Perseus ( el, Περσεύς; c. 150 BC) was an

ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic p ...

geometer A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * ...

, who invented the concept of

spiric section In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form :(x^2+y^2)^2=dx^2+ey^2+f. \, Equivalently, spiric sections can be defined as bicircular quartic curves that are symme ...

s, in analogy to the

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...

s studied by Apollonius of Perga.

Life

Few details of Perseus' life are known, as he is mentioned only by Proclus and

Geminus Geminus of Rhodes ( el, Γεμῖνος ὁ Ῥόδιος), was a Greek astronomer and mathematician, who flourished in the 1st century BC. An astronomy work of his, the ''Introduction to the Phenomena'', still survives; it was intended as an int ...

; none of his own works have survived.

Spiric sections

The spiric sections result from the intersection of a

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

with a

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

that is parallel to the rotational symmetry axis of the torus. Consequently, spiric sections are fourth-order ( quartic)

plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...

s, whereas the

s are second-order ( quadratic)

s. Spiric sections are a special case of a toric section, and were the first toric sections to be described.

Examples

The most famous spiric section is the

Cassini oval In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points ( foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of t ...

, which is the

locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...

of points having a constant ''product'' of distances to two foci. For comparison, an ellipse has a constant sum of focal distances, a

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...

has a constant difference of focal distances, and a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

has a constant ratio of focal distances.

References

* Tannery P. (1884) "Pour l'histoire des lignes et de surfaces courbes dans l'antiquité", ''Bull. des sciences mathématique et astronomique'', 8, 19–30. * Heath TL. (1931) ''A history of Greek mathematics'', vols. I & II, Oxford. * {{DEFAULTSORT:Perseus Ancient Greek geometers 2nd-century BC Greek people 2nd-century BC mathematicians