# 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.

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Contributed by: Erik Mahieu (December 2016)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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:

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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 .

Reference

[1] Kinematic Models for Design Digital Libary. "Reuleaux Collection, Cornell: Cycloid Rolling Models." (Dec 5, 2016) kmoddl.library.cornell.edu/model.php?cat=R.

## Permanent Citation