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