Tschirnhausen cubic

Light falling on a parabolic mirror square to its axis is reflected into a bright curve, and in 1690 Tschirnhaus showed that it is this cubic. It is also the envelope of the lines drawn through each point of the parabola square to the line from that point to the focus, the construction shown here. Its loop crosses itself at 60°, and Bouguer met it again in 1732 as the path of a dog that runs twice as fast as its walking master, always straight at him.

27ay² = (a - x)(8a + x)²

On mathcurve.com as Cubique de Tschirnhausen