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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, and as a quadric of dimension 1. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a non-circular 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 being 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 have been named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
Menaechmus and early works
It is believed that the first definition of a conic section is due to Menaechmus (died 320 BC). His work did not survive 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, (a generatrix), that generates the cone as a surface of revolution. 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 ( fl. 300 BC ) is said to have written four books on conics but these were lost as well.Archimedes (died c. 212 BC) 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 (died c. 190 BC), whose eight-volume Conic Sections or Conics summarized and greatly extended existing knowledge. 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 of Alexandria (died c. 350 CE) is credited with discovering the importance of the concept of a conic's focus, and with 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 Al-Kuhi.
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 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.
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 section||equation||eccentricity (e)||linear eccentricity (c)||semi-latus rectum (ℓ)||focal parameter (p)|
The non-circular 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 (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 from the focus (or one of the two foci) to the directrix.
The following relations hold:
There are many methods to construct a conic. One of them, that is useful in engineering applications, being , where a conic is constructed point by point by means of connecting certain equally spaced points on horizontal line and vertical line.
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.
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.:p. 203
Furthermore, a conic is determined by any combination of k points in general position that it passes through and 5–k lines that are tangent to it, for 0≤k≤5.
Intersection at infinity
- ellipses intersect the line at infinity in 0 points—rather, in 0 real points, but in 2 complex points, which are conjugate;
- parabolas intersect the line at infinity in 1 double point, corresponding to the axis—they are tangent to the line at infinity, and close at infinity, as distended ellipses;
- hyperbolas intersect the line at infinity in 2 points, corresponding to the asymptotes—hyperbolas pass through infinity, with a twist. Going to infinity along one branch passes through the point at infinity corresponding to the asymptote, then re-emerges on the other branch at the other side but with the inside of the hyperbola (the direction of curvature) on the other side – left vs. right (corresponding to the non-orientability of the real projective plane)—and then passing through the other point at infinity returns to the first branch. Hyperbolas can thus be seen as ellipses that have been pulled through infinity and re-emerged on the other side, flipped.
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; 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 (the focus), a line (the directrix) not containing and a nonnegative real number (the eccentricity). The corresponding conic section consists of the locus of all points whose distance to equals times their distance to . For we obtain an ellipse, for a parabola, and for 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 of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is .
The circle is a limiting case and is not defined by a focus and directrix in the plane. However, if one were to consider the directrix to be infinitely far from the center (the line at infinity), then by taking the eccentricity to be a circle will have the focus-directrix property, but is still not defined by that property. One must be careful in this situation to correctly use the definition of eccentricity as the ratio of the distance of a point on the circle to the focus (length of a radius) to the distance of that point to the directrix (this distance is infinite) which gives the limiting value of zero.
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.
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 −1 is a square; precisely, the ellipse becomes a hyperbola under the substitution geometrically a complex rotation, yielding – 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.
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.
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, l, 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.
- 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.
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
As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space
The conic sections described by this equation can be classified with the discriminant
If the conic is non-degenerate, then:
- if , the equation represents an ellipse;
- if and , the equation represents a circle, which is a special case of an ellipse;
- if , the equation represents a parabola;
- if , the equation represents a hyperbola;
- if we also have , 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.:p.63
Moreover, in the case of a non-degenerate ellipse (with and ∆≠0), we have a real ellipse if C∆ < 0 but an imaginary ellipse if C∆ > 0. An example is , 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.
The above equation can be written in matrix notation as
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
where a,c, and G satisfy and .
The quadratic can also be written as
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
can be rearranged by taking the affine linear part to the other side, yielding
In this form, a conic section is realized exactly as the intersection of the graph of the quadratic form and the plane Degenerate conics correspond to degenerate intersections, such as taking slices such as of a positive-definite form.
Eccentricity in terms of parameters of the quadratic form
When the conic section is written algebraically as
the eccentricity can be written as a function of the parameters of the quadratic equation. 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
where η = 1 if the determinant of the 3×3 matrix is negative and η = −1 if that determinant is positive.
Through a change of coordinates (a rotation of axes and a translation of axes) these equations can be put into standard forms:
- Circle: x2 + y2 = a2
- Ellipse: x2/a2 + y2/b2 = 1
- Parabola: y2 = 4ax, x2 = 4ay
- Hyperbola: x2/a2 − y2/b2 = 1, x2/b2 − y2/a2 = −1
- Rectangular hyperbola: xy = c2
The first four of these forms are symmetric about both the x-axis and y-axis (for the circle, ellipse and hyperbola), or about either but not both (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines y = x and y = −x.
These standard forms can be written as parametric equations,
- 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: (ct , c/t).
Invariants of conics
The trace and determinant of are both invariant with respect to both rotation of axes and translation of the plane (movement of the origin).
The constant term F is invariant under rotation only.
|This section does not cite any references (sources). (June 2010)|
For some practical applications, it is important to re-arrange the standard form so that the focal-point can be placed at the origin. The mathematical formulation for a general conic section, with the other focus if any placed at a positive value (for an ellipse) or a negative value (for a hyperbola) on the horizontal axis, is then given in the polar form by
and in the Cartesian form by
From the above equation, the linear eccentricity (c) is given by .
From the general equations given above, different conic sections can be represented as shown below:
In homogeneous coordinates a conic section can be represented as:
Or in matrix notation
The matrix is called the matrix of the conic section.
is called the determinant of the conic section. If Δ = 0 then the conic section is said to be degenerate; this means that the conic section is either a union of two straight lines, a repeated line, a point or the empty set.
For example, the conic section reduces to the union of two lines:
Similarly, a conic section sometimes reduces to a (single) repeated line:
is called the discriminant of the conic section. If δ = 0 then the conic section is a parabola, if δ < 0, it is an hyperbola and if δ > 0, it is an ellipse. A conic section is a circle if δ > 0 and A1 = A2 and B1 = 0, it is an rectangular hyperbola if δ < 0 and A1 = −A2. It can be proven that in the complex projective plane CP2 two conic sections have four points in common (if one accounts for multiplicity), so there are never more than 4 intersection points and there is always one intersection point (possibilities: four distinct intersection points, two singular intersection points and one double intersection points, two double intersection points, one 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. 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 (the intersection points with the line at infinity). If these points are real, the conic section must be a hyperbola, if they are imaginary conjugated, the conic section must be an ellipse, if the conic section has one double point at infinity it is a parabola. If the points at infinity are (1,i,0) and (1,-i,0), the conic section is a circle (see circular points at infinity). If a conic section has one real and one imaginary point at infinity or it has two imaginary points that are not conjugated then it not a real conic section (its coefficients are complex).
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 (see above). 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.
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. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.
In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.
A pencil of conics can represented algebraically in the following way. Let C1 and C2 be two distinct conics in a projective plane defined over an algebraically closed field K. For every pair λ, μ of elements of K, not both zero, the expression:
represents a conic in the pencil determined by C1 and C2. This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.) Thinking of C1, say, as a ternary quadratic form, then C1 = 0 is the equation of the "conic C1". Another concrete realization would be obtained by thinking of C1 as the 3×3 symmetric matrix which represents it. If C1 and C2 have such concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant.
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.
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.
- Circumconic and inconic
- Conic Sections Rebellion
- Dandelin spheres
- Director circle
- Elliptic coordinate system
- Equidistant set
- Focus (geometry), an overview of properties of conic sections related to the foci
- Lambert conformal conic projection
- Matrix representation of conic sections
- Nine-point conic
- Parabolic coordinates
- Projective conics
- Quadratic function
- Quadrics, the higher-dimensional analogs of conics
- Akopyan, A.V. and Zaslavsky, A.A. (2007). Geometry of Conics. American Mathematical Society. p. 134. ISBN 0-8218-4323-0.
- Eves, Howard (1963), A Survey of Geometry (Volume One), Boston: Allyn and Bacon
- Faulkner, T. E. (1952), Projective Geometry (2nd ed.), Edinburgh: Oliver and Boyd
- Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042
- Samuel, Pierre (1988), Projective Geometry, Undergraduate Texts in Mathematics (Readings in Mathematics), New York: Springer-Verlag, ISBN 0-387-96752-4