Evolute II x = a cosθ +θsinθ y = a sinθ−θcosθ ′ x = dx dθ = a −sinθ

Evolute
II
x = a (cosθ + θ sin θ )
y = a (sin θ − θ cos θ )
dx
= a (−sin θ + sin θ + θ cos θ ) = a (θ cos θ )
dθ
d 2x
x ′′ = 2 = a (cosθ − θ sin θ )
dθ
dy
y′ =
= a (cosθ − cos θ + θ sin θ ) = a (θ sin θ )
dθ
d2y
y ′′ = 2 = a (sin θ + θ cos θ )
dθ
x′ =
The
curvature,
€
d 2 y dx dy d 2 x
−
2
2
κ = dθ dθ dθ dθ3
 2  2 2
 dx  +  dy  
 dθ   dθ  
=
a (sin θ + θ cos θ ) a (θ cosθ ) − a (θ sin θ ) a (cosθ − θ sin θ )
3
2 2
2
 2

a θ cos θ ) + a 2 (θ sin θ ) 
 (

=
a 2θ 2
1
=
3 3
aθ
aθ
and
therefore
the
radius
of
curvature
is
aθ .
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