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ARCHYTAS CURVE
Archytas
of Tarentum (430350 B.C.) : Greek general, scholar and statesman.
This curve is supposed to be the first non planar curve studied in history. 
System of Cartesian equations: .
Algebraic curve of degree 8 (3D biquartic). Cartesian parametrization: 
The Archytas curve is the intersection between a horn torus and a cylinder of revolution with axis perpendicular to the central circle of the torus and with the same radius. 

It was studied by Archytas because it is a duplicatrix:
indeed, therefore, if , then .
This can be generalized to any torus and we can consider the intersection between the torus with major radius a and minor radius b, and the cylinder with radius b .
As a is larger, the curve tends to a bicylindrical curve (case of the double ellipse). 
Compare to bitorics.
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© Robert FERRÉOL 2018