2
).
1
κ
0
−
(
x
κ
0
sin
α
0
2
x
0
)+
1
+
cos
α
0
κ
0
y
−
y
0
=
−
−
(23.148)
If
α
0
=0
,x
0
=0
,y
0
=
κ
0
,then
r
0
−
y
=
−
x
2
covers the Southern half circle.
End of Corollary.
1
κ
0
=0
,κ
0
=0).
Example 23.2 (the straight line
L
α
(
s
)=
α
0
x − x
0
=
s
0
cos
α
0
(
s
)
ds
=cos
α
0
(
s − s
0
)
y − y
0
=
s
0
sin
α
0
(
s
)
ds
=sin
α
0
(
s − s
0
)
End of Example.
1
2
).
Corollary 23.5 (
L
⊂
R
x − x
0
=cos
α
0
(
s − s
0
)
,y− y
0
=sin
α
0
(
s − s
0
)
y − y
0
=tan
α
0
(
x − x
0
)
(23.149)
End of Corollary.
Examples
23.1
and
23.2
have clearly documented that the fundamental local representa-
tion (
23.139
), (
23.140
)aswellas(
23.145
)ofthe
clothoid
contains the circle and the straight line
as special cases. Indeed this has been one target function why we developed (
23.139
), (
23.140
),
and (
23.145
).
Case studies
For the case studies we have collected the initial value data of Table
23.5
for various degrees
of approximation.
This contribution is based on
Grafarend
(
2002
).
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