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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.
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In the frame with centre O such that A(a,0):
Polar equation: Cartesian equation: Rational sextic (double point at O, triple at A). |
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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: |
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The complex parametrization above shows that Freeth's nephroid is a tritrochoid. |
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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: 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