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TORPEDO CURVE

Curve studied by G. Gohierre de Longchamps in 1884 (geometrie analytique, tome 2, 511-515).
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:
Polar equation: .
Cartesian equation: .


 
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: .

Rational quartics.
 

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

Tardy dickoid


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© Robert FERRÉOL  2017