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:  where a is the diameter of  the cylinder and the torus. Système cylindrical equations : . Système spherical equations : , where  is the latitude. 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. Construction of the curve: The torus is produced by a circle (CQ) of horizontal diameter [OQ], the point Q having a movement circular around O, and the cylinder, by a vertical line (D) passing through P, the point P describing a fixed circle of diameter [OA] (OQ = OA). Taking P on (OQ), the point M of intersection of (D) with (CQ) describes the Archytas curve. Analytically: .                     Latitude  and longitude  are connected by .

It was studied by Archytas because it is a duplicatrix:
indeed,  therefore, if , then ; In this configuration, the two numbers  and   appear simultaneously as ratios of lengths, but to obtain
this configuration, it is necessary to know  which is not constructible.

 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.