11552

# Elliptic Epitrochoid

This Demonstration traces the path of a point (known as the pole or generator) fixed to an ellipse that rolls without slipping around a stationary base ellipse.
If the circumference ratio between the ellipses is the rational number, a closed curve is obtained after complete revolutions of the rolling ellipse around the base. By then the rolling ellipse will have made revolutions around its center.
In this Demonstration, the circumference ratios are either integers () or of the form (). Consequently, a curve closes after one or two revolutions of the rolling ellipse around the base ellipse.
Moving the pole inside or outside the rolling ellipse makes the elliptic epitrochoid either curtate or prolate.
Changing the eccentricity of either ellipse creates a great variety of curves.

### DETAILS

With the rolling ellipse in its initial angular position , define two points:
1. The point on the base ellipse is at an arc length from its intersection with the positive axis.
2. The point on the rolling ellipse is at an arc length from the intersection with its semimajor axis.
Also define two angles:
3. is the angle that subtends an arc of length on the base ellipse.
4. is the angle between the tangent line on the rolling ellipse at and the axis.
Increasing rolls the ellipse around the base ellipse by means of two geometric transformations on points on the ellipse, performed by the Wolfram Language function transfoEC(ϕ,{x,y},e,n), which consists of a translation by the vector E C and a rotation around through the angle .

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.