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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 epitrochoid
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 = 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