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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a conic section, quadratic curve or conic is a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
obtained as the intersection of the
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
of a
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. A cone is formed by a set of line segments, half-lines, or lines con ...
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' ...
. The three types of conic section are the
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, cal ...
, the
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
, and the
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
; the
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 const ...
is a special case of the ellipse, though historically it was sometimes called a fourth type. The
ancient Greek mathematicians Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
studied conic sections, culminating around 200 BC with
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
's systematic work on their properties. The conic sections in the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of those points whose distances to some particular point, called a ''
focus Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ...
'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
''. The type of conic is determined by the value of the eccentricity. In
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
, a conic may be defined as a
plane algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
of degree 2; that is, as the set of points whose coordinates satisfy a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
in two variables which can be written in the form Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0. The geometric properties of the conic can be deduced from its equation. In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
, the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity. Further extension, by expanding the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
coordinates to admit
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
coordinates, provides the means to see this unification algebraically.


Euclidean geometry

The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
.


Definition

A conic is the curve obtained as the intersection of 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' ...
, called the ''cutting plane'', with the surface of a double
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. A cone is formed by a set of line segments, half-lines, or lines con ...
(a cone with two ''nappes''). It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called ''degenerate'' conics and some authors do not consider them to be conics at all. Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic. There are three types of conics: the
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
,
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
, and
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, cal ...
. The
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 const ...
is a special kind of ellipse, although historically Apollonius considered it a fourth type. Ellipses arise when the intersection of the cone and plane is a
closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone; for a right cone, this means the cutting plane is perpendicular to the axis. If the cutting plane is
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...
to exactly one generating line of the cone, then the conic is unbounded and is called a ''parabola''. In the remaining case, the figure is a ''hyperbola'': the plane intersects ''both'' halves of the cone, producing two separate unbounded curves. Compare also spheric section (intersection of a plane with a sphere, producing a circle or point), and
spherical conic In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in th ...
(intersection of an elliptic cone with a concentric sphere).


Eccentricity, focus and directrix

Alternatively, one can define a conic section purely in terms of plane geometry: it 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 all points whose distance to a fixed point (called the ''
focus Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ...
'') is a constant multiple (called the ''
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
'' ) of the distance from to a fixed line (called the ''directrix''). For we obtain an ellipse, for a parabola, and for a hyperbola. A circle is a limiting case and is not defined by a focus and directrix in the Euclidean plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but its directrix can only be taken as the line at infinity in the projective plane. The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular. If the angle between the surface of the cone and its axis is \beta and the angle between the cutting plane and the axis is \alpha, the eccentricity is \frac. 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 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. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane ...
. Alternatively, an ellipse can be defined in terms of two focus points, as the locus of points for which the sum of the distances to the two foci is ; while a hyperbola is the locus for which the difference of distances is . (Here is the semi-major axis defined below.) A parabola may also be defined in terms of its focus and latus rectum line (parallel to the directrix and passing through the focus): it is the locus of points whose distance to the focus plus or minus the distance to the line is equal to ; plus if the point is between the directrix and the latus rectum, minus otherwise.


Conic parameters

In addition to the eccentricity (), foci, and directrix, various geometric features and lengths are associated with a conic section. The ''principal axis'' is the line joining the foci of an ellipse or hyperbola, and its midpoint is the curve's ''center''. A parabola has no center. The ''linear eccentricity'' () is the distance between the center and a focus. The ''latus rectum'' is the chord parallel to the directrix and passing through a focus; its half-length is the ''semi-latus rectum'' (). The ''focal parameter'' () is the distance from a focus to the corresponding directrix. The ''major axis'' is the chord between the two vertices: the longest chord of an ellipse, the shortest chord between the branches of a hyperbola. Its half-length is the ''
semi-major axis 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 two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
'' (). When an ellipse or hyperbola are in standard position as in the equations below, with foci on the -axis and center at the origin, the vertices of the conic have coordinates and , with non-negative. The ''minor axis'' is the shortest diameter of an ellipse, and its half-length is the ''semi-minor axis'' (), the same value as in the standard equation below. By analogy, for a hyperbola the parameter in the standard equation is also called the semi-minor axis. The following relations hold: * \ \ell = p e * \ c = a e * \ p+c = \frac ae For conics in standard position, these parameters have the following values, taking a ,b > 0.


Standard forms in Cartesian coordinates

After introducing Cartesian coordinates, the focus-directrix property can be used to produce the equations satisfied by the points of the conic section. By means of a change of coordinates (
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
and
translation of axes In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...
) these equations can be put into ''standard forms''. For ellipses and hyperbolas a standard form has the -axis as principal axis and the origin (0,0) as center. The vertices are and the foci . Define by the equations for an ellipse and for a hyperbola. For a circle, so , with ''
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
'' . For the parabola, the standard form has the focus on the -axis at the point and the directrix the line with equation . In standard form the parabola will always pass through the origin. For a ''rectangular'' or ''equilateral'' hyperbola, one whose asymptotes are perpendicular, there is an alternative standard form in which the asymptotes are the coordinate axes and the line is the principal axis. The foci then have coordinates and . * Circle: *: x^2 + y^2 = a^2, * Ellipse: *: \frac + \frac = 1, * Parabola: *: y^2 = 4ax, \text a > 0, * Hyperbola: *: \frac - \frac = 1, * Rectangular hyperbola: *:xy = c^2. The first four of these forms are symmetric about both the -axis and -axis (for the circle, ellipse and hyperbola), or about the -axis only (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines and . These standard forms can be written
parametrically A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
as, *
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 const ...
: *: (a \cos\theta, a \sin\theta), *
Ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
: *: (a \cos\theta, b \sin\theta), *
Parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
: *: (at^2, 2at), *
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, cal ...
: *: (a\sec\theta, b\tan\theta), \text (\pm a\cosh \psi, b \sinh \psi), *
Rectangular 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, cal ...
: *: \left(dt, \frac dt\right), \text d = \frac c.


General Cartesian form

In the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
in two variables is always a conic section (though it may be
degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
), and all conic sections arise in this way. The most general equation is of the form :Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0, with all coefficients
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s and not all zero.


Matrix notation

The above equation can be written in matrix notation as : \left (\beginx & y \end\right) \left( \beginA & B/2\\B/2 & C\end\right) \left( \beginx\\y\end\right) + \left(\beginD & E \end\right) \left(\beginx\\y\end\right) +F= 0. The general equation can also be written as : \left(\beginx & y & 1\end\right) \left( \beginA & B/2 & D/2\\B/2 & C & E/2\\D/2&E/2&F\end\right) \left(\beginx\\y\\1\end\right) = 0. This form is a specialization of the homogeneous form used in the more general setting of projective geometry (see below).


Discriminant

The conic sections described by this equation can be classified in terms of the value B^2 - 4AC, called the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
of the equation. Thus, the discriminant is where is the
matrix determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
\left, \beginA & B/2\\B/2 & C\end\. If the conic is
non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent defin ...
, then: * if , the equation represents an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
; ** if and , the equation represents 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 const ...
, which is a special case of an ellipse; * if , the equation represents a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
; * if , the equation represents 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, cal ...
; ** if , the equation represents a
rectangular 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, cal ...
. In the notation used here, and are polynomial coefficients, in contrast to some sources that denote the semimajor and semiminor axes as and .


Invariants

The discriminant of the conic section's quadratic equation (or equivalently the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the 2 × 2 matrix) and the quantity (the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
of the 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes, as is the determinant of the 3 × 3 matrix above. The constant term and the sum are invariant under rotation only.


Eccentricity in terms of coefficients

When the conic section is written algebraically as :Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0, \, the eccentricity can be written as a function of the coefficients of the quadratic equation. If the conic is a parabola and its eccentricity equals 1 (provided it is non-degenerate). Otherwise, assuming the equation represents either a non-degenerate hyperbola or ellipse, the eccentricity is given by :e=\sqrt, where if the determinant of the 3 × 3 matrix above is negative and if that determinant is positive. It can also be shownSpain, B.
''Analytical Conics''
(Mineola, NY: Dover, 2007). Originally published in 1957 by
Pergamon Pergamon or Pergamum ( or ; grc-gre, Πέργαμον), also referred to by its modern Greek form Pergamos (), was a rich and powerful ancient Greece, ancient Greek city in Mysia. It is located from the modern coastline of the Aegean Sea on a ...
.
that the eccentricity is a positive solution of the equation :\Delta e^4+ A+C)^2-4\Delta^2- A+C)^2-4\Delta0, where again \Delta =AC-\frac. This has precisely one positive solution—the eccentricity— in the case of a parabola or ellipse, while in the case of a hyperbola it has two positive solutions, one of which is the eccentricity.


Conversion to canonical form

In the case of an ellipse or hyperbola, the equation :Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0 \, can be converted to canonical form in transformed variables \tilde, \tilde as :\frac+\frac = 1, or equivalently :\frac+\frac = 1, where \lambda_1 and \lambda_2 are the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the matrix \left( \beginA & B/2\\B/2 & C\end\right) — that is, the solutions of the equation :\lambda^2-(A+C)\lambda +(AC-(B/2)^2) = 0 — and S is the determinant of the 3 × 3 matrix above, and \Delta = \lambda_1\lambda_2 is again the determinant of the 2 × 2 matrix. In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form.


Polar coordinates

In
polar coordinates 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. The reference point (analogous to the or ...
, a conic section with one focus at the origin and, if any, the other at a negative value (for an ellipse) or a positive value (for a hyperbola) on the -axis, is given by the equation : r=\frac, where is the eccentricity and is the semi-latus rectum. As above, for , the graph is a circle, for the graph is an ellipse, for a parabola, and for a hyperbola. The polar form of the equation of a conic is often used in dynamics; for instance, determining the orbits of objects revolving about the Sun.


Properties

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). There are additional subtleties for conics that do not exist for lines, and thu ...
. Formally, given any five points in the plane in
general linear position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
, meaning no three
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
, 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. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion. Four points in the plane in general linear position determine a unique conic passing through the first three points and having the fourth point as its center. Thus knowing the center is equivalent to knowing two points on the conic for the purpose of determining the curve. Whitworth, William Allen. ''Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions'', Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866), p. 203. Furthermore, a conic is determined by any combination of ''k'' points in general position that it passes through and 5 – ''k'' lines that are tangent to it, for 0≤''k''≤5. Any point in the plane is on either zero, one or two
tangent line 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. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s of a conic. A point on just one tangent line is on the conic. A point on no tangent line is said to be an ''interior point'' (or inner point) of the conic, while a point on two tangent lines is an ''exterior point'' (or outer point). All the conic sections share a ''reflection property'' that can be stated as: All mirrors in the shape of a non-degenerate conic section reflect light coming from or going toward one focus toward or away from the other focus. In the case of the parabola, the second focus needs to be thought of as infinitely far away, so that the light rays going toward or coming from the second focus are parallel.
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 (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined ...
concerns the collinearity of three points that are constructed from a set of six points on any non-degenerate conic. The theorem also holds for degenerate conics consisting of two lines, but in that case it is known as Pappus's theorem. Non-degenerate conic sections are always "
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
". This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure
laminar flow In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
and to prevent
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
.


History


Menaechmus and early works

It is believed that the first definition of a conic section was given by
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.'' Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, w ...
(died 320 BC) as part of his solution of the Delian problem ( Duplicating the cube). His work did not survive, not even the names he used for these curves, and is only known through secondary accounts. The definition used at that time differs from the one commonly used today. Cones were constructed by rotating a right triangle about one of its legs so the hypotenuse generates the surface of the cone (such a line is called a
generatrix In geometry, a generatrix () or describent is a point, curve or surface that, when moved along a given path, generates a new shape. The path directing the motion of the generatrix motion is called a directrix or dirigent. Examples A cone can be ...
). Three types of cones were determined by their vertex angles (measured by twice the angle formed by the hypotenuse and the leg being rotated about in the right triangle). The conic section was then determined by intersecting one of these cones with a plane drawn perpendicular to a generatrix. The type of the conic is determined by the type of cone, that is, by the angle formed at the vertex of the cone: If the angle is acute then the conic is an ellipse; if the angle is right then the conic is a parabola; and if the angle is obtuse then the conic is a hyperbola (but only one branch of the curve).
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
(fl. 300 BC) is said to have written four books on conics but these were lost as well.
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
(died BC) is known to have studied conics, having determined the area bounded by a parabola and a chord in ''Quadrature of the Parabola''. His main interest was in terms of measuring areas and volumes of figures related to the conics and part of this work survives in his book on the solids of revolution of conics, ''On Conoids and Spheroids''.


Apollonius of Perga

The greatest progress in the study of conics by the ancient Greeks is due to
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
(died BC), whose eight-volume ''Conic Sections'' or ''Conics'' summarized and greatly extended existing knowledge. Apollonius's study of the properties of these curves made it possible to show that any plane cutting a fixed double cone (two napped), regardless of its angle, will produce a conic according to the earlier definition, leading to the definition commonly used today. Circles, not constructible by the earlier method, are also obtainable in this way. This may account for why Apollonius considered circles a fourth type of conic section, a distinction that is no longer made. Apollonius used the names 'ellipse', 'parabola' and 'hyperbola' for these curves, borrowing the terminology from earlier Pythagorean work on areas.
Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
(died AD) is credited with expounding on the importance of the concept of a conic's focus, and detailing the related concept of a directrix, including the case of the parabola (which is lacking in Apollonius's known works).


Al-Kuhi

An instrument for drawing conic sections was first described in 1000 AD by the Islamic mathematician Al-Kuhi.


Omar Khayyám

Apollonius's work was translated into Arabic, and much of his work only survives through the Arabic version. Persians found applications of the theory, most notably the Persian mathematician and poet Omar Khayyám, who found a geometrical method of solving
cubic equations In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
using conic sections.


Europe

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
extended the theory of conics through the "
principle of continuity The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite". Kepler use ...
", a precursor to the concept of limits. Kepler first used the term 'foci' in 1604.
Girard Desargues Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are ...
and
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
developed a theory of conics using an early form of
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
and this helped to provide impetus for the study of this new field. In particular, Pascal discovered a theorem known as the hexagrammum mysticum from which many other properties of conics can be deduced.
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
and
Pierre Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
both applied their newly discovered
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
to the study of conics. This had the effect of reducing the geometrical problems of conics to problems in algebra. However, it was
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
in his 1655 treatise who first defined the conic sections as instances of equations of second degree. Written earlier, but published later,
Jan de Witt Johan de Witt (; 24 September 1625 – 20 August 1672), ''lord of Zuid- en Noord-Linschoten, Snelrewaard, Hekendorp en IJsselvere'', was a Dutch statesman and a major political figure in the Dutch Republic in the mid-17th century, the Fi ...
's starts with Kepler's
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
construction of the conics and then develops the algebraic equations. This work, which uses Fermat's methodology and Descartes' notation has been described as the first textbook on the subject.. De Witt invented the term 'directrix'.


Applications

Conic sections are important in
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
: the
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
s of two massive objects that interact according to
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
are conic sections if their common
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See
two-body problem In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
. The reflective properties of the conic sections are used in the design of searchlights, radio-telescopes and some optical telescopes. A searchlight uses a parabolic mirror as the reflector, with a bulb at the focus; and a similar construction is used for a parabolic microphone. The 4.2 meter Herschel optical telescope on La Palma, in the Canary islands, uses a primary parabolic mirror to reflect light towards a secondary hyperbolic mirror, which reflects it again to a focus behind the first mirror.


In the real projective plane

The conic sections have some very similar properties in the Euclidean plane and the reasons for this become clearer when the conics are viewed from the perspective of a larger geometry. The Euclidean plane may be embedded in the
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. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
and the conics may be considered as objects in this projective geometry. One way to do this is to introduce
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
and define a conic to be the set of points whose coordinates satisfy an irreducible quadratic equation in three variables (or equivalently, the zeros of an irreducible
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
). More technically, the set of points that are zeros of a quadratic form (in any number of variables) is called a
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
, and the irreducible quadrics in a two dimensional projective space (that is, having three variables) are traditionally called conics. The Euclidean plane is embedded in the real projective plane by adjoining a
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. The ...
(and its corresponding
points at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
) so that all the lines of a parallel class meet on this line. On the other hand, starting with the real projective plane, a Euclidean plane is obtained by distinguishing some line as the line at infinity and removing it and all its points.


Intersection at infinity

In a
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
over any division ring, but in particular over either the real or complex numbers, all non-degenerate conics are equivalent, and thus in projective geometry one speaks of "a conic" without specifying a type. That is, there is a projective transformation that will map any non-degenerate conic to any other non-degenerate conic. The three types of conic sections will reappear in the affine plane obtained by choosing a line of the projective space to be the line at infinity. The three types are then determined by how this line at infinity intersects the conic in the projective space. In the corresponding affine space, one obtains an ellipse if the conic does not intersect the line at infinity, a parabola if the conic intersects the line at infinity in one
double point In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curv ...
corresponding to the axis, and a hyperbola if the conic intersects the line at infinity in two points corresponding to the asymptotes.


Homogeneous coordinates

In
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
a conic section can be represented as: :Ax^2 + Bxy + Cy^2 +Dxz + Eyz + Fz^2 = 0. Or in
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
notation : \left(\beginx & y & z\end\right) \left( \beginA & B/2 & D/2\\B/2 & C & E/2\\D/2&E/2&F\end\right) \left(\beginx\\y\\z\end\right) = 0. The 3 × 3 matrix above is called ''the matrix of the conic section''. Some authors prefer to write the general homogeneous equation as :Ax^2 + 2Bxy + Cy^2 +2Dxz + 2Eyz + Fz^2 = 0, (or some variation of this) so that the matrix of the conic section has the simpler form, :M = \left( \beginA & B & D\\B & C & E\\D&E&F\end\right), but this notation is not used in this article. If the determinant of the matrix of the conic section is zero, the conic section is
degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
. As multiplying all six coefficients by the same non-zero scalar yields an equation with the same set of zeros, one can consider conics, represented by as points in the five-dimensional
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
\mathbf^5.


Projective definition of a circle

Metrical concepts of Euclidean geometry (concepts concerned with measuring lengths and angles) can not be immediately extended to the real projective plane. They must be redefined (and generalized) in this new geometry. This can be done for arbitrary
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
s, but to obtain the real projective plane as the extended Euclidean plane, some specific choices have to be made. Fix an arbitrary line in a projective plane that shall be referred to as the ''absolute line''. Select two distinct points on the absolute line and refer to them as ''absolute points''. Several metrical concepts can be defined with reference to these choices. For instance, given a line containing the points and , the ''midpoint'' of line segment is defined as the point which is the
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: :Given three collinear points , let be a point not lying on their join and let any line t ...
of the point of intersection of and the absolute line, with respect to and . A conic in a projective plane that contains the two absolute points is called a ''circle''. Since five points determine a conic, a circle (which may be degenerate) is determined by three points. To obtain the extended Euclidean plane, the absolute line is chosen to be the line at infinity of the Euclidean plane and the absolute points are two special points on that line called the
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. Coordinates ...
. Lines containing two points with real coordinates do not pass through the circular points at infinity, so in the Euclidean plane a circle, under this definition, is determined by three points that are not
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
. It has been mentioned that circles in the Euclidean plane can not be defined by the focus-directrix property. However, if one were to consider the line at infinity as the directrix, then by taking the eccentricity to be a circle will have the focus-directrix property, but it is still not defined by that property. One must be careful in this situation to correctly use the definition of eccentricity as the ratio of the distance of a point on the circle to the focus (length of a radius) to the distance of that point to the directrix (this distance is infinite) which gives the limiting value of zero.


Steiner's projective conic definition

A synthetic (coordinate-free) approach to defining the conic sections in a projective plane was given by
Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
in 1867. * Given two pencils B(U),B(V) of lines at two points U,V (all lines containing U and V resp.) and a projective but not perspective mapping \pi of B(U) onto B(V). Then the intersection points of corresponding lines form a non-degenerate projective conic section. A ''perspective'' mapping \pi of a pencil B(U) onto a pencil B(V) is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
(1-1 correspondence) such that corresponding lines intersect on a fixed line a, which is called the ''axis'' of the perspectivity \pi. A ''projective'' mapping is a finite sequence of perspective mappings. As a projective mapping in a projective plane over a field (
pappian plane In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of Collinearity, collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line (mathemati ...
) is uniquely determined by prescribing the images of three lines, for the Steiner generation of a conic section, besides two points U,V only the images of 3 lines have to be given. These 5 items (2 points, 3 lines) uniquely determine the conic section.


Line conics

By the Principle of Duality in a projective plane, the dual of each point is a line, and the dual of a locus of points (a set of points satisfying some condition) is called an ''envelope'' of lines. Using Steiner's definition of a conic (this locus of points will now be referred to as a ''point conic'') as the meet of corresponding rays of two related pencils, it is easy to dualize and obtain the corresponding envelope consisting of the joins of corresponding points of two related ranges (points on a line) on different bases (the lines the points are on). Such an envelope is called a ''line conic'' (or dual conic). In the real projective plane, a point conic has the property that every line meets it in two points (which may coincide, or may be complex) and any set of points with this property is a point conic. It follows dually that a line conic has two of its lines through every point and any envelope of lines with this property is a line conic. At every point of a point conic there is a unique tangent line, and dually, on every line of a line conic there is a unique point called a ''point of contact''. An important theorem states that the tangent lines of a point conic form a line conic, and dually, the points of contact of a line conic form a point conic.


Von Staudt's definition

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. Life and influence Karl was born in the Free Imperial City of Rothenburg, which is n ...
defined a conic as the point set given by all the absolute points of a ''polarity'' that has absolute points. Von Staudt introduced this definition in ''Geometrie der Lage'' (1847) as part of his attempt to remove all metrical concepts from projective geometry. A ''polarity'', , of a projective plane is an involutory
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between the points and the lines of that preserves the
incidence relation In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element ...
. Thus, a polarity associates a point with a line by and . Following
Gergonne Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician. Life In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion becau ...
, is called the ''polar'' of and the ''pole'' of . An ''absolute point'' (or ''line'') of a polarity is one which is incident with its polar (pole). A von Staudt conic in the real projective plane is equivalent to a
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. The usual d ...
.


Constructions

No continuous arc of a conic can be constructed with straightedge and compass. However, there are several straightedge-and-compass constructions for any number of individual points on an arc. One of them is based on the converse of Pascal's theorem, namely, ''if the points of intersection of opposite sides of a hexagon are collinear, then the six vertices lie on a conic.'' Specifically, given five points, and a line passing through , say , a point that lies on this line and is on the conic determined by the five points can be constructed. Let meet in , meet in and let meet at . Then meets at the required point . By varying the line through ,as many additional points on the conic as desired can be constructed. Another method, based on Steiner's construction and which is useful in engineering applications, is the parallelogram method, where a conic is constructed point by point by means of connecting certain equally spaced points on a horizontal line and a vertical line. Specifically, to construct the ellipse with equation , first construct the rectangle with vertices and . Divide the side into equal segments and use parallel projection, with respect to the diagonal , to form equal segments on side (the lengths of these segments will be times the length of the segments on ). On the side label the left-hand endpoints of the segments with to starting at and going towards . On the side label the upper endpoints to starting at and going towards . The points of intersection, for will be points of the ellipse between and . The labeling associates the lines of the pencil through with the lines of the pencil through projectively but not perspectively. The sought for conic is obtained by this construction since three points and and two tangents (the vertical lines at and ) uniquely determine the conic. If another diameter (and its conjugate diameter) are used instead of the major and minor axes of the ellipse, a parallelogram that is not a rectangle is used in the construction, giving the name of the method. The association of lines of the pencils can be extended to obtain other points on the ellipse. The constructions for hyperbolas and parabolas are similar. Yet another general method uses the polarity property to construct the tangent envelope of a conic (a line conic).


In the complex projective plane

In the complex plane , ellipses and hyperbolas are not distinct: one may consider a hyperbola as an ellipse with an imaginary axis length. For example, the ellipse x^2+y^2=1 becomes a hyperbola under the substitution y=iw, geometrically a complex rotation, yielding x^2-w^2=1. Thus there is a 2-way classification: ellipse/hyperbola and parabola. Extending the curves to the complex projective plane, this corresponds to intersecting the
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. The ...
in either 2 distinct points (corresponding to two asymptotes) or in 1 double point (corresponding to the axis of a parabola); thus the real hyperbola is a more suggestive real image for the complex ellipse/hyperbola, as it also has 2 (real) intersections with the line at infinity. Further unification occurs in the
complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
: the non-degenerate conics cannot be distinguished from one another, since any can be taken to any other by a
projective linear transformation 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. It is a bijection that maps line (geometry), lines to lines, and thus a collineati ...
. It can be proven that in , two conic sections have four points in common (if one accounts for
multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using mult ...
), so there are between 1 and 4 intersection points. The intersection possibilities are: four distinct points, two singular points and one double point, two double points, one singular point and one with multiplicity 3, one point with multiplicity 4. If any intersection point has multiplicity > 1, the two curves are said to be
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. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
. If there is an intersection point of multiplicity at least 3, the two curves are said to be osculating. If there is only one intersection point, which has multiplicity 4, the two curves are said to be ''superosculating''. Furthermore, each
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
intersects each conic section twice. If the intersection point is double, the line is a
tangent line 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. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
. Intersecting with the line at infinity, each conic section has two points at infinity. If these points are real, the curve is 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, cal ...
; if they are imaginary conjugates, it is an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
; if there is only one double point, it is a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
. If the points at infinity are the
cyclic points 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. Coordinates ...
and , the conic section is 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 const ...
. If the coefficients of a conic section are real, the points at infinity are either real or
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. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.


Degenerate cases

What should be considered as a ''degenerate'' case of a conic depends on the definition being used and the geometric setting for the conic section. There are some authors who define a conic as a two-dimensional nondegenerate quadric. With this terminology there are no degenerate conics (only degenerate quadrics), but we shall use the more traditional terminology and avoid that definition. In the Euclidean plane, using the geometric definition, a degenerate case arises when the cutting plane passes through the
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex, ...
of the cone. The degenerate conic is either: a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
, when the plane intersects the cone only at the apex; a
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
, when the plane is tangent to the cone (it contains exactly one generator of the cone); or a pair of intersecting lines (two generators of the cone). These correspond respectively to the limiting forms of an ellipse, parabola, and a hyperbola. If a conic in the Euclidean plane is being defined by the zeros of a quadratic equation (that is, as a quadric), then the degenerate conics are: the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
, a point, or a pair of lines which may be parallel, intersect at a point, or coincide. The empty set case may correspond either to a pair of
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. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
parallel lines such as with the equation x^2+1=0, or to an ''imaginary ellipse'', such as with the equation x^2 +y^2+1=0. An imaginary ellipse does not satisfy the general definition of a degeneracy, and is thus not normally considered as degenerated. The two lines case occurs when the quadratic expression factors into two linear factors, the zeros of each giving a line. In the case that the factors are the same, the corresponding lines coincide and we refer to the line as a ''double'' line (a line with
multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using mult ...
2) and this is the previous case of a tangent cutting plane. In the real projective plane, since parallel lines meet at a point on the line at infinity, the parallel line case of the Euclidean plane can be viewed as intersecting lines. However, as the point of intersection is the apex of the cone, the cone itself degenerates to a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
, i.e. with the apex at infinity. Other sections in this case are called ''cylindric sections''. The non-degenerate cylindrical sections are ellipses (or circles). When viewed from the perspective of the complex projective plane, the degenerate cases of a real quadric (i.e., the quadratic equation has real coefficients) can all be considered as a pair of lines, possibly coinciding. The empty set may be the line at infinity considered as a double line, a (real) point is the intersection of two complex conjugate lines and the other cases as previously mentioned. To distinguish the degenerate cases from the non-degenerate cases (including the empty set with the latter) using matrix notation, let be the determinant of the 3 × 3 matrix of the conic section—that is, ; and let be the discriminant. Then the conic section is non-degenerate if and only if . If we have a point when , two parallel lines (possibly coinciding) when , or two intersecting lines when .


Pencil of conics

A (non-degenerate) conic is completely determined by five points in general position (no three
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a ''pencil of conics''. The four common points are called the ''base points'' of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a
pencil of circles In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane. Although the definiti ...
.


Intersecting two conics

The solutions to a system of two second degree equations in two variables may be viewed as the coordinates of the points of intersection of two generic conic sections. In particular two conics may possess none, two or four possibly coincident intersection points. An efficient method of locating these solutions exploits the homogeneous
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. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relatio ...
, i.e. a 3 × 3
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
which depends on six parameters. The procedure to locate the intersection points follows these steps, where the conics are represented by matrices: * given the two conics C_1 and C_2, consider the pencil of conics given by their linear combination \lambda C_1 + \mu C_2. * identify the homogeneous parameters (\lambda,\mu) which correspond to the degenerate conic of the pencil. This can be done by imposing the condition that \det(\lambda C_1 + \mu C_2) = 0 and solving for \lambda and \mu. These turn out to be the solutions of a third degree equation. * given the degenerate conic C_0, identify the two, possibly coincident, lines constituting it. * intersect each identified line with either one of the two original conics; this step can be done efficiently using the dual conic representation of C_0 * the points of intersection will represent the solutions to the initial equation system.


Generalizations

Conics may be defined over other fields (that is, in other pappian plane, pappian geometries). However, some care must be used when the field has Characteristic (field), characteristic 2, as some formulas can not be used. For example, the matrix representations used #General Cartesian form, above require division by 2. A generalization of a non-degenerate conic in a projective plane is an oval (projective plane), oval. An oval is a point set that has the following properties, which are held by conics: 1) any line intersects an oval in none, one or two points, 2) at any point of the oval there exists a unique tangent line. Generalizing the focus properties of conics to the case where there are more than two foci produces sets called generalized conics. The intersection of an Cone#Elliptic cone, elliptic cone with a sphere is a
spherical conic In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in th ...
, which shares many properties with planar conics.


In other areas of mathematics

The classification into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a field into sharply distinct subfields. The classification mostly arises due to the presence of a quadratic form (in two variables this corresponds to the associated
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
), but can also correspond to eccentricity. Quadratic form classifications: ;Quadratic forms: Quadratic forms over the reals are classified by Sylvester's law of inertia, namely by their positive index, zero index, and negative index: a quadratic form in ''n'' variables can be converted to a diagonal form, as x_1^2+x_2^2+\cdots+x_k^2-x_^2-\cdots-x_^2, where the number of +1 coefficients, ''k,'' is the positive index, the number of −1 coefficients, ''ℓ'', is the negative index, and the remaining variables are the zero index ''m,'' so k+\ell+m=n. In two variables the non-zero quadratic forms are classified as: :* x^2+y^2 – positive-definite (the negative is also included), corresponding to ellipses, :* x^2 – degenerate, corresponding to parabolas, and :* x^2-y^2 – indefinite, corresponding to hyperbolas. :In two variables quadratic forms are classified by discriminant, analogously to conics, but in higher dimensions the more useful classification is as ''definite,'' (all positive or all negative), ''degenerate,'' (some zeros), or ''indefinite'' (mix of positive and negative but no zeros). This classification underlies many that follow. ;Curvature: The Gaussian curvature of a surface (mathematics), surface describes the infinitesimal geometry, and may at each point be either positive – elliptic geometry, zero –
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
(flat, parabola), or negative – hyperbolic geometry; infinitesimally, to second order the surface looks like the graph of x^2+y^2, x^2 (or 0), or x^2-y^2. Indeed, by the uniformization theorem every surface can be taken to be globally (at every point) positively curved, flat, or negatively curved. In higher dimensions the Riemann curvature tensor is a more complicated object, but manifolds with constant sectional curvature are interesting objects of study, and have strikingly different properties, as discussed at sectional curvature. ;Second order PDEs: Partial differential equations (PDEs) of Partial differential equation#Linear equations of second order, second order are classified at each point as elliptic, parabolic, or hyperbolic, accordingly as their second order terms correspond to an elliptic, parabolic, or hyperbolic quadratic form. The behavior and theory of these different types of PDEs are strikingly different – representative examples is that the Poisson equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic. Eccentricity (mathematics)#Analogous classifications, Eccentricity classifications include: ;Möbius transformations: Real Möbius transformations (elements of PSL2(R), or its 2-fold cover, SL2(R), ) are Möbius transformation#Classification, classified as elliptic, parabolic, or hyperbolic accordingly as their half-trace is 0\leq , \operatorname, /2 < 1, , \operatorname, /2 = 1, or , \operatorname, /2 > 1, mirroring the classification by eccentricity. ;Variance-to-mean ratio: The variance-to-mean ratio classifies several important families of discrete probability distributions: the constant distribution as circular (eccentricity 0), binomial distributions as elliptical, Poisson distributions as parabolic, and negative binomial distributions as hyperbolic. This is elaborated at Cumulant#Cumulants of some discrete probability distributions, cumulants of some discrete probability distributions.


See also

* Circumconic and inconic * Conic Sections Rebellion, protests by Yale university students * Director circle * Elliptic coordinate system * Equidistant set * Nine-point conic * Parabolic coordinates * Quadratic function * Spherical conic


Notes


References


Bibliography

* * * * * * * * * * (PDF; 891 kB). * * * * * * * * *


External links


Can You Really Derive Conic Formulae from a Cone?archive 2007-07-15
Gary S. Stoudt (Indiana University of Pennsylvania
Conic sections
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* Se
Conic Sections
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cut-the-knot
for a sharp proof that any finite conic section is an ellipse an

for a similar treatment of other conics.

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An interactive Java conics grapher; uses a general second-order implicit equation. {{Authority control Conic sections, * Euclidean solid geometry Algebraic curves Birational geometry Analytic geometry