Spherical Trochoid

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This Demonstration simulates the generation of a spherical trochoid by a point attached to a circle rolling without sliding along the edge of another circle (the base circle) on the same sphere (the base sphere).

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A spherical cycloid is traced by a point on the rolling circle's edge; a spherical trochoid is drawn by a point attached to the circle at a distance greater than or less than its radius. A spherical trochoid becomes a spherical cycloid if the distance of the generating point to the rolling circle's center is equal to its radius.

is the angle between the planes of the base circle and the rolling circle.

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


Snapshots


Details

Let be the radius of the base circle centered at , the radius of the rolling circle and the distance of the generating point to its center; is the angle between the - plane and the plane of the rolling circle.

Let be the angular displacement of the rolling circle along the edge of the base circle. Since the rolling circle rolls without sliding, its angular displacement around its center is .

The point on a copy of the rolling circle 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] H. M. Jeffery, "On Spherical Cycloidal and Trochoidal Curves," The Quarterly Journal of Pure and Applied Mathematics, 19(73), 1882 pp. 45–66. gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0019%7CLOG_0012.



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