Apollonian cubic


last updated: 2003-08-01

The Appolonian cubic can be constructed as follows. Given two line segments AB and CD, the curve is the collection of the points P from which the angles viewing the segments are equal.
The curve is a generalization of the Apollonian circle

V found some interesting properties of the curve, in 1829. In 1852 Steiner formulated the corresponding problem, unaware of the work of van Rees.
It was Gomes Teixeira who remarked this equivalence (in 1915) between the work of Steiner and van Rees.
These cubic curves got quite popular among mathematicians, e.g. Brocard, Chasles, Dandelin, Darboux and Salmon studied them.

Nowadays, at the University of Crete, a group around Paris Pamfilos is working on these cubics. They gave the curve the name Apollonian cubic or isoptic cubic and they constructed a tool, named Isoptikon (796 kB) to draw an Apollonian cubic, given the two segments. An Apollonian cubic can have two parts or just one.
Also the theory about the isoptic cubics has been included in the package, and many characteristics of the curve. It is shown, for instance, that the unifying concept is the Abelian group structure defined on a cubic, as discovered by Jacobi (1835).