The parabola is a member of the family of conics. The caustic curve construction with respect to the variable radiant point K gives rise to a number of rather complicated curves. One curve is of special interest.

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 caustic curves. Investigate this situation for various positions of the radiant point K. Obvious points to look at closely are the origin, the interesections of the directrix with the x-axis, the focus, and when K lies ‘at infinity’. (Drag K right along the x-axis, beyond the bounds of the applet window, and to the edge of your monitor. Afterwards, refresh your window to retrieve the point K !) Can you describe these caustic curves?

For most positions of K the caustic curves here are rather complicated. when K lies at the focus F, the whole plane is generated by the resulting lines to the x-axis. When K lies at the oriogin O, the resulting cautic appears close to the parabola. When K is at ‘infinity’ on the x-axis, the lines PK become parallel to the axis of the parabola, the caustic generating lines all pass through the focus F, and the caustic curve has a special name: it is called Tschirnhausen’s cubic.



Bibliography

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