Spherical Cycloids Generated by One Cone Rolling on Another

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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.


Contributed by: Erik Mahieu (December 2016)
Open content licensed under CC BY-NC-SA



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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