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

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