Spherical Cycloids Generated by One Cone Rolling on Another

In this Demonstration, we generate a spherical trochoid with a cone that rolls without slipping on another stationary cone. The generated curve is called a spherical cycloid or spherical trochoid.
Let and be the base circles of the stationary and rolling cones, respectively, with radii and . Let be the distance of the generating point to the center of .
A spherical cycloid is traced by a point on the edge of , that is, ; a spherical trochoid is traced if .
A closed curve is obtained if is rational.
Let be the angle between the planes of and . For a spherical hypotrochoid, , and for a spherical epitrochoid, .
In the extreme cases or , we get a planar hypotrochoid or epitrochoid, respectively.


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Let be the angular displacement of along the edge of . Since rolls without sliding, its angular displacement around its center is .
The point on a copy of centered at in the - plane and at a distance from its center is:
First rotate this circle by around the axis:
Now translate the circle over a distance along the axis to get:
Finally, rotate this circle by an angle around the axis:
This gives the parametric equation of the spherical trochoid:
The spherical trochoid is on a sphere with center at and radius .
[1] Kinematic Models for Design Digital Libary. "Reuleaux Collection, Cornell: Cycloid Rolling Models." (Dec 5, 2016) kmoddl.library.cornell.edu/model.php?cat=R.
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