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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.


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.
GENERALIZATION
The Freeth's nephroid is the case n = 4 of the
family of curves of complex parameterization:
, therefore of polar equation .
The case n = 3 gives the limaçon trisectrix and when n tends to infinity, the limit curve is the cochleoid:. 
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© Robert FERRÉOL 2020