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
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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:
.
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.
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