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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.

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© Robert FERRÉOL 2017