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FREETH'S NEPHROID
Curve studied by Freeth in 1879.
T. J. Freeth (1819  1904): British mathematician. Loria p. 329. 
Freeth's nephroid is the strophoid of a circle with respect to two points O and A, A being on the circle and O the centre of the circle: when a point M_{0} describes the circle, the curve is the locus of the points M of the line (AM_{0}) such that M_{0}M
= M_{0}A.
See here an animation of this construction.


In the frame with centre O such that A(a,0):
Polar equation: . Cartesian equation: . Rational sextic (double point at O, triple at A). 

In the frame centred on A such that O(a,0):
Pedal equation: . Complex parametrization: (). Area of the domain delimited by the external part: . 
The first equation shows that Freeth's nephroid is a conchoid of the Dürer folium.
But Freeth's nephroid is also the pedal of the cardioid: with respect to the point (a, 0). 

The complex parametrization above shows that Freeth's nephroid is a tritrochoid. 

For , ; Freeth's nephroid enables to construct the regular heptagon.
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© Robert FERRÉOL 2017