next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
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:
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 (C_{Q}) 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.
next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2023