Epicycles: Circles Rolling on Circles

Tomas Garza

Description

In this notebook we consider the situation where a circle (green in the image below) rolls on the circumference of a fixed circle (in white), and, at the same time, another circle (in orange) rolls on the circumference of the green circle. A given point (green, in the image) on this circumference describes on its motion an epicycloid (cf. Tomas Garza, “Epicycloids: Using Vector Functions” from the Notebook Archive (2023), https://notebookarchive.org/2023-02-47osxcf), and now we consider the curve described by a given point on the circumference of the second rolling circle (in orange).

To this effect, we use the same approach that has shown great effectiveness is similar problems, namely, the use of elementary vector methods. In essence, we look separately at the vector components of the motion: a) a circle that rolls at a given rate, measured in terms of the angle attained by a given radius at each moment, and b) the position of the center of the circle as it rolls (this depends on the position of the green circle). These two components are independent, and the curve described by the orange point is then the sum of the two respective vectors.

Generating the image

We present below the three circles involved, namely the fixed white circle, and the two moving ones, the green and the orange, respectively. A control is provided to show a) only the main elements of the problem (i.e., the three circles and two additional circles (in blurred white) which correspond to the centers of the rolling circles), or b) in addition, the vectors involved in the construction of the solution).

In the constructive view, the white and gray vectors at the center of the image show the angles of the rolling motion of the green and orange circles, respectively, since we allow the orange circle to roll at a speed different from that of the green one. The settings correspond to the number of full turns of the orange circle at the end. Showing the path of the yellow point on the green circle is optional. Finally, the radius of the orange circle may take different values.

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