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NODAL CURVE

Curve studied by La Gournerie in 1851.

 
Polar equation:  (or  ) with n > 0.

The nodal curves are the Brocard transforms of the Kappa, when the pole is at the centre of the Kappa.
Each curve is composed of an infinite branch, the base, obtained for :

and all its images by the rotations of angle  when k is an integer.

If n is rational and its numerator is p and its denominator q, then the curve is composed of 2p branches, images of the base branch by rotation when q is odd, and p branches when it is even.

Examples:

n = 1 : Kappa

n  = 2: windmill

n = 3 

n = 4

n = 5

n = 1/2:
right strophoid

 n = 3/2

n = 5/2

n = 7/2

n = 9/2

n  = 1/3 

n = 2/3

n = 4/3

n = 5/3

n = 7/3

n = 1/4

n = 3/4

n = 5/4

n = 7/4

n = 9/4

n = 1/5

n = 2/5

n = 3/5

n = 4/5

n = 6/5

All nodal curves are stereographic projections of the clelia.
The inverse (with centre O and radius a2) of a nodal curve is the same curve turned by .
Asymptotic line of a 3D helicoid????

Compare to the epispiral.
 
 
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© Robert FERRÉOL  2017