 
Apollonian cubic


last updated: 20030801 
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).
