Perseus (geometer)
Encyclopedia
Perseus was an ancient Greek
geometer, who invented the concept of spiric section
s, in analogy to the conic section
s studied by Apollonius of Perga
.
Few details of Perseus' life are known, as he is mentioned only by Proclus
and Geminus
; none of his own works have survived.
The spiric sections result from the intersection of a torus
with a plane
that is parallel to the rotational symmetry axis of the torus. Consequently, spiric sections are fourth-order (quartic) plane curve
s, whereas the conic section
s are second-order (quadratic
) plane curve
s. Spiric sections are a special case of a toric section
, and were the first toric sections to be described.
The most famous spiric section is the Cassini oval
, which is the locus of points having a constant product of distances to two foci. For comparison, an ellipse
has a constant sum of focal distances, a hyperbola
has a constant difference of focal distances and a circle
has a constant ratio of focal distances.
Greece
Greece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....
geometer, who invented the concept of spiric section
Spiric section
In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form^2=dx^2+ey^2+f....
s, in analogy to the conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s studied by Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
.
Few details of Perseus' life are known, as he is mentioned only by Proclus
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
and Geminus
Geminus
Geminus of Rhodes , 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 introductory astronomy book for students. He also wrote a work on mathematics, of which only...
; none of his own works have survived.
The spiric sections result from the intersection of a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
with a plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
that is parallel to the rotational symmetry axis of the torus. Consequently, spiric sections are fourth-order (quartic) plane curve
Plane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....
s, whereas the conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s are second-order (quadratic
Quadratic
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...
) plane curve
Plane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....
s. Spiric sections are a special case of a toric section
Toric section
A toric section is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone.-Mathematical formulae:In general, toric sections are fourth-order plane curves of the form...
, and were the first toric sections to be described.
The most famous spiric section is the Cassini oval
Cassini oval
A Cassini oval is a plane curve defined as the set of points in the plane such that the product of the distances to two fixed points is constant. This is related to an ellipse, for which the...
, which is the locus of points having a constant product of distances to two foci. For comparison, an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
has a constant sum of focal distances, a hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
has a constant difference of focal distances and a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
has a constant ratio of focal distances.