Quadratrix of Hippias
The quadratrix or trisectrix of Hippias (also called the quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is traced out by the crossing point of two lines, one moving by translation at a uniform speed, and the other moving by rotation around one of its points at a uniform speed. An alternative definition as a parametric curve leads to an equivalence between the quadratrix, the image of the Lambert W function, and the graph of the function .
The discovery of this curve is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem, hence its name as a trisectrix. Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle, hence its name as a quadratrix. Dinostratus's theorem, used in this attempt, relates an endpoint of the curve to the value of π. Both angle trisection and squaring the circle can be solved using a compass, a straightedge, and a given copy of this curve, but not by compass and straightedge alone. Although a dense set of points on the curve can be constructed by compass and straightedge, allowing these problems to be approximated, the whole curve cannot be constructed in this way.
The quadratrix of Hippias is a transcendental curve. It is one of several curves used in Greek mathematics for squaring the circle.