Conic section

In mathematics, a conic section (or simply conic) is

a curve obtained as the intersection of the surface of a
cone with a plane. The three types of conic section are
the hyperbola, the parabola, and the ellipse; the circle
is a special case of the ellipse, though historically it
was sometimes called a fourth type. The ancient Greek
mathematicians studied conic sections, culminating
around 200 BC with Apollonius of Perga's systematic
work on their properties.

The conic sections in the Euclidean plane have various Types of conic sections:
distinguishing properties, many of which can be used 1. Parabola
as alternative definitions. One such property defines a 2. Circle and ellipse
non-circular conic [1] to be the set of those points 3. Hyperbola
whose distances to some particular point, called a
focus, and some particular line, called a directrix, are
in a fixed ratio, called the eccentricity. The type of conic is
determined by the value of the eccentricity. In analytic
geometry, a conic may be defined as a plane algebraic curve
of degree 2; that is, as the set of points whose coordinates
satisfy a quadratic equation in two variables. This equation
may be written in matrix form, and some geometric
properties can be studied as algebraic conditions.

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, 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 coordinates
to admit complex coordinates, provides the means to see this
unification algebraically.

Contents Table of conics, Cyclopaedia, 1728

Euclidean geometry
Definition
Eccentricity, focus and directrix
Conic parameters
 Standard forms in Cartesian coordinates
General Cartesian form
Matrix notation
Discriminant
Invariants
Eccentricity in terms of coefficients
Conversion to canonical form
Polar coordinates
Properties
History
Menaechmus and early works
Apollonius of Perga
Al-Kuhi
Omar Khayyám
Europe
Applications
In the real projective plane
Intersection at infinity
Homogeneous coordinates
Projective definition of a circle
Steiner's projective conic definition
Line conics
Von Staudt's definition
Constructions
In the complex projective plane
Degenerate cases
Pencil of conics
Intersecting two conics
Generalizations
In other areas of mathematics
See also
Notes
References
External links

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.

Definition
A conic is the curve obtained as the intersection of a
plane, called the cutting plane, with the surface of a
double cone (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, parabola,

and hyperbola. The circle is a special kind of ellipse,
although historically Apollonius considered as a
fourth type. Ellipses arise when the intersection of the The black boundaries of the colored regions are
cone and plane is a closed curve. The circle is obtained conic sections. Not shown is the other half of the
when the cutting plane is parallel to the plane of the hyperbola, which is on the unshown other half of
generating circle of the cone; for a right cone, this the double cone.
means the cutting plane is perpendicular to the axis. If
the cutting plane is parallel 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.

Eccentricity, focus and directrix

Alternatively, one can define a conic section purely in terms of plane geometry: it is the locus of all
points P whose distance to a fixed point F (called the focus) is a constant multiple (called the
eccentricity e) of the distance from P to a fixed line L (called the directrix). For 0 < e < 1 we
obtain an ellipse, for e = 1 a parabola, and for e > 1 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.[2]

The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being
circular.[3]:844
If the angle between the surface of the cone and its axis is and the angle between the cutting plane
and the axis is the eccentricity is[4]

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.[5]

Conic parameters
In addition to the eccentricity (e), 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 (c) 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 (ℓ).

The focal parameter (p) 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 (a). When an ellipse or hyperbola
are in standard position as in the equations below, with
foci on the x-axis and center at the origin, the vertices of Circle (e=0), ellipse (e=1/2), parabola (e=1) and
hyperbola (e=2) with fixed focus F and directrix
the conic have coordinates (−a, 0) and (a, 0), with a
(e=∞).
non-negative.

The minor axis is the shortest diameter of an ellipse,

and its half-length is the semi-minor axis (b), the same value b as in the standard equation below. By
analogy, for a hyperbola we also call the parameter b in the standard equation, the semi-minor axis.

The following relations hold:[6]

For conics in standard position, these parameters have the following values, taking .
 Conic parameters in the case of an ellipse

conic eccentricity linear eccentricity semi-latus rectum focal parameter

equation
section (e) (c) (ℓ ) ( p)

circle

ellipse

parabola N/A

hyperbola

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.[7] By means of a
change of coordinates
(rotation and translation
of axes) these equations
Standard forms of an ellipse
can be put into standard
Standard forms of a parabola
forms.[8] For ellipses and
hyperbolas a standard
form has the x-axis as principal axis and the origin (0,0) as center. The
vertices are (±a, 0) and the foci (±c, 0). Define b by the equations c2 = a2 − b2 for an ellipse and
c2 = a2 + b2 for a hyperbola. For a circle, c = 0 so a2 = b2. For the parabola, the standard form has
the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. 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 x = y is the principal
axis. The foci then have coordinates (c, c) and
(−c, −c).[9]

Circle: x2 + y2 = a2
x2 y2
Ellipse: 2 + 2 = 1
a b
Parabola: y2 = 4ax with a > 0
x2 y2
Hyperbola: 2 − 2 = 1
a b
c2 Standard forms of a hyperbola
Rectangular hyperbola:[10] xy =
2

The first four of these forms are symmetric about both the x-axis and y-axis (for the circle, ellipse and
hyperbola), or about the x-axis only (for the parabola). The rectangular hyperbola, however, is instead
symmetric about the lines y = x and y = −x.

These standard forms can be written parametrically as,

Circle: (a cos θ, a sin θ),

Ellipse: (a cos θ, b sin θ),
Parabola: (at2, 2at),
Hyperbola: (a sec θ, b tan θ) or (±a cosh u, b sinh u),

Rectangular hyperbola: where

General Cartesian form

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a
conic section (though it may be degenerate[11]), and all conic sections arise in this way. The most
general equation is of the form[12]

with all coefficients real numbers and A, B, C not all zero.

Matrix notation
The above equation can be written in matrix notation as[13]

The general equation can also be written as

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 , called
the discriminant of the equation. [14] Thus, the discriminant is − 4Δ where Δ is the matrix
determinant

If the conic is non-degenerate, then:[15]

if B2 − 4AC < 0, the equation represents an ellipse;

if A = C and B = 0, the equation represents a circle, which is a special case of an ellipse;

if B2 − 4AC = 0, the equation represents a parabola;
if B2 − 4AC > 0, the equation represents a hyperbola;
if A + C = 0, the equation represents a rectangular hyperbola.

In the notation used here, A and B are polynomial coefficients, in contrast to some sources that
denote the semimajor and semiminor axes as A and B.

Invariants

The discriminant B2 – 4AC of the conic section's quadratic equation (or equivalently the determinant
AC – B2/4 of the 2×2 matrix) and the quantity A + C (the trace of the 2×2 matrix) are invariant
under arbitrary rotations and translations of the coordinate axes,[15][16][17] as is the determinant of
the 3×3 matrix above.[18]:pp. 60–62 The constant term F and the sum D2+E2 are invariant under
rotation only.[18]:pp. 60–62

Eccentricity in terms of coefficients

When the conic section is written algebraically as

the eccentricity can be written as a function of the coefficients of the quadratic equation.[19] If
4AC = B2 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

where η = 1 if the determinant of the 3×3 matrix above is negative and η = −1 if that determinant is
positive.

It can also be shown[18]:p. 89 that the eccentricity is a positive solution of the equation

where again 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

can be converted to canonical form in transformed variables as[20]

or equivalently

where and are the eigenvalues of the matrix — that is, the solutions of the

equation
— and is the determinant of the 3×3 matrix above, and 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, 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 x-axis, is
given by the equation

where e is the eccentricity and l is the semi-latus rectum.

As above, for e = 0, the graph is a circle, for 0 < e < 1the

graph is an ellipse, for e = 1 a parabola, and for e > 1 a
hyperbola.
Development of the conic section as the
The polar form of the equation of a conic is often used in
eccentricity e increases
dynamics; for instance, determining the orbits of objects
revolving about the Sun. [21]

Properties
Just as two (distinct) points determine a line, five points determine a conic. Formally, given any five
points in the plane in general linear position, meaning no three collinear, there is a unique conic
passing through them, which will be non-degenerate; this is true in both the Euclidean plane and its
extension, the real projective plane. 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.[22]

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.[23]

Any point in the plane is on either zero, one or two tangent lines 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.[24][25]

Pascal's theorem 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". This is important for many applications, such as
aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence.

History

Menaechmus and early works

It is believed that the first definition of a conic section was given by Menaechmus (died 320 BCE) as
part of his solution of the Delian problem (Duplicating the cube).[26][27] His work did not survive, not
even the names he used for these curves, and is only known through secondary accounts.[28] 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). 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).[29]

Euclid (fl. 300 BCE) is said to have written four books on conics but these were lost as well.[30]
Archimedes (died c. 212 BCE) 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.[31]

Apollonius of Perga
The greatest progress in the study of conics by the ancient Greeks
is due to Apollonius of Perga (died c. 190 BCE), whose eight-
volume Conic Sections or Conics summarized and greatly
extended existing knowledge.[32] 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 Diagram from Apollonius' Conics, in
the earlier method, are also obtainable in this way. This may a 9th-century Arabic translation
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.[33]

Pappus of Alexandria (died c. 350 CE) 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).[34]

Al-Kuhi
An instrument for drawing conic sections was first described in 1000 CE by the Islamic
mathematician Al-Kuhi.[35]:30[36]

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[37] mathematician and
poet Omar Khayyám, who used conic sections to solve algebraic equations of no higher a degree than
three.[38][39]

Europe
Johannes Kepler extended the theory of conics through the "principle of continuity", a precursor to
the concept of limits. Kepler first used the term foci in 1604.[40]

Girard Desargues and Blaise Pascal developed a theory of conics using an early form of projective
geometry and this helped to provide impetus for the study of this new field. In particular, Pascal
discovered a theorem known as the hexagrammum mysticum from which many other properties of
conics can be deduced.

René Descartes and Pierre Fermat both applied their newly discovered analytic geometry 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 in his 1655 treatise Tractatus de sectionibus conicis who first defined the
conic sections as instances of equations of second degree.[41] Written earlier, but published later, Jan
de Witt's Elementa Curvarum Linearum starts with Kepler's kinematic 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.[42] De Witt invented the term
directrix.[42]

Applications
Conic sections are important in astronomy: the orbits of two massive objects that interact according
to Newton's law of universal gravitation are conic sections if their common center of mass is
considered to be at rest. 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.
The reflective properties of the conic sections are used in the design of searchlights, radio-telescopes
and some optical telescopes.[43] 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
and the conics may be considered as objects in this projective
The paraboloid shape of
geometry. One way to do this is to introduce homogeneous
Archeocyathids produces conic
coordinates and define a conic to be the set of points whose sections on rock faces
coordinates satisfy an irreducible quadratic equation in three
variables (or equivalently, the zeros of an irreducible quadratic
form). More technically, the set of points that are zeros of a quadratic form (in any number of
variables) is called a quadric, and the irreducible quadrics in a two dimensional projective space (that
is, having three variables) are traditionally called conics.

The Euclidean plane R2 is embedded in the real projective plane by adjoining a line at infinity (and its
corresponding points at infinity) 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 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 simply 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.[44]

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 corresponding to the axis, and a hyperbola if the conic intersects the line
at infinity in two points corresponding to the asymptotes.[45]

Homogeneous coordinates
In homogeneous coordinates a conic section can be represented as:
Or in matrix notation

The 3 × 3 matrix above is called the matrix of the conic section.

Some authors prefer to write the general homogeneous equation as

(or some variation of this) so that the matrix of the conic section has the simpler form,

but this notation is not used in this article.[46]

If the determinant of the matrix of the conic section is zero, the conic section is degenerate.

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 (A, B, C, D, E, F) as points in the five-dimensional
projective space

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.[47] They must be redefined (and
generalized) in this new geometry. This can be done for arbitrary projective planes, but to obtain the
real projective plane as the extended Euclidean plane, some specific choices have to be made.[48]

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 A and
B, the midpoint of line segment AB is defined as the point C which is the projective harmonic
conjugate of the point of intersection of AB and the absolute line, with respect to A and B.

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. 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.[49]:72

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 e = 0 a circle will have the focus-directrix property, but it is still not defined by that
property.[50] 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 in 1867.

Given two pencils of lines at two points (all

lines containing and resp.) and a projective but not
perspective mapping of onto . Then the
intersection points of corresponding lines form a non-
degenerate projective conic section.[51][52][53][54]

A perspective mapping of a pencil onto a pencil is a

bijection (1-1 correspondence) such that corresponding lines
intersect on a fixed line , which is called the axis of the
perspectivity .

A projective mapping is a finite sequence of perspective Definition of the Steiner generation of

mappings. a conic section

As a projective mapping in a projective plane over a field

(pappian plane) is uniquely determined by prescribing the images of three lines,[55] for the Steiner
generation of a conic section, besides two points 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.[56]:48–49

Von Staudt's definition

Karl Georg Christian von Staudt 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, π, of a projective plane, P, is an involutory (i.e., of order two) bijection between the points
and the lines of P that preserves the incidence relation. Thus, a polarity relates a point Q with a line q
and, following Gergonne, q is called the polar of Q and Q the pole of q.[57] An absolute point
(line) of a polarity is one which is incident with its polar (pole).[58]

A von Staudt conic in the real projective plane is equivalent to a Steiner conic.[59]

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, A, B, C, D, E and a line passing through E, say EG, \a point F that lies on this line and is on
the conic determined by the five points can be constructed. Let AB meet DE in L, BC meet EG in M
and let CD meet LM at N. Then AN meets EG at the required point F.[60]:52–53 By varying the line
through E,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.[61] Specifically, to construct the
x2 y2
ellipse with equation a2 + b2 = 1, first construct the rectangle ABCD with vertices
A(a, 0), B(a, 2b), C(−a, 2b) and D(−a, 0). Divide the side BC into n equal segments and use
parallel projection, with respect to the diagonal AC, to form equal segments on side AB (the lengths
b
of these segments will be a times the length of the segments on BC). On the side BC label the left-
hand endpoints of the segments with A1 to An starting at B and going towards C. On the side AB label
the upper endpoints D1 to Dn starting at A and going towards B. The points of intersection,
AAi ∩ DDi for 1 ≤ i ≤ n will be points of the ellipse between A and P(0, b). The labeling associates
the lines of the pencil through A with the lines of the pencil
through D projectively but not perspectively. The sought for conic
is obtained by this construction since three points A, D and P
and two tangents (the vertical lines at A and D) 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[62] and parabolas[63] are
similar.

Yet another general method uses the polarity property to Parallelogram method for
construct the tangent envelope of a conic (a line conic).[64] constructing an ellipse

In the complex projective plane

In the complex plane C2, ellipses and hyperbolas are not distinct: one may consider a hyperbola as an
ellipse with an imaginary axis length. For example, the ellipse becomes a hyperbola
under the substitution geometrically a complex rotation, yielding . 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 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 CP2: the non-degenerate conics cannot be
distinguished from one another, since any can be taken to any other by a projective linear
transformation.

It can be proven that in CP2, two conic sections have four points in common (if one accounts for
multiplicity), 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. 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.[65]

Furthermore, each straight line intersects each conic section twice. If the intersection point is double,
the line is a tangent line. Intersecting with the line at infinity, each conic section has two points at
infinity. If these points are real, the curve is a hyperbola; if they are imaginary conjugates, it is an
ellipse; if there is only one double point, it is a parabola. If the points at infinity are the cyclic points
(1, i, 0) and (1, –i, 0), the conic section is a circle. If the coefficients of a conic section are real, the
points at infinity are either real or complex conjugate.

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 of the cone. The degenerate conic is either: a point, when the plane
intersects the cone only at the apex; a straight line, 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).[66]
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, 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 parallel lines such as with the equation or to an imaginary ellipse,
such as with the equation An imaginary ellipse does not satisfy the general
definition of a degeneracy, and is thus not normally considered as degenerated.[67] 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 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, i.e. with the apex at infinity. Other
sections in this case are called cylindric sections.[68] 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 β be the determinant of the 3×3 matrix of the conic section—that is,
2 2 − AE2
β = (AC − B4 )F + BED − CD 4 ; and let α = B2 − 4AC be the discriminant. Then the conic
section is non-degenerate if and only if β ≠ 0. If β = 0 we have a point when α < 0, two parallel lines
(possibly coinciding) when α = 0, or two intersecting lines when α > 0.[69]

Pencil of conics
A (non-degenerate) conic is completely determined by five points in general position (no three
collinear) 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.[70]:64 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.[71]:127
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, i.e. a 3x3
symmetric matrix 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 and , consider the pencil of conics given by their linear combination

identify the homogeneous parameters which correspond to the degenerate conic of the
pencil. This can be done by imposing the condition that and solving for
and . These turn out to be the solutions of a third degree equation.
given the degenerate conic , 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
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 geometries). However, some care
must be used when the field has characteristic 2, as some formulas can not be used. For example, the
matrix representations used above require division by 2.

A generalization of a non-degenerate conic in a projective plane is an 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.

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), 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 where the number of +1
coefficients, k, is the positive index, the number of −1 coefficients, , is the negative index, and
the remaining variables are the zero index m, so In two variables the non-zero
quadratic forms are classified as:

– positive-definite (the negative is also included), corresponding to ellipses,

– degenerate, corresponding to parabolas, and
– 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 describes the infinitesimal geometry, and may at each
point be either positive – elliptic geometry, zero – Euclidean geometry (flat, parabola), or
negative – hyperbolic geometry; infinitesimally, to second order the surface looks like the graph
of (or 0), or . 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 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 classifications include:

Möbius transformations
Real Möbius transformations (elements of PSL2(R) or its 2-fold cover, SL2(R)) are classified as
elliptic, parabolic, or hyperbolic accordingly as their half-trace is or
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 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

Notes
1. Eves 1963, p. 319
2. Brannan, Esplen & Gray 1999, p. 13
3. Cohen, D., Precalculus: With Unit Circle Trigonometry (Stamford: Thomson Brooks/Cole, 2006),
p. 844 (https://books.google.com/books?id=_cIT_pRdfJIC&pg=PA844).
4. Thomas & Finney 1979, p. 434
5. Brannan, Esplen & Gray 1999, p. 19; Kendig 2005, pp. 86, 141
6. Brannan, Esplen & Gray 1999, pp. 13–16
7. Brannan, Esplen & Gray 1999, pp. 11–16
8. Protter & Morrey 1970, pp. 314–328, 585–589
9. Protter & Morrey 1970, pp. 290–314
10. Wilson & Tracey 1925, p. 130
11. the empty set is included as a degenerate conic since it may arise as a solution of this equation
12. Protter & Morrey 1970, p. 316
13. Brannan, Esplen & Gray 1999, p. 30
14. Fanchi, John R. (2006), Math refresher for scientists and engineers (https://books.google.com/boo
ks?id=75mAJPcAWT8C), John Wiley and Sons, pp. 44–45, ISBN 0-471-75715-2, Section 3.2,
page 45 (https://books.google.com/books?id=75mAJPcAWT8C&pg=PA45) In this interactive SVG, move left and
right over the SVG image to rotate
15. Protter & Morrey 1970, p. 326 the double cone
16. Wilson & Tracey 1925, p. 153
17. Pettofrezzo, Anthony, Matrices and Transformations, Dover Publ., 1966, p. 110.
18. Spain, Barry, Analytical Conics, Dover, 2007 (originally published 1957 by Pergamon Press).
19. Ayoub, Ayoub B., "The eccentricity of a conic section," The College Mathematics Journal 34(2),
March 2003, 116–121.
20. Ayoub, A. B., "The central conic sections revisited", Mathematics Magazine 66(5), 1993, 322–325.
21. Brannan, Esplen & Gray 1999, p. 17
22. 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.
23. Paris Pamfilos, "A gallery of conics by five elements", Forum Geometricorum 14, 2014, 295–348.
http://forumgeom.fau.edu/FG2014volume14/FG201431.pdf
24. Brannan, Esplen & Gray 1999, p. 28
25. Downs 2003, pp. 36ff.
26. According to Plutarch this solution was rejected by Plato on the grounds that it could not be
achieved using only straightedge and compass, however this interpretation of Plutarch's
statement has come under criticism.Boyer 2004, p.14, footnote 14
27. Boyer 2004, pp. 17–18
28. Boyer 2004, p. 18
29. Katz 1998, p. 117
30. Heath, T.L., The Thirteen Books of Euclid's Elements, Vol. I, Dover, 1956, pg.16
31. Eves 1963, p. 28
32. Apollonius of Perga, Treatise on Conic Sections (https://books.google.com/books?id=hRtaAQAA
QBAJ&printsec=frontcover), edited by T. L. Heath (Cambridge: Cambridge University Press,
2013).
33. Eves 1963, p. 30
34. Boyer 2004, p. 36
35. Stillwell, John (2010). Mathematics and its history (3rd ed.). New York: Springer. p. 30 (https://boo
ks.google.cz/books?id=DOYlBQAAQBAJ&lpg=PP1&hl=de&pg=PA30&redir_esc=y#v=onepage&q
&f=false). ISBN 978-1-4419-6052-8.
36. "Apollonius of Perga Conics Books One to Seven" (http://www.math.psu.edu/katok_s/Commentari
es-new.pdf) (PDF). Retrieved 10 June 2011.
37. Turner, Howard R. (1997). Science in medieval Islam: an illustrated introduction (https://books.goo
gle.com/books?id=3VfY8PgmhDMC). University of Texas Press. p. 53. ISBN 0-292-78149-0.,
Chapter , p. 53 (https://books.google.com/books?id=3VfY8PgmhDMC&pg=PA53)
38. Boyer, C. B., & Merzbach, U. C., A History of Mathematics (Hoboken: John Wiley & Sons, Inc.,
1968), p. 219 (https://books.google.com/books?id=V6RUDwAAQBAJ&pg=PA219).
39. Van der Waerden, B. L., Geometry and Algebra in Ancient Civilizations (Berlin/Heidelberg:
Springer Verlag, 1983), p. 73 (https://books.google.cz/books?id=-pTzCAAAQBAJ&pg=PA73#v=on
epage&q&f=false).
40. Katz 1998, p. 126
41. Boyer 2004, p. 110
42. Boyer 2004, p. 114
43. Brannan, Esplen & Gray 1999, p. 27
44. Artzy 2008, p. 158, Thm 3-5.1
45. Artzy 2008, p. 159
46. This form of the equation does not generalize to fields of characteristic two (see below)
47. Consider finding the midpoint of a line segment with one endpoint on the line at infinity.
48. Faulkner 1952, p. 71
49. Faulkner 1952, p. 72 (https://books.google.com/books?id=3TwCIg_O2yMC&lpg=PP1&hl=cs&pg=
PA72)
50. Eves 1963, p. 320
51. Coxeter 1993, p. 80
52. Hartmann, p. 38
53. Merserve 1983, p. 65
54. Jacob Steiner’s Vorlesungen über synthetische Geometrie, B. G. Teubner, Leipzig 1867 (from
Google Books: (German) Part II follows Part I (https://archive.org/details/bub_gb_jCgPAAAAQAAJ
)) Part II, pg. 96
55. Hartmann, p. 19
56. Faulkner 1952, pp. 48–49 (https://books.google.com/books?id=3TwCIg_O2yMC&lpg=PP1&hl=cs
&pg=PA48).
57. Coxeter 1964, p. 60
58. Coxeter and several other authors use the term self-conjugate instead of absolute.
59. Coxeter 1964, p. 80
60. Faulkner 1952, pp. 52–53 (https://books.google.com/books?id=3TwCIg_O2yMC&lpg=PP1&hl=cs
60. Faulkner 1952, pp. 52–53 (https://books.google.com/books?id=3TwCIg_O2yMC&lpg=PP1&hl=cs
&pg=PA52)
61. Downs 2003, p. 5
62. Downs 2003, p. 14
63. Downs 2003, p. 19
64. Akopyan & Zaslavsky 2007, p. 70
65. Wilczynski, E. J. (1916), "Some remarks on the historical development and the future prospects of
the differential geometry of plane curves", Bull. Amer. Math. Soc., 22 (7): 317–329,
doi:10.1090/s0002-9904-1916-02785-6 (https://doi.org/10.1090%2Fs0002-9904-1916-02785-6).
66. Brannan, Esplen & Gray 1999, p. 6
67. Korn, G. A., & Korn, T. M., Mathematical Handbook for Scientists and Engineers: Definitions,
Theorems, and Formulas for Reference and Review (Mineola, NY: Dover Publications, 1961), p.
42 (https://books.google.com/books?id=A4XCAgAAQBAJ&pg=PA42).
68. "MathWorld: Cylindric section" (http://mathworld.wolfram.com/CylindricSection.html).
69. Lawrence, J. Dennis (1972), A Catalog of Special Plane Curves (https://archive.org/details/catalog
ofspecial00lawr/page/63), Dover, p. 63 (https://archive.org/details/catalogofspecial00lawr/page/63
), ISBN 0-486-60288-5
70. Faulkner 1952, pg. 64 (https://books.google.com/books?id=3TwCIg_O2yMC&lpg=PP1&hl=cs&pg
=PA64).
71. Berger, M., Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry
(Berlin/Heidelberg: Springer, 2010), p. 127 (https://books.google.com/books?id=pN0iAVavPR8C&
pg=PA127#v=onepage&q&f=false).
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Akopyan, A.V.; Zaslavsky, A.A. (2007). Geometry of Conics. American Mathematical Society.
ISBN 978-0-8218-4323-9.
Artzy, Rafael (2008) [1965], Linear Geometry, Dover, ISBN 978-0-486-46627-9
Boyer, Carl B. (2004) [1956], History of Analytic Geometry (https://books.google.com/books?id=2T
4i5fXZbOYC&printsec=frontcover), Dover, ISBN 978-0-486-43832-0
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999), Geometry (https://books.google.c
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Coxeter, H.S.M. (1964), Projective Geometry (https://books.google.com/books?id=gjAAI4FW0tsC
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parabolas and hyperbolas (https://books.google.com/books?id=PE7CAgAAQBAJ&printsec=frontc
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External links
Conic section (Geometry) (https://www.britannica.com/EBchecked/topic/132684) at the
 Encyclopædia Britannica
Can You Really Derive Conic Formulae from a Cone? (http://www.maa.org/press/periodicals/conv
ergence/can-you-really-derive-conic-formulae-from-a-cone-introduction) archive 2007-07-15 (https
://web.archive.org/web/20070715064142/http://mathdl.maa.org/convergence/1/?pa=content&sa=v
iewDocument&nodeId=196&bodyId=60) Gary S. Stoudt (Indiana University of Pennsylvania
Conic sections (http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html)
at Special plane curves (http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html).
Weisstein, Eric W. "Conic Section" (https://mathworld.wolfram.com/ConicSection.html).
MathWorld.
Occurrence of the conics. Conics in nature and elsewhere (https://web.archive.org/web/20060406
010638/http://britton.disted.camosun.bc.ca/jbconics.htm).
See Conic Sections (http://www.cut-the-knot.org/proofs/conics.shtml) at cut-the-knot (http://www.c
ut-the-knot.org) for a sharp proof that any finite conic section is an ellipse and Xah Lee (http://xahl
ee.org/PageTwo_dir/more.html) for a similar treatment of other conics.
Eight Point Conic (https://web.archive.org/web/20091025083524/http://math.kennesaw.edu/~mde
villi/eightpointconic.html) at Dynamic Geometry Sketches (https://web.archive.org/web/200903210
24112/http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm)
Second-order implicit equation locus (https://web.archive.org/web/20151024031726/http://archive.
geogebra.org/en/upload/files/nikenuke/conics04b.html) An interactive Java conics grapher; uses a
general second-order implicit equation.

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