Geography Reference
In-Depth Information
Case 1:
f
2
|
X
−
x
(
x
)
positive
X
−
x
(
x
)=
h
c
f
2
if
X
−
x
(
x
)
=
h
c
>
0
(23.175)
Case 2:
f
2
|
X
−
x
(
x
)
negative
X
x
(
x
)=
h
c
f
2
if
X
−
x
(
x
)=
|h
c
| ,h
c
<
0
−
(23.176)
In case one, the point
X
is
outside
the clothoid, which in case two inside as illustrated by
Fig.
23.36
. The orthogonal projection of the point
X
∈
R
2
to
x ∈
C
has created the
clothoid
height h
c
which may be represented by (Fig.
23.37
)
h
c
=
X − x
=
(
X − x
)
2
+(
Y − y
(
x
))
2
(23.177)
L
1
,
κ
0
=0
,κ
0
=0
Fig. 23.36.
Minimal distance mapping of a point close to the straight line
2nd condition (“sucient
)
1
st version: 2nd derivative of the Lagrangean
d
2
L
dx
2
(
x
)=0
2
nd version: Hesse scalar of second derivatives
h
2
(
x
)=
dx
dx
(
x
)
X
dx
2
(
x
)
>
0
d
2
x
dx
dx
(
x
)
|
−
−
x
(
x
)
|
(23.178)
The
Hesse scalar h
2
of first and second derivatives of the clothoidal placement vector
x
(
x
)
parameterized with respect to
x,
namely to be
positive
, constitutes a special form of the
su
ciency
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