3D Equivalent of Cycloid, Epicycloid and Hypocycloid
- Author:
- Eduardo Toffolo
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3D Equivalent of the Cycloid
3D Equivalent of the Epicycloid
3D Equivalent of the Hypocycloid
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?
3D Equivalent of the Cycloid
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.
Well, and about the epicycloid?
3D Equivalent of 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.
Finally, only the hypocycloid remains.
3D Equivalent of the Hypocycloid
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.