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 M0 describes the circle, the curve is the locus of the points M of the line (AM0) such that M0M = M0A.

	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