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CURVE WITH SINUSOIDAL RADIUS
Curve studied by L. Bieberbach in 1932. 
The curves studied on this page are the curves the radius of curvature of which is a sinusoidal function of the curvilinear abscissa.
Intrinsic equation 1: .
Intrinsic equation 2 when 0
< 1:
Intrinsic equation 2 when
= 1:
Intrinsic equation 2 when
> 1: .


curve when lambda = 1 
curve when n =1 (i.e. lambda = sqrt(2)) 
curve when n =3/2 (i.e. lambda = sqrt(13/9)) 
curve when n =2 (i.e. lambda = sqrt(5)) 
curve when n =3 (i.e. lambda = sqrt(10)) 
If the value of n and lambda are made independent from one another in , we get aesthetically pleasing curves reminding in some cases the hypotrochoids.
If n is rational, they are Goursat curves of order the numerator of n.

If, now, we consider that the amplitude of the sinusoid varies, we can study the family curves with intrinsic equation 1 ,
that are parametrized by
Opposite, an animation with k ranging from 0 to 3, with a stop for k =1 which corresponds the case =0 above. 
The curves the curvature of which varies as a sinusoidal function of the curvilinear abscissa are the meander curves.
Other curves defined by their intrinsic equation: the clothoid, the curve of constant gyration, the syntractrix curve.
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© Robert FERRÉOL 2017