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TORPEDO CURVE
Curve studied by G. Gohierre de Longchamps
in 1884 (geometrie analytique, tome
2, 511515).
Homemade name (G. de Longchamps called this curve the right trifolium, but this name now refers to a more general family). 

Polar equation: .
Cartesian equation: . Cartesian parametrization: (t = q / 4). Complex parametrization: . Rational quartic. In a frame turned by p/4:

The torpedo is the strophoid of a circle (here the circle with centre (a/4, 0) passing by O) with respect to a point O on the circle and a point A located at infinity in the direction of the diameter passing by O.
See also, on this page, a beautiful animation of another construction of the torpedo, mistakenly called Cartesian folium. 

It is also a special case of right trifolium (pedal of a deltoid with respect to the point on the inscribed circle that is the symmetric image of the vertex of the deltoid  see folium), of "fish curve" and of tritrochoid (cf the complex parametrization). It is the locus of the isobarycentre of 3 circular motions with same radius, two of which have opposite speeds while the other has double speed. 

Like all "fish" curves, the torpedo is a planar projection of Viviani's window. 

It can also be obtained as the orthopolar of a circle.
See equilateral trefoil for a succinct study of its inverse.
A very close variation of the torpedo is called the
"Cramer trifolium":

Cartesian equation: .
Polar equation: . 
Who knows why certain mathematicians studied variations of the torpedo and called them "dickoids":


Tardy dickoid 


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© Robert FERRÉOL 2017