Dinostratus quadratrix

DINOSTRATUS QUADRATRIX

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 .

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