Geography Reference
In-Depth Information
The
Frenet equations
are the derivational equations
f
=
e
(
R
)
∗
=
f
R
(
R
)
∗
=
Ω
∗
where
Ω
:=
R
× R
∗
denotes the
Cartan matrix,
as an antisymmetric matric subject to the
SO
(2)
algebra.
Ω
∈
R
2
×
2
as an
antisymmetric
matrix is structured by only one nonvanishing element,
namely
ω
12
=
κ
(
s
), called curvature of the planar curve.
=
0
κ
(
s
)
,
f
1
(
s
)=
κ
(
s
)
f
2
,
f
2
(
s
)=
Ω
−
κ
(
s
)
f
1
(23.126)
−κ
(
s
)0
We are going to derive the angular representation of curvature
κ
(
s
). An explicit writing of the
identity
f
=
eR
∗
is
f
1
=
e
1
cos
α
+
e
2
sin
α
↔
f
1
cos
α
−
f
2
sin
α
=
e
1
(23.127)
f
2
=
−
e
1
sin
α
+
e
2
cos
α
↔
f
1
sin
α
+
f
2
cos
α
=
e
2
differentiated to
f
1
=
e
1
α
sin
α
+
e
2
α
cos
α
=
f
2
α
−
(23.128)
f
2
=
f
1
α
e
1
α
cos
α
+
e
2
α
sin
α
=
−
−
Indeed prime differentiation refers to differentiation with respect to
arc lengths.
The final result
of the differentiation we collect in
Corollary 23.1 (angular representation of curvature).
κ
(
s
)=
α
(
s
)
(23.129)
End of Corollary.
For the proof we just have to identity
ω
12
=
k
(
s
) within
f
1
and
f
2
, respectively. A
clothoid
may now be defined as such a curve whose product of curvature radius
r
(
s
):=1
/k
(
s
) and its
arc lengths
is a positive constant, namely
rs
=
a
2
. Conversely we take advantage of
Definition 23.1 (“clothoid”).
A planar curve is called clothoid
C
if its curvature is positively proportional to the arc lengths,
in particular
κ
(
s
)=
s/a
2
(23.130)
End of Definition.
Corollary 23.2 (circular motion of the tangent vector and the normal vector of the clothoid).
The initial value problem (i) α
=s
/a
2
,s∈
C
+
, (ii) α
0
=
α
(
s
0
)
,α
0
=
α
(
s
0
)
is solved by
s
0
)+
1
s
0
)+
1
s
0
)
2
=
α
0
+
k
0
(
s
s
0
)
2
(s) =
α
0
+
α
0
(
s
2!
α
0
(
s
α
−
−
−
2!
k
0
(
s
−
(23.131)
subject to
κ
0
=
k
(
s
0
)=s
/a
2
,κ
0
=
κ
(
s
0
)=1
/α
2
(23.132)
End of Corollary.
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