ellipse : evolute

ELLIPSE : Evolute

As is our pattern, we take the ellipse as a point locus, and construct the normal at each point of the curve. The envelope of these normals then generates the required evolute.



Click the linked diagram below, and then click the ‘Animate’ button to generate the evolute of the ellipse. Clicking the little red x which appears at bottom right will clear the drawing from the applet window. Now drag the red point Q slowly around the circle. Can you describe the construction? Do you recognize the evolute?

The curve we obtain here for the evolute is a stretched astroid, sometimes known as a Lamé curve. Notice that, as expected, it has the same symmetry about the x-axis as the ellipse itself.

This evolute has the following interesting property. From a point inside the evolute, four normals can be drawn to the ellipse, from a point on the evolute precisely, three normals can be drawn, and from a point outside, only two normals can be drawn.