# Persian curve

*spiric curve*

A plane algebraic curve of order four that is the line of intersection between the surface of a torus and a plane parallel to its axis (see Fig. a, Fig. b, Fig. c). The equation in rectangular coordinates is

$$(x^2+y^2+p^2+d^2-r^2)^2=4d^2(x^2+p^2),$$

where $r$ is the radius of the circle describing the torus, $d$ is the distance from the origin to its centre and $p$ is the distance from the axis of the torus to the plane. The following are Persian curves: the Booth lemniscate, the Cassini oval and the Bernoulli lemniscate.

Figure: p072400a

$d>r$.

Figure: p072400b

$d=r$.

Figure: p072400c

$d<r$.

The name is after the Ancient Greek geometer Persei (2nd century B.C.), who examined it in relation to research on various ways of specifying curves.

#### References

[1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |

#### Comments

#### References

[a1] | F. Gomez Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971) |

[a2] | K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962) |

**How to Cite This Entry:**

Persian curve.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Persian_curve&oldid=31952