Conic section

conicconic sectionsconicsdirectrixsemi-latus rectumlatus rectumconic equationdirectricesstandard form(projective) conic sections
[[File:Conic sections with plane.svg|right|300px|thumb|Types of conic sections:wikipedia
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Parabola

parabolicparabolic curveparabolic arc
1. Parabola
One description of a parabola involves a point (the focus) and a line (the directrix).

Ellipse

ellipticalellipticeccentricity
2. Circle and ellipse
Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure).

Hyperbola

hyperbolicrectangular hyperbolahyperbolas
3. Hyperbola]]
The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone.

Apollonius of Perga

ApolloniusApollonius of PergeApollonian
The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.
Apollonius of Perga (Apollonius Pergaeus; late 3rd – early 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections.

Eccentricity (mathematics)

eccentricityeccentriceccentricities
One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity.
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.

Focus (geometry)

focifocusfocal points
One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity.
For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola.

Mathematics

mathematicalmathmathematician
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hipparchus of Nicaea (2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD).

Cone

conicalconesconic
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
If the cone is right circular the intersection of a plane with the lateral surface is a conic section.

Degenerate conic

degeneratedegenerate casedegenerate cases
If the conic is non-degenerate, then:
In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve.

Circle

circularcircles360 degrees
2. Circle and ellipse
In homogeneous coordinates, each conic section with the equation of a circle has the form

Five points determine a conic

Braikenridge–Maclaurin constructionunambiguously determine
Just as two (distinct) points determine a line, five points determine a conic.
In Euclidean and projective geometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a degree-2 plane curve).

Dandelin spheres

A proof that the above curves defined by the focus-directrix property are the same as those obtained by planes intersecting a cone is facilitated by the use of Dandelin spheres.
The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.

Quadratic equation

quadratic equationsquadraticquadratic formula
In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables.
The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.

Omar Khayyam

Omar KhayyámKhayyamOmar Khayam
Persians found applications of the theory, most notably the Persian mathematician and poet Omar Khayyám, who used conic sections to solve algebraic equations of no higher a degree than three.
As a mathematician, he is most notable for his work on the classification and solution of cubic equations, where he provided geometric solutions by the intersection of conics.

Conical surface

coneconicalcone directrix
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
More generally, when the directrix C is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of C, one obtains a conical quadric, which is a special case of a quadric surface.

Pascal's theorem

hexagrammum mysticumGeneration of Conic SectionsHexagrammum Mysticum Theorem
Pascal's theorem concerns the collinearity of three points that are constructed from a set of six points on any non-degenerate conic.
In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem) states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon.

Analytic geometry

analytical geometryCartesian geometrycoordinate geometry
In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. René Descartes and Pierre Fermat both applied their newly discovered analytic geometry to the study of conics.
In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.

Euclid

Euclid of AlexandriaEuklidGreek Mathematician
Euclid (fl.
Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.

Two-dimensional space

Euclidean planetwo-dimensional2D
The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions.
There are an infinitude of other curved shapes in two dimensions, notably including the conic sections: the ellipse, the parabola, and the hyperbola.

General position

general linear positiongenericallyin general position
Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true in both the Euclidean plane and its extension, the real projective plane.
This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations (e.g. conic sections).

Blaise Pascal

PascalPascal, BlaisePascalian
Girard Desargues and Blaise Pascal developed a theory of conics using an early form of projective geometry and this helped to provide impetus for the study of this new field.
Particularly of interest to Pascal was a work of Desargues on conic sections.

Quadric

quadric surfacequadric surfacesquadric hypersurface
More technically, the set of points that are zeros of a quadratic form (in any number of variables) is called a quadric, and the irreducible quadrics in a two dimensional projective space (that is, having three variables) are traditionally called conics.
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).

Menaechmus

It is believed that the first definition of a conic section was given by Menaechmus (died 320 BCE) as part of his solution of the Delian problem (Duplicating the cube).
Menaechmus (Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher Plato and for his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube using the parabola and hyperbola.

René Descartes

DescartesCartesianRene Descartes
René Descartes and Pierre Fermat both applied their newly discovered analytic geometry to the study of conics.
Together they worked on free fall, catenary, conic section, and fluid statics.

Orbit

orbitsorbital motionplanetary motion
Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest.
Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections (this assumes that the force of gravity propagates instantaneously).