The parabola is a member of the family of conics. The evolute in this case looks rather like the Lamé curve which occurs with the ellipse and the hyperbola, but it is a different curve.


Click the diagram below to activate the applet, and then click the ‘Animate’ button, or manually drag the red driver point along the x-axis. We have extended the range of this point well beyond the edges of the applet window in order to generate the evolute. How would you describe the evolute?

The evolute of the parabola is known as the semicubical parabola, sometimes known a Neile’s parabola. It has a cusp lying on the x-axis.

From a point to the right of the evolute, three normals can be drawn to the parabola, while only one normal can be drawn to the parabola from a point to the left of the evolute.





Bibliography

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