next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

PLANE SPIRIC CURVE


The torus was called "speira" by the Greeks: spiric is thus equivalent to toric.

 
Cartesian equation:  with A ¹ B.

The plane spirics are the planar sections of a torus; they are called spirics of Perseus when the plane is parallel to the axis of the torus.
For a torus with centre O, axis Oz, major and minor radii a and b, cut by the plane located at distance d to O, forming an angle a with xOy and cutting xOy along a parallel of Ox, we get, in a frame with origin the projection of O on the plane, the above Cartesian equation with .
sign problem in A.
When the plane is bitangent to the torus and not perpendicular to its axis, we get a Villarceau circle of the torus.
To obtain all the cases of the Cartesian equation given in the header, the torus must sometimes be considered as complex.

The spirics also are the isoptics of the centred conics.
 
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL 2017