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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. 
Cartesianpolar parametrization: . Polar equation: , where . 
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 nsectrix; 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:
Cartesianpolar parametrization: . Polar equation: , where . Cartesian equation: . 
Case where q_{0} = p/2 , with equation . 
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© Robert FERRÉOL 2017