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FERMAT'S SPIRAL
Curve studied by Menelaus in the end of the first century
and by Fermat in 1636.
Pierre de Fermat (16011655): French mathematician. 
Polar equation: .
Cartesian equation: . Transcendental curve. Curvilinear abscissa: . 
The Fermat spiral
is a special case of parabolic
spiral.
It is a closed curve without double points dividing the
plane into two connected regions, symmetrical about O.
The blue region opposite corresponds to . 

If the curve is traced only for nonnegative values of , the area between two consecutive coils is constant equal to . 
Its inverse with respect to O is the lituus.
The curve on which it rolls in such a way that the movement of its centre is linear is a cubic parabola. 
PreColumbian work, museum of archaeology, Mexico City.
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© Robert FERRÉOL 2017