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ARCHYTAS CURVE

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Archytas of Tarentum (430-350 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