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# Conic section

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

141 relations: Abū Sahl al-Qūhī, Addison-Wesley, Algebraic curve, Algebraically closed field, Analytic geometry, Apollonian circles, Apollonius of Perga, Archimedes, Astronomy, Bijection, Binomial distribution, Blaise Pascal, Cartesian coordinate system, Center of mass, Characteristic (algebra), Chord (geometry), Circle, Circular points at infinity, Circumconic and inconic, Collinearity, Complex conjugate, Complex conjugate line, Complex number, Complex projective plane, Cone, Conic Sections Rebellion, Conical surface, Curve, Cylinder, Dandelin spheres, Degeneracy (mathematics), Degenerate conic, Determinant, Diagonal form, Director circle, Discriminant, Doubling the cube, Duality (projective geometry), Dynamics (mechanics), Eccentricity (mathematics), Eigenvalues and eigenvectors, Ellipse, Elliptic coordinate system, Elliptic geometry, Empty set, Equidistant set, Euclid, Euclidean geometry, Five points determine a conic, Focus (geometry), ... Expand index (91 more) »

## Abū Sahl al-Qūhī

(ابوسهل بیژن کوهی Abusahl Bijan-e Koohi) was a Persian mathematician, physicist and astronomer.

Addison-Wesley is a publisher of textbooks and computer literature.

## Algebraic curve

In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

## Algebraically closed field

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

## Analytic geometry

In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.

## Apollonian circles

Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa.

## Apollonius of Perga

Apollonius of Perga (Ἀπολλώνιος ὁ Περγαῖος; Apollonius Pergaeus; late 3rdearly 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections.

## Archimedes

Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

## Astronomy

Astronomy (from ἀστρονομία) is a natural science that studies celestial objects and phenomena.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Binomial distribution

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q.

## Blaise Pascal

Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, writer and Catholic theologian.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Center of mass

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero, or the point where if a force is applied it moves in the direction of the force without rotating.

## Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

## Chord (geometry)

A chord of a circle is a straight line segment whose endpoints both lie on the circle.

## Circle

A circle is a simple closed shape.

## Circular points at infinity

In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.

## Circumconic and inconic

In triangle geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.

## Collinearity

In geometry, collinearity of a set of points is the property of their lying on a single line.

## Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

## Complex conjugate line

In complex geometry, the complex conjugate line of a straight line is the line that it becomes by taking the complex conjugate of each point on this line.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Complex projective plane

In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space.

## Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.

## Conic Sections Rebellion

The Conic Sections Rebellion, also known as the Conic Section Rebellion, refers primarily to an incident which occurred at Yale University in 1830, as a result of changes in the methods of mathematics education.

## Conical surface

In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point &mdash; the apex or vertex &mdash; and any point of some fixed space curve &mdash; the directrix &mdash; that does not contain the apex.

## Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

## Cylinder

A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.

## Dandelin spheres

In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane.

## Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class.

## Degenerate conic

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.

## Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

## Diagonal form

In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates.

## Director circle

In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.

## Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.

## Doubling the cube

Doubling the cube, also known as the Delian problem, is an ancient geometric problem.

## Duality (projective geometry)

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept.

## Dynamics (mechanics)

Dynamics is the branch of applied mathematics (specifically classical mechanics) concerned with the study of forces and torques and their effect on motion, as opposed to kinematics, which studies the motion of objects without reference to these forces.

## Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section.

## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

## Ellipse

In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

## Elliptic coordinate system

In geometry, the elliptic(al) coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae.

## Elliptic geometry

Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Equidistant set

In mathematics, an equidistant set (also called a midset, or a bisector) is a set each of whose elements has the same distance (measured using some appropriate distance function) from two or more sets.

## Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

## Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

## 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).

## Focus (geometry)

In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed.

## Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere.

## General position

In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects.

## Generalized conic

In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of some defining property of the classical conic.

## Girard Desargues

Girard Desargues (21 February 1591 &ndash; September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Gravity

Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.

## Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

## Homogeneous coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.

## Homography

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive.

## Hyperbola

In mathematics, a hyperbola (plural hyperbolas or hyperbolae) 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.

## Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

## Incidence (geometry)

In geometry, an incidence relation is a binary relation between different types of objects that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used.

## Index of dispersion

In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a probability distribution: it is a measure used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard statistical model.

## Indiana University of Pennsylvania

Indiana University of Pennsylvania (IUP) is a public research university in Indiana County, Pennsylvania, United States.

## Jakob Steiner

Jakob Steiner (18 March 1796 &ndash; 1 April 1863) was a Swiss mathematician who worked primarily in geometry.

## Johan de Witt

Johan de Witt or Jan de Witt, heer van Zuid- en Noord-Linschoten, Snelrewaard, Hekendorp and IJsselveere (24 September 1625 &ndash; 20 August 1672) was a key figure in Dutch politics in the mid-17th century, when its flourishing sea trade in a period of globalisation made the United Provinces a leading European power during the Dutch Golden Age.

## Johannes Kepler

Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer.

## John Wallis

John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.

## Joseph Diez Gergonne

Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician.

## Karl Georg Christian von Staudt

Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician who used synthetic geometry to provide a foundation for arithmetic.

## Kinematics

Kinematics is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the mass of each or the forces that caused the motion.

## Laminar flow

In fluid dynamics, laminar flow (or streamline flow) occurs when a fluid flows in parallel layers, with no disruption between the layers.

## Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

## Line at infinity

In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane.

## Line–line intersection

In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line.

## Locus (mathematics)

In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.

## Mathematical Association of America

The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mathematics Magazine

Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America.

## Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

## Matrix representation of conic sections

In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections.

## Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

## Menaechmus

Menaechmus (Μέναιχμος, 380–320 BC) was an ancient Greek mathematician and geometer born in Alopeconnesus 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.

## Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

## Multiplicity (mathematics)

In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.

## Negative binomial distribution

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs.

## Nine-point conic

In geometry, the nine-point conic of a complete quadrangle is a conic that passes through the three diagonal points and the six midpoints of sides of the complete quadrangle.

## Omar Khayyam

Omar Khayyam (عمر خیّام; 18 May 1048 – 4 December 1131) was a Persian mathematician, astronomer, and poet.

## Orbit

In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet.

## Orbital elements

Orbital elements are the parameters required to uniquely identify a specific orbit.

## Oval (projective plane)

In projective geometry an oval is a circle-like pointset (curve) in a plane that is defined by incidence properties.

## Pappus of Alexandria

Pappus of Alexandria (Πάππος ὁ Ἀλεξανδρεύς; c. 290 &ndash; c. 350 AD) was one of the last great Greek mathematicians of Antiquity, known for his Synagoge (Συναγωγή) or Collection (c. 340), and for Pappus's hexagon theorem in projective geometry.

## Pappus's hexagon theorem

In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the Pappus line.

## Parabola

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.

## Parabolic coordinates

Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas.

## Parallel (geometry)

In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.

## Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.

## Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

## Pascal's theorem

In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem) states that if six arbitrary points are chosen on a conic (i.e., ellipse, parabola or hyperbola) 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 in three points which lie on a straight line, called the Pascal line of the hexagon.

## Perspectivity

In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point.

## Pierre de Fermat

Pierre de Fermat (Between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

## Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

## Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

## Point at infinity

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.

## Poisson distribution

In probability theory and statistics, the Poisson distribution (in English often rendered), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event.

## Poisson's equation

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics.

## Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

## Pole and polar

In geometry, the pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.

## Probability distribution

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

## Projective geometry

Projective geometry is a topic in mathematics.

## Projective harmonic conjugate

In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: The point D does not depend on what point L is taken initially, nor upon what line through C is used to find M and N. This fact follows from Desargues theorem.

## Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane.

## Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form where represents an unknown, and,, and represent known numbers such that is not equal to.

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree.

In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.

## René Descartes

René Descartes (Latinized: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist.

## Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.

## Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.

## Semi-major and semi-minor axes

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter.

## Singular point of a curve

In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter.

## SL2(R)

In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: a & b \\ c & d \end \right): a,b,c,d\in\mathbf\mboxad-bc.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

## Steiner conic

The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field.

## Surface (mathematics)

In mathematics, a surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero.

## Sylvester's law of inertia

Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis.

## Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

## Synthetic geometry

Synthetic geometry (sometimes referred to as axiomatic or even pure geometry) is the study of geometry without the use of coordinates or formulas.

## Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

## The College Mathematics Journal

The College Mathematics Journal, published by the Mathematical Association of America, is an expository journal aimed at teachers of college mathematics, particular those teaching the first two years.

## Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.

## Turbulence

In fluid dynamics, turbulence or turbulent flow is any pattern of fluid motion characterized by chaotic changes in pressure and flow velocity.

## Two-body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other.

## Two-dimensional space

Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).

## Uniformization theorem

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.

## Wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves.

## William Herschel Telescope

The William Herschel Telescope (WHT) is a optical/near-infrared reflecting telescope located at the Observatorio del Roque de los Muchachos on the island of La Palma in the Canary Islands, Spain.

## References

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