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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a conic section (or just conic) is a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 obtained by intersecting a cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
 (more precisely, a circular conical surface
Conical surface

In geometry, a conical surface is the unbounded surface formed by the union of all the straight line that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex....
) with a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
. A conic section is therefore a restriction of a quadric
Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
 surface to the plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
.






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Table of Conics, Cyclopaedia, Volume 1, P 304, 1728
In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a conic section (or just conic) is a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 obtained by intersecting a cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
 (more precisely, a circular conical surface
Conical surface

In geometry, a conical surface is the unbounded surface formed by the union of all the straight line that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex....
) with a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
. A conic section is therefore a restriction of a quadric
Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
 surface to the plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga
Apollonius of Perga

Apollonius of Perga [Pergaeus] was a Greeks geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and Ren? Descartes....
 undertook a systematic study of their properties.

Types of conics

The three types of conics are the hyperbola
Hyperbola

In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
, ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
, and parabola
Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
. The circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 can be considered as a fourth type (as it was by Apollonius) or as a kind of ellipse. The circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 and the ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 arise when the intersection of cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone. If the cutting plane is parallel
Parallel

From Greek language: pa???????? Parallel may refer to:...
 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. In this case, the plane will intersect both halves (nappes) of the cone, producing two separate unbounded curves, though often one is ignored.

Degenerate cases

There are multiple degenerate cases, in which the plane passes through the apex of the cone. The intersection in these cases can be a straight line (when the plane is tangential to the surface of the cone); a point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 (when the angle between the plane and the axis of the cone is larger than tangential); or a pair of intersecting lines (when the angle is smaller).

Where the cone is a cylinder
Cylinder

Cylinder may refer to:* Cylinder , a three-dimensional geometric shape* Cylinder , the cartesian product of a set with its superset* Cylinder , the space within which a piston travels in an engine...
, i.e. with the vertex at infinity, cylindric sections are obtained. Although these yield mostly ellipses (or circles) as usual, a degenerate case of two parallel lines, known as a ribbon, can also be produced, and it is also possible for there to be no intersection at all.

Eccentricity

Eccentricity
The four defining conditions above can be combined into one condition that depends on a fixed point F (the focus), a line L (the directrix) not containing F and a nonnegative real number e (the eccentricity
Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or , is a parameter associated with every Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular....
). The corresponding conic section consists of all points whose distance to F equals e times their distance to L. For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.

For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is , where is the semi-major axis
Semi-major axis

In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
 of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is .

In the case of a circle, the eccentricity e = 0, and one can imagine the directrix to be infinitely far removed from the center. However, the statement that the circle consists of all points whose distance is e times the distance to L is not useful, because we get zero times infinity.

The eccentricity of a conic section is thus a measure of how far it deviates from being circular.

For a given , the closer is to 1, the smaller is the semi-minor axis
Semi-minor axis

In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis....
.

Cartesian coordinates

In the Cartesian coordinate system
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, the graph
Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian coordinate system, together with Cartesian axes, etc....
 of a quadratic equation
Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
 in two variables is always a conic section, and all conic sections arise in this way. The equation will be of the form with , , not all zero. then:
  • if , the equation represents an ellipse
    Ellipse

    In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
     (unless the conic is degenerate, for example );
    • if and , the equation represents a circle
      Circle

      A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
      ;
  • if , the equation represents a parabola
    Parabola

    In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
    ;
  • if , the equation represents a hyperbola
    Hyperbola

    In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
    ;
    • if we also have , the equation represents a rectangular hyperbola
      Hyperbola

      In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
      .


Note that A and B are just polynomial coefficients, not the lengths of semi-major/minor axis as defined in the following sections.

Through change of coordinates these equations can be put in standard forms:
  • Circle:
  • Ellipse: ,
  • Parabola: ,
  • Hyperbola: ,
  • Rectangular Hyperbola:


Such forms will be symmetrical about the x-axis and for the circle, ellipse and hyperbola symmetrical about the y-axis.
The rectangular hyperbola however is only symmetrical about the lines and . Therefore its inverse function is exactly the same as its original function.

These standard forms can be written as parametric equation
Parametric equation

In mathematics, parametric equations are a method of defining a curve. A simple kinematics example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
s,
  • Circle: ,
  • Ellipse: ,
  • Parabola: ,
  • Hyperbola: or .
  • Rectangular Hyperbola:


Homogeneous coordinates

In homogeneous coordinates
Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand M?bius in his 1827 work Der barycentrische Calc?l, allow affine transformations to be easily represented by a matrix....
 a conic section can be represented as:

Or in matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 notation



The matrix is called the matrix of the conic section.

is called the determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
 of the conic section. If ? = 0 then the conic section is said to be degenerate, this means that the conic section is in fact a union of two straight lines. A conic section that intersects itself is always degenerate, however not all degenerate conic sections intersect themselves, if they do not they are straight lines.

For example, the conic section reduces to the union of two lines:

.

Similarly, a conic section sometimes reduces to a (single) line:

.

is called the discriminant
Discriminant

In algebra, the discriminant of a polynomial with real number or complex number coefficients is a certain expression in the coefficients of the polynomial which is equal to zero if and only if the polynomial has a multiple Root in the complex numbers....
 of the conic section. If d = 0 then the conic section is a parabola
Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
, if d<0, it is an hyperbola
Hyperbola

In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
 and if d>0, it is an ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
. A conic section is a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 if d>0 and A1 = A2, it is an rectangular hyperbola if d<0 and A1 = -A2. It can be proven that in the complex projective plane
Complex projective plane

In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. It is a complex manifold described by three complex coordinates...
 CP2 two conic sections have four points in common (if one accounts for multiplicity), so there are never more than 4 intersection
Line-line intersection

In Euclidean geometry, the intersection of a line and a line can be the empty set,a point , ora line. Distinguishing these cases, and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection....
 points and there is always 1 intersection point (possibilities: 4 distinct intersection points, 2 singular intersection points and 1 double intersection points, 2 double intersection points, 1 singular intersection point and 1 with multiplicity 3, 1 intersection point with multiplicity 4). If there exists at least one intersection point with multiplicity > 1, then the two conic sections are said to be tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
. If there is only one intersection point, which has multiplicity 4, the two conic sections are said to be osculating.

Furthermore each straight line intersects each conic section twice. If the intersection point is double, the line is said to be tangent and it is called the tangent line. Because every straight line intersects a conic section twice, each conic section has two points at infinity
Real projective plane

In mathematics, the real projective plane is the space of lines in R3 passing through the origin. It is a non-Orientability two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedding in our usual three-dimensional space without intersecting itself....
 (the intersection points with the line at infinity
Line at infinity

In geometry and topology, the line at infinity is a line which is added to the real plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane....
). If these points are real, the conic section must be a hyperbola
Hyperbola

In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
, if they are imaginary conjugated, the conic section must be an ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
, if the conic section has one double point at infinity it is a parabola
Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
. If the points at infinity are (1,i,0) and (1,-i,0), the conic section is a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
. If a conic section has one real and one imaginary point at infinity or it has two imaginary points that are not conjugated it is neither a parabola nor an ellipse nor a hyperbola.

Polar coordinates

In polar coordinates
Polar coordinate system

In mathematics, the polar coordinate system is a dimension coordinate system in which each point on a plane is determined by an angle and a distance....
, a conic section with one focus at the origin and, if any, the other on the x-axis, is given by the equation
,
where e is the eccentricity and l is the semi-latus rectum (see below). As above, for e = 0, we have a circle, for 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.

Parameters

Various parameters can be associated with a conic section.

conic section equation eccentricity (e) linear eccentricity (c) semi-latus rectum (l) focal parameter (p)
circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 		;
parabola
Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
 		;
hyperbola
Hyperbola

In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
 		;


For every conic section, there exist a fixed point F, a fixed line L and a non-negative number e such that the conic section consists of all points whose distance to F equals e times their distance to L. e is called the eccentricity
Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or , is a parameter associated with every Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular....
 of the conic section.

The linear eccentricity (c) is the distance between the center and the focus
Focus (geometry)

In geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola....
 (or one of the two foci).

The latus rectum (2l) is the chord parallel to the directrix and passing through the focus (or one of the two foci).

The semi-latus rectum (l) is half the latus rectum.

The focal parameter (p) is the distance
Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
 from the focus (or one of the two foci) to the directrix.

The relation holds.

Properties

Conic sections are always "smooth". More precisely, they never contain any inflection point
Inflection point

In differential calculus, an inflection point, or point of inflection is a point on a curve at which the curvature changes Negative and non-negative numbers....
s. This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow
Laminar flow

Laminar flow, sometimes known as Streamlines, streaklines and pathlines flow, occurs when a fluid flows in parallel layers, with no disruption between the layers....
 and to prevent turbulence
Turbulence

In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time....
.

Applications

Conic sections are important in astronomy
Astronomy

Astronomy is the science of Astronomical object and Phenomenon that originate outside the Earth's atmosphere . It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the physical cosmology....
: the orbit
ORBit

ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
s of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 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
N-body problem

The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i.e., Newton's laws of motion and gravity....
.

In projective geometry
Projective geometry

In mathematics projective geometry is the study of geometric properties which are invariant under projective transformations. The field of projective geometry is itself divided into many subfields, two examples of which are projective algebraic geometry and projective differential geometry ....
, the conic sections in the projective plane are equivalent to each other up to
Up to

In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e....
 projective transformation
Projective transformation

A projective transformation is a Transformation used in projective geometry: it is the composition of a pair of perspective projections. It describes what happens to the perceived positions of observed objects when the point of view of the observer changes....
s.

For specific applications of each type of conic section, see the articles circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
, ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
, parabola
Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
, and hyperbola
Hyperbola

In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
.

Intersecting two conics

The solutions to a two second degree equations system in two variables may be seen as the coordinates of the intersections of two generic conic sections. In particular two conics may possess none, two, four possibly coincident intersection points. The best method of locating these solutions exploits the homogeneous matrix representation of conic sections
Matrix representation of conic sections

In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis of rotation, vertex , Focus , tangents, and the relative position of a given point....
, i.e. a 3x3 symmetric matrix
Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix, A, that is equal to its transposeThe entries of a symmetric matrix are symmetric with respect to the main diagonal ....
 which depends on six parameters.

The procedure to locate the intersection points follows these steps:
  • given the two conics and consider the pencil of conics given by their linear combination
  • identify the homogeneous parameters which corresponds to the degenerate conic of the pencil. This can be done by imposing that , which turns out to be the solution to a third degree equation.
  • given the degenerate cone , identify the two, possibly coincident, lines constituting it
  • intersects each identified line with one of the two original conic; this step can be done efficiently using the dual conic representation of
  • the points of intersection will represent the solution to the initial equation system


Dandelin spheres

See Dandelin spheres
Dandelin spheres

In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus:This concept is named in honor of Germinal Pierre Dandelin....
 for a short elementary argument showing that the characterization of these curves as intersections of a plane with a cone is equivalent to the characterization in terms of foci, or of a focus and a directrix.

See also

  • Focus (geometry)
    Focus (geometry)

    In geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola....
    , an overview of properties of conic sections related to the foci
  • Lambert conformal conic projection
    Lambert conformal conic projection

    A Lambert conformal conic projection is a Conic section map projection, which is often used for aeronautical charts. In essence, the projection superimposes a cone over the sphere of the Earth, with two reference Circle of latitudes Secant line to the globe and intersecting it....
  • Matrix representation of conic sections
    Matrix representation of conic sections

    In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis of rotation, vertex , Focus , tangents, and the relative position of a given point....
  • Quadric
    Quadric

    In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
    s, the higher-dimensional analogs of conics
  • Quadratic function
    Quadratic function

    A quadratic function, in mathematics, is a polynomial function of the form , where . The graph of a function of a quadratic function is a parabola whose major axis is parallel to the y-axis....
  • Rotation of Axes
    Rotation of Axes

    Rotation of Axes is a form of Euclidean transformation in which the entire xy-coordinate system is rotated in the counter-clockwise direction with respect to the origin through a scalar quantity denoted by ?....
  • Dandelin spheres
    Dandelin spheres

    In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus:This concept is named in honor of Germinal Pierre Dandelin....
  • Projective conics
    Projective harmonic conjugates

    In projective geometry, a pair of harmonic conjugate points on the real projective line is defined by the following harmonic construction:So is the harmonic construction introduced by Goodstein and Primrose ....


External links

  • at
  • at .
  • .
  • . An essay on conics and how they are generated.
  • See at for a sharp proof that any finite conic section is an ellipse and for a similar treatment of other conics.