Cissoid of Diocles

It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix.

Converting the polar form to Cartesian coordinates produces A compass-and-straightedge construction of various points on the cissoid proceeds as follows.

While this construction produces arbitrarily many points on the cissoid, it cannot trace any continuous segment of the curve.

Let ∠BST be a right angle which moves so that ST equals the distance from B to J and T remains on J, while the other leg BS slides along B.

Note that, as with the double projection construction, this can be adapted to produce a mechanical device that generates the curve.

The Greek geometer Diocles used the cissoid to obtain two mean proportionals to a given ratio.

As a special case, this can be used to solve the Delian problem: how much must the length of a cube be increased in order to double its volume?

Note however that this solution does not fall within the rules of compass and straightedge construction since it relies on the existence of the cissoid.

If they are connected by line segments, then the construction will be well-defined, but it will not be an exact cissoid of Diocles, but only an approximation.

Likewise, if the dots are connected with circular arcs, the construction will be well-defined, but incorrect.

An alternative is to keep adding constructed points to the cissoid which get closer and closer to the intersection with line PC, but the number of steps may very well be infinite, and the Greeks did not recognize approximations as limits of infinite steps (so they were very puzzled by Zeno's paradoxes).

One could also construct a cissoid of Diocles by means of a mechanical tool specially designed for that purpose, but this violates the rule of only using compass and straightedge.

[3] The geometrical properties of pedal curves in general produce several alternate methods of constructing the cissoid.