ASTROID : Evolute

To obtain the evolute of the astroid, we will look at the envelope of the normals to the curve. The nice four-fold symmetry of the astroid leads us to expect an evolute with similar symmetry.

Click on the diagram below to open the applet, and then click on the 'Animate' button to generate the evolute of the astroid. Clicking the little red x at bottom right will clear the drawing from the applet window. Now drag the red point Q slowly around the circle. This will generate a rather denser picture of the evolute which is less helpful. Do you recognize the evolute of the astroid? Can you describe it?



We see that the evolute of the astroid is another astroid, with double the scale of the original, and with its axes at 45° to those of the original.

Obviously in this case the evolute has the same four-fold symmetry as the original curve.


Bibliography

http://mathworld.wolfram.com/AstroidEvolute.html