*If you wanna copy one (or more) of these applets, here are the links:*[br] [url=https://www.geogebra.org/m/nfmzdqgp]3D Equivalent of the Cycloid[/url][br] [url=https://www.geogebra.org/m/fsyrvvxd]3D Equivalent of the Epicycloid[/url][br] [url=https://www.geogebra.org/m/uunayyms]3D Equivalent of the Hypocycloid[/url][br][br]The cycloid is the locus of a point on a circle that rolls on a straight line. But how can we define its equivalent in a three-dimensional environment?

Note that the intersection between the generated surface and the y = k plane (for k in (-r,r) - plane parallel to the translation movement of the sphere and perpendicular to the plane in which it rolls) is always a trochoid, more specifically a cycloid for k = 0 and a shortened trochoid for the other values.[br][br]Well, and about the epicycloid?

Again, note that the intersection between the generated surface and the z = k plane (for k in (-r,r) - plane parallel to the sphere's orbit) is always a epitrochoid, more specifically a epicycloid for k = 0 and a shortened epitrochoid for the other values.[br][br]Finally, only the hypocycloid remains.

Once more, note that the intersection between the generated surface and the z = k plane (for k in (-r,r) - plane parallel to the sphere's orbit) is always a hypotrochoid, more specifically a hypocycloid for k = 0 and a shortened hypotrochoid for the other values.