Spherical Cycloids Generated by One Cone Rolling on Another

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.

[more]

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.

[less]

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:

.

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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send