Curve studied by Hippias of Elis in 430 BC and by Dinostratus in 350 BC. Dinostratus (4th century BC): Greek mathematician. Other name: Hippias sectrix.

 Cartesian-polar parametrization: .  Polar equation: , where . Cartesian equation: .

 The Dinostratus quadratrix is the locus of the intersection points between a line in a uniform translation and a line in a uniform rotation, the two lines coinciding at some moment. In that capacity, it is a limit case of Maclaurin sectrix, when one of the poles is at infinity. It is the inverse with respect to O of a cochleoid. Consider the complex parametrization: . We have the following property: . Any sequence defined by  is therefore traced on the quadratrix ; the limit of such a sequence is .

The Dinostratus quadratrix also is the projection on a plane perpendicular to the axis of the section of a right helicoid by a plane containing a generatrix of the helicoid.

As you can tell by the name, this curve is a quadratrix; indeed:.
But it was first considered by Hippias as a trisectrix and even an n-sectrix; indeed .

If we study the general case of the loci of intersection points between a line in uniform translation and a line in uniform rotation, we get the following curves, that also are multisectrices:

 Cartesian-polar parametrization: .  Polar equation: , where . Cartesian equation: . Case where q0 = p/2 , with equation .

© Robert FERRÉOL  2017