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CAPAREDA CURVE


Curves studied by Levi Capareda in 2010.

The Capareda curves are the curves traced on a sphere the projection of which on an equatorial plane of the sphere is a hypo- or epi-trochoid inscribed in the equator. In fact, they are precisely the same curve, with a different presentation, as the satellite curves.
 
Cartesian parametrization in the case of the hypotrochoid:
with  (q > 1; radius of the sphere = (q - 1 + k )a ).
In the case of the epitrochoid:
with  (q > 0; radius of the sphere = (q +1+ k )a ).

Case of the hypotrochoid:
For k = 0, we get the equatorial circle (or a cylindric sine wave for any choice of b).
For k = q - 1, we get the clelias of index  > 1 (case where the poles are multiple points; the equatorial projection is a rose).
 
q = 6 k = 4

q = 6, k = 1

q = 4, k = 3
(clelia)

q = 4 , k = 1
(seam line of a tennis ball)

q = 3, k = 2
(clelia)

q = 3, k = 1

 
 
 
 
Case q = 8, k = 4,3 modelled by Levi Capareda with a gear belt during an Industrial Sciences lecture...

Case of the epitrochoid:
For k = 0, we get the equatorial circle (or a cylindric sine wave for any choice of b).
For k = 1, we get the spherical helices.
For k = q + 1, we get the clelias of index  < 1 (case where the poles are multiple points; the equatorial projection is a rose).
 
 

q = 4, k = 1 
(spherical helix)

q = 4, k = 2

q = 4, k = 5
(clelia)

q = 6, k = 7


 Some examples with the equatorial projections, by Alain Esculier


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© Robert FERRÉOL  2018