Epicycloids and Epitrochoids
Hypocycloids and Hypotrochoids
An epicycloid is the locus of a point on a circle rolling around a second circle on the OUTSIDE. Epitrochoids are the loci of points some distance d from the center of the rolling circle other than r, the radius. On the right, the ratio of the two cirles is 3:1. The dark blue curve is an epicycloid. The purple curve is a curate epitrochoid. The light blue curve with the loops is a prolate epitrochoid. Click HERE for a GSP file.
The hypocycloid is the locus of a point on a circle rolling around a second circle on the INSIDE. Hypotrochoids are the loci of points some distance d from the center of the rolling circle other than r, the radius. On the right is a more complex picture. The ratio of the radii of the green circle to the red circle is 3:1. TWO hypocycloids are drawn be taking the locus simultaneously at each end of a diameter, so two points ON the red circle are traced. These are the red curves with three cusps. Two curate hypotrochoids are in light green. Points the same distance from the center of the circle but in opposite directions were traced. Finally two prolate (with loops) hypotrochoids are show in dark brown. Click HERE for this GSP file.
Construct GSP files to generate curves similar to these.
Most elementary explorations will explore pairs of circles in the ratios 3:1, 2:1, or 1:1.
Click HERE for a coordinated animation of circles in a ratio 3:1.
The case for the ratio of 2:1