In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity.
Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 be hyperbolas. In the focusdirectrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well.
The conic sections were named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
It is believed that the first definition of a conic section is due to Menaechmus. This work does not survive, however, and is only known through secondary accounts. The definition used at that time differs from the one commonly used today in that it requires the plane cutting the cone to be perpendicular to one of the lines that generate the cone as a surface of revolution (a generatrix). Thus the shape of the conic is determined by the angle formed at the vertex of the cone (between two opposite generatrices): 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. Note that the circle cannot be defined this way and was not considered a conic at this time.
Euclid is said to have written four books on conics but these were lost as well.^{[1]} Archimedes is known to have studied conics, having determined the area bounded by a parabola and an ellipse. The only part of this work to survive is a book on the solids of revolution of conics.
The greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga, whose eight volume Conic Sections summarized the existing knowledge at the time and greatly extended it. Apollonius's major innovation was to characterize a conic using properties within the plane and intrinsic to the curve; this greatly simplified analysis. With this tool, it was now possible to show that any plane cutting the cone, regardless of its angle, will produce a conic according to the earlier definition, leading to the definition commonly used today.
Pappus is credited with discovering importance of the concept of a focus of a conic, and the discovery of the related concept of a directrix.
An instrument for drawing conic sections was first described in 1000 CE by the Islamic mathematician AlKuhi.^{[2]} ^{[3]}
Apollonius's work was translated into Arabic (the technical language of the time) and much of his work only survives through the Arabic version. Persians found applications to the theory; the most notable of these was the Persian^{[4]} mathematician and poet Omar Khayyám who used conic sections to solve algebraic equations.
Johannes Kepler extended the theory of conics through the "principle of continuity", a precursor to the concept of limits. 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. Meanwhile, René Descartes applied his newly discovered Analytic geometry to the study of conics. This had the effect of reducing the geometrical problems of conics to problems in algebra.
thumbalt=Diagram of conic sectionsConics are of three types: parabolas, ellipses, including circles, and hyperbolas.
The three types of conics are the ellipse, parabola, and hyperbola. The circle can be considered as a fourth type (as it was by Apollonius) or as a kind of ellipse. The circle and the ellipse 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 – for a right cone as in the picture at the top of the page this means that the cutting plane is perpendicular to the symmetry axis of the cone. 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. In this case, the plane will intersect both halves (nappes) of the cone, producing two separate unbounded curves.
Various parameters are associated with a conic section, as shown in the following table. (For the ellipse, the table gives the case of a>b, for which the major axis is horizontal; for the reverse case, interchange the symbols a and b. For the hyperbola the eastwest opening case is given. In all cases, a and b are positive.)
conic section  equation  eccentricity (e)  linear eccentricity (c)  semilatus rectum (ℓ)  focal parameter (p)  

circle  x^{2+y}^{2=a}^{2}  0  0  a  infty  
ellipse 
=1 
 \sqrt{a^{2b}^{2}} 

 
parabola  y^{2=4ax}  1  a  2a  2a  
hyperbola 
=1 
 \sqrt{a^{2+b}^{2}} 


Conic sections are exactly those curves that, for a point F, a line L not containing F and a nonnegative number e, are the locus of points whose distance to F equals e times their distance to L. F is called the focus, L the directrix, and e the eccentricity.
The linear eccentricity (c) is the distance between the center and the focus (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 semilatus rectum (ℓ) is half the latus rectum.
The focal parameter (p) is the distance from the focus (or one of the two foci) to the directrix.
The following relations hold:
pe=\ell
ae=c.
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 nondegenerate; this is true over both the affine plane and 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.
Irreducible 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.
An algebrogeometrically intrinsic form of this classification is by the intersection of the conic with the line at infinity, which gives further insight into their geometry:
There are five degenerate cases: three in which the plane passes through apex of the cone, and three that arise when the cone itself degenerates to a cylinder (a doubled line can occur in both cases).
When the plane passes through the apex, the resulting conic is always degenerate, and is either: a point (when the angle between the plane and the axis of the cone is larger than tangential); a straight line (when the plane is tangential to the surface of the cone); or a pair of intersecting lines (when the angle is smaller than the tangential). These correspond respectively to degeneration of an ellipse, parabola, and a hyperbola, which are characterized in the same way by angle. The straight line is more precisely a double line (a line with multiplicity 2) because the plane is tangent to the cone, and thus the intersection should be counted twice.
Where the cone is a cylinder, i.e. with the vertex at infinity, cylindric sections are obtained;^{[5]} this corresponds to the apex being at infinity. Cylindrical sections are ellipses (or circles), unless the plane is vertical (which corresponds to passing through the apex at infinity), in which case three degenerate cases occur: two parallel lines, known as a ribbon (corresponding to an ellipse with one axis infinite and the other axis real and nonzero, the distance between the lines), a double line (an ellipse with one infinite axis and one axis zero), and no intersection (an ellipse with one infinite axis and the other axis imaginary).
The four defining conditions above can be combined into one condition that depends on a fixed point
F
L
F
e
F
e
L
0<e<1
e=1
e>1
For an ellipse and a hyperbola, two focusdirectrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is
a/e
a
ae
In the case of a circle, the eccentricity
e=0
The eccentricity of a conic section is thus a measure of how far it deviates from being circular.
For a given
a
e
Conics may be defined over other fields, and may also be classified in the projective plane rather than in the affine plane.
Over the complex numbers ellipses and hyperbolas are not distinct, since there is no meaningful difference between 1 and 1; precisely, the ellipse
x^{2+y}^{2=1}
y=iw,
x^{2w}^{2=1}
In projective space, over any division ring, but in particular over either the real or complex numbers, all nondegenerate conics are equivalent, and thus in projective geometry one simply speaks of "a conic" without specifying a type, as type is not meaningful. Geometrically, the line at infinity is no longer special (distinguished), so while some conics intersect the line at infinity differently, this can be changed by a projective transformation – pulling an ellipse out to infinity or pushing a parabola off infinity to an ellipse or a hyperbola.
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:
2,  
x  
k+l 
k+l+m=n.
x^{2+y}^{2,}
x^{2}
x^{2y}^{2}
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.
x^{2+y}^{2,}
x^{2}
x^{2y}^{2.}
Eccentricity classifications include:
0\leq\operatorname{tr}/2<1,
\operatorname{tr}/2=1,
\operatorname{tr}/2>1,
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form
Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0withA,B,Cnotallzero.
P^{5.}
The conic sections described by this equation can be classified with the discriminant^{[6]}
B^{2}4AC.
B^{2}4AC<0
A=C
B=0
B^{2}4AC=0
B^{2}4AC>0
A+C=0
To distinguish the degenerate cases from the nondegenerate cases, let ∆ be the determinant of the 3×3 matrix [''A'', ''B''/2, ''D''/2 ; ''B''/2, ''C'', ''E''/2 ; ''D''/2, ''E''/2, ''F'' ]: that is, ∆ = (AC  B^{2}/4)F + BED/4  CD^{2}/4  AE^{2}/4. Then the conic section is nondegenerate if and only if ∆ ≠ 0. If ∆=0 we have a point ellipse, two parallel lines (possibly coinciding with each other) in the case of a parabola, or two intersecting lines in the case of a hyperbola.^{[7]}
Moreover, in the case of a nondegenerate ellipse (with
B^{2}4AC<0
x^{2}+y^{2}+10=0
Note that A and B are polynomial coefficients, not the lengths of semimajor/minor axis as defined in some sources.
See main article: Matrix representation of conic sections.
The above equation can be written in matrix notation as
\begin{bmatrix}x&y\end{bmatrix}.\begin{bmatrix}A&B/2\\B/2&C\end{bmatrix}.\begin{bmatrix}x\\y\end{bmatrix}+Dx+Ey+F=0.
The type of conic section is solely determined by the determinant of middle matrix: if it is positive, zero, or negative then the conic is an ellipse, parabola, or hyperbola respectively (see geometric meaning of a quadratic form). If both the eigenvalues of the middle matrix are nonzero (i.e. it is an ellipse or a hyperbola), we can do a transformation of variables to obtain
\left(\begin{array}{c} xa\\ yc\end{array}\right)^{T}\left(\begin{array}{cc} A&
B  \\  
2 
B  
2 
&C\end{array}\right)\left(\begin{array}{c} xa\\ yc\end{array}\right)=G
where a,c, and G satisfy
D+2aA+Bc=0,E+2Cc+Ba=0,
G=Aa^{2}+Cc^{2}+BacF
The quadratic can also be written as
\begin{bmatrix}x&y&1\end{bmatrix}.\begin{bmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{bmatrix}.\begin{bmatrix}x\\y\\1\end{bmatrix}=0.
If the determinant of this 3×3 matrix is nonzero, the conic section is not degenerate. If the determinant equals zero, the conic is a degenerate parabola (two parallel or coinciding lines), a degenerate ellipse (a point ellipse), or a degenerate hyperbola (two intersecting lines).
Note that in the centered equation with constant term G, G equals minus one times the ratio of the 3×3 determinant to the 2×2 determinant.
The equation
Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0
Ax^{2}+Bxy+Cy^{2}=(Dx+Ey+F).
z=Ax^{2}+Bxy+Cy^{2}
z=(Dx+Ey+F).
D=E=0
z=1
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 parameters of the quadratic equation.^{[8]} If 4AC = B^{2} the conic is a parabola and its eccentricity equals 1 (if it is nondegenerate). Otherwise, assuming the equation represents either a nondegenerate hyperbola or a nondegenerate, nonimaginary ellipse, the eccentricity is given by
e=\sqrt{  2\sqrt{(AC)^{2}+B^{2} 