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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 ) 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 0 = κ ( s 0 )=1 2
(23.132)
End of Corollary.
 
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