Conic section explained

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 focus-directrix 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.

History

Menaechmus

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.

Apollonius of Perga

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.

Al-Kuhi

An instrument for drawing conic sections was first described in 1000 CE by the Islamic mathematician Al-Kuhi.[2] [3]

Omar Khayyám

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.

Europe

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.

Features

thumb|alt=Diagram of conic sections|Conics 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 east-west opening case is given. In all cases, a and b are positive.)

conic sectionequationeccentricity (e)linear eccentricity (c)semi-latus rectum ()focal parameter (p)
circle

x2+y2=a2

0

0

a

infty

ellipse
x2+
a2
y2
b2

=1

\sqrt{1-b2
a2
}

\sqrt{a2-b2}

b2
a
b2
\sqrt{a2-b2
}
parabola

y2=4ax

1

a

2a

2a

hyperbola
x2-
a2
y2
b2

=1

\sqrt{1+b2
a2
}

\sqrt{a2+b2}

b2
a
b2
\sqrt{a2+b2
}

Conic sections are exactly those curves that, for a point F, a line L not containing F and a non-negative 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 (2) is the chord parallel to the directrix and passing through the focus (or one of the two foci).

The semi-latus rectum () 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.

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 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.

Intersection at infinity

An algebro-geometrically intrinsic form of this classification is by the intersection of the conic with the line at infinity, which gives further insight into their geometry:

Degenerate cases

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 non-zero, 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).

Eccentricity, focus and directrix

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). The corresponding conic section consists of the locus 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

a/e

, where

a

is the semi-major axis of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is

ae

.

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 to F 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

a

, the closer

e

is to 1, the smaller is the semi-minor axis.

Generalizations

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

x2+y2=1

becomes a hyperbola under the substitution

y=iw,

geometrically a complex rotation, yielding

x2-w2=1

– a hyperbola is simply an ellipse with an imaginary axis length. Thus there is a 2-way classification: ellipse/hyperbola and parabola. Geometrically, 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), and thus the real hyperbola is a more suggestive image for the complex ellipse/hyperbola, as it also has 2 (real) intersections with the line at infinity.

In 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, 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.

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
2,
x
k+l
where the number of +1 coefficients, k, is the positive index, the number of -1 coefficients, l, is the negative index, and the remaining variables are the zero index m, so

k+l+m=n.

In two variables the non-zero quadratic forms are classified as:

x2+y2,

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

x2

– degenerate, corresponding to parabolas, and

x2-y2

– 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

x2+y2,

,

x2

(or 0), or

x2-y2.

. 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

0\leq|\operatorname{tr}|/2<1,

|\operatorname{tr}|/2=1,

or

|\operatorname{tr}|/2>1,

mirroring the classification by eccentricity.
Variance-to-mean ratio
  • The variance-to-mean ratio classifies several important families of discrete probability distributions: the constant distribution as circular (eccentricity 0), binomial distributions as elliptical, Poisson distributions as parabolic, and negative binomial distributions as hyperbolic. This is elaborated at cumulants of some discrete probability distributions.

    Cartesian coordinates

    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

    Ax2+Bxy+Cy2+Dx+Ey+F=0withA,B,Cnotallzero.

    As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space

    P5.

    Discriminant classification

    The conic sections described by this equation can be classified with the discriminant[6]

    B2-4AC.

    If the conic is non-degenerate, then:

    B2-4AC<0

    , the equation represents an ellipse;

    A=C

    and

    B=0

    , the equation represents a circle, which is a special case of an ellipse;

    B2-4AC=0

    , the equation represents a parabola;

    B2-4AC>0

    , the equation represents a hyperbola;

    A+C=0

    , the equation represents a rectangular hyperbola.

    To distinguish the degenerate cases from the non-degenerate 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 - B2/4)F + BED/4 - CD2/4 - AE2/4. Then the conic section is non-degenerate 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 non-degenerate ellipse (with

    B2-4AC<0

    and ≠0), we have a real ellipse if C∆ < 0 but an imaginary ellipse if C∆ > 0. An example is

    x2+y2+10=0

    , which has no real-valued solutions.

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

    Matrix notation

    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 non-zero (i.e. it is an ellipse or a hyperbola), we can do a transformation of variables to obtain

    \left(\begin{array}{c} x-a\\ y-c\end{array}\right)T\left(\begin{array}{cc} A&

    B\\
    2
    B
    2

    &C\end{array}\right)\left(\begin{array}{c} x-a\\ y-c\end{array}\right)=G

    where a,c, and G satisfy

    D+2aA+Bc=0,E+2Cc+Ba=0,

    and

    G=Aa2+Cc2+Bac-F

    .

    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 non-zero, 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.

    As slice of quadratic form

    The equation

    Ax2+Bxy+Cy2+Dx+Ey+F=0

    can be rearranged by taking the affine linear part to the other side, yielding

    Ax2+Bxy+Cy2=-(Dx+Ey+F).

    In this form, a conic section is realized exactly as the intersection of the graph of the quadratic form

    z=Ax2+Bxy+Cy2

    and the plane

    z=-(Dx+Ey+F).

    Parabolas and hyperbolas can be realized by a horizontal plane (

    D=E=0

    ), while ellipses require that the plane be slanted. Degenerate conics correspond to degenerate intersections, such as taking slices such as

    z=-1

    of a positive-definite form.

    Eccentricity in terms of parameters of the quadratic form

    When the conic section is written algebraically as

    Ax2+Bxy+Cy2+Dx+Ey+F=0,

    the eccentricity can be written as a function of the parameters of the quadratic equation.[8] If 4AC = B2 the conic is a parabola and its eccentricity equals 1 (if it is non-degenerate). Otherwise, assuming the equation represents either a non-degenerate hyperbola or a non-degenerate, non-imaginary ellipse, the eccentricity is given by

    e=\sqrt{2\sqrt{(A-C)2+B2
    }

    Notes and References

    1. Heath, T.L., The Thirteen Books of Euclid's Elements, Vol. I, Dover, 1956, pg.16
    2. Book: Stillwell, John. Mathematics and its history. 2010. Springer. New York. 144196052X. 30. 3rd ed..
    3. Web site: Apollonius of Perga Conics Books One to Seven. 10 June 2011.
    4. Book: Science in medieval Islam: an illustrated introduction. Howard R.. Turner. University of Texas Press. 1997. 0-292-78149-0. 53., Chapter, p. 53
    5. Web site: MathWorld: Cylindric section.
    6. , Section 3.2, page 45
    7. Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.
    8. Ayoub, Ayoub B., "The eccentricity of a conic section," The College Mathematics Journal 34(2), March 2003, 116-121.