In mathematics, a conic section (or just conic) is a curve obtained by intersecting 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. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
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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 the line 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; 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. 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.
Apollonius's work was translated into Arabic and much of his work only survives through the Arabic version. Muslims 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.
Johann 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 help 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 hyperbola, ellipse, and parabola. 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, though often one is ignored.
There are multiple degenerate cases, in which the plane passes through the apex of the cone. The intersection in these cases can be a straight line (when the plane is tangential to the surface of the cone); a point (when the angle between the plane and the axis of the cone is larger than tangential); or a pair of intersecting lines (when the angle is smaller).
Where the cone is a cylinder, i.e. with the vertex at infinity, cylindric sections are obtained. Although these yield mostly ellipses (or circles) as usual, a degenerate case of two parallel lines, known as a ribbon, can also be produced, and it is also possible for there to be no intersection at all.^{[1]}
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 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 focusdirectrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is a / e, where is the semimajor 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 .
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 , the closer is to 1, the smaller is the semiminor axis.
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. The equation will be of the form
then:
Note that A and B are just polynomial coefficients, not the lengths of semimajor/minor axis as defined in the following sections.
In matrix notation the equation above becomes:
or
and
Through change of coordinates these equations can be put in
standard forms:
Such forms will be symmetrical about the xaxis and for
the circle, ellipse and hyperbola symmetrical about the
yaxis.
The rectangular hyperbola however is only symmetrical about the
lines y = x and
y = − x. Therefore
its inverse function is exactly the same as its original
function.
These standard forms can be written as parametric equations,
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 in fact a union of two straight lines. A conic section that intersects itself is always degenerate, however not all degenerate conic sections intersect themselves, if they do not they are straight lines.
For example, the conic section reduces to the union of two lines:
.
Similarly, a conic section sometimes reduces to a (single) line:
.
is called the discriminant 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 A_{1} = A_{2} and B_{1} = 0, it is an rectangular hyperbola if δ<0 and A_{1} = A_{2}. It can be proven that in the complex projective plane CP^{2} 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 1 intersection point (possibilities: 4 distinct intersection points, 2 singular intersection points and 1 double intersection points, 2 double intersection points, 1 singular intersection point and 1 with multiplicity 3, 1 intersection point with multiplicity 4). If there exists at least one intersection point with multiplicity > 1, then the two conic sections are said to be tangent. If there is only one intersection point, which has multiplicity 4, the two conic sections are said to be osculating.^{[2]}
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. If a conic section has one real and one imaginary point at infinity or it has two imaginary points that are not conjugated it is neither a parabola nor an ellipse nor a hyperbola.
In polar coordinates, a conic section with one focus at the origin and, if any, the other on the xaxis, is given by the equation
where e is the eccentricity and l is the semilatus rectum (see below). As above, for e = 0, we have a circle, for 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.
Various parameters are associated with a conic section.
conic section  equation  eccentricity (e)  linear eccentricity (c)  semilatus rectum (ℓ)  focal parameter (p) 

circle  0  0  
ellipse  
parabola  y^{2} = 4ax  1  
hyperbola 
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:
Irreducible conic sections are always "smooth". More precisely, they never contain any inflection points. This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence.
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 twobody problem.
In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations.
For specific applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola.
The solutions to a two second degree equations system in two variables may be seen as the coordinates of the intersections of two generic conic sections. In particular two conics may possess none, two or four possibly coincident intersection points. The best 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:
See Dandelin spheres for a short elementary argument showing that the characterization of these curves as intersections of a plane with a cone is equivalent to the characterization in terms of foci, or of a focus and a directrix.
CONIC SECTION, or briefly Conic, a curve in which a plane intersects a cone. In ancient geometry the name was restricted to the three particular forms now designated the ellipse, parabola and hyperbola, and this sense is still retained in general works. But in modern geometry, especially in the analytical and projective methods, the "principle of continuity" renders advisable the inclusion of the other forms of the section of a cone, viz. the circle, and two lines (and also two points, the reciprocal of two lines) under the general title conic. The definition of conics as sections of a cone was employed by the Greek geometers as the fundamental principle of their researches in this subject; but the subsequent development of geometrical methods has brought to light many other means for defining these curves. One definition, which is of especial value in the geometrical treatment of the conic sections (ellipse, parabola and hyperbola) in piano, is that a conic is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio. This ratio, known as the eccentricity, determines the nature of the curve; if it be greater than unity, the conic is a hyperbola; if equal to unity, a parabola; and if less than unity, an ellipse. In the case of the circle, the centre is the focus, and the line at infinity the directrix; we therefore see that a circle is a conic of zero eccentricity.
In projective geometry it is convenient to define a conic section as the projection of a circle. The particular conic into which the circle. is projected depends upon the relation of the "vanishing line" to the circle; if it intersects it in real points, then the projection is a hyperbola, if in imaginary points an ellipse, and if it touches the circle, the projection is a parabola. These results may be put in another way, viz. the line at infinity intersects the hyperbola in real points, the ellipse in imaginary points, and the parabola in coincident real points. A conic may also be regarded as the polar reciprocal of a circle for a point; if the point be without the circle the conic is an ellipse, if on the circle a parabola, and if within the circle a hyperbola. In analytical geometry the conic is represented by an algebraic equation of the second degree, and the species of conic is solely determined by means of certain relations between the coefficients. Confocal conics are conics having the same foci. If one of the foci be at infinity, the conics are confocal parabolas, which may also be regarded as parabolas having a common focus and axis. An important property of confocal systems is that only two confocals can be drawn through a specified point, one being an ellipse, the other a hyperbola, and they intersect orthogonally.
The definitions given above reflect the intimate association of these curves, but it frequently happens that a particular conic is defined by some special property (as the ellipse, which is the locus of a point such that the sum of its distances from two fixed points is constant); such definitions and other special properties are treated in the articles Ellipse, Hyperbola and Parabola. In this article we shall consider the historical development of the geometry of conics, and refer the reader to the article Geometry: Analytical and Projective, for the special methods of investigation.
The invention of the conic sections is to be assigned to the school of geometers founded by Plato at Athens about the 4th century B.C. Under the guidance and inspiration of this philosopher much attention was given to the geometry of solids, and it is probable that while investigating the cone, Menaechrnus, an associate of Plato, pupil of Eudoxus, and brother of Dinostratus (the inventor of the quadratrix), discovered and investigated the various curves made by truncating a cone. Menaechmus discussed three species of cones (distinguished by the magnitude of the vertical angle as obtuseangled, rightangled and acuteangled), and the only section he treated was that made by a plane perpendicular to a generator of the cone; according to the species of the cone, he obtained the curves now known as the hyperbola, parabola and ellipse. That he made considerable progress in the study of these curves is evidenced by Eutocius, who flourished about the 6th century A.D., and who assigns to Menaechmus two solutions of the problem of duplicating the cube by means of intersecting conics. On the authority of the two great commentators Pappus and Proclus, Euclid wrote four books on conics, but the originals are now lost, and all we have is chiefly to be found in the works of Apollonius of Perga. Archimedes contributed to the knowledge of these curves by determining the area of the parabola, giving both a geometrical and a mechanical solution, and also by evaluating the ratio of elliptic to circular spaces. He probably wrote a book on conics, but it is now lost. In his extant Conoids and Spheroids he defines a conoid to be the solid formed by the revolution of the parabola and hyperbola about its axis, and a spheroid to be formed similarly from the ellipse; these solids he discussed with great acumen, and effected their cubature by his famous "method of exhaustions." But the greatest Greek writer on the conic sections was Apollonius of Perga, and it is to his Conic Sections that we are indebted for a review of the early history of this subject. Of the eight books which made up his original treatise, only seven are certainly known, the first four in the original Greek, the next three are found in Arabic translations, and the eighth was restored by Edmund Halley in 1710 from certain introductory lemmas of Pappus. The first four books, of which the first three are dedicated to Eudemus, a pupil of Aristotle and author of the original Eudemian Summary, contain little that is original, and are principally based on the earlier works of Menaechmus, Aristaeus (probably a senior contemporary of Euclid, flourishing about a century later than Menaechmus), Euclid and Archimedes. The remaining books are strikingly original and are to be regarded as embracing Apollonius's own researches.
The first book, which is almost entirely concerned with the construction of the three conic sections, contains one of the most brilliant of all the discoveries of Apollonius. Prior to his time, a right cone of a definite vertical angle was required for the generation of any particular conic; Apollonius showed that the sections could all be produced from one and the same cone, which may be either right or oblique, by simply varying the inclination of the cutting plane. The importance of this generalization cannot be overestimated; it is of more than historical interest, for it remains the basis upon which certain authorities introduce the study of these curves. To comprehend more exactly the discovery of Apollonius, imagine an oblique cone on a circular base, of which the line joining the vertex to the centre of the base is the axis. The section made by a plane containing the axis and perpendicular to the base is a triangle contained by two generating lines of the cone and a diameter of the basal circle. Apollonius considered sections of the cone made by planes at any inclination to the plane of the circular base and perpendicular to the triangle containing the axis. The points in which the cutting plane intersects the sides of the triangle are the vertices of the curve; and the line joining these points is a diameter which Apollonius named the latus transversum. He discriminated the three species of conics as follows:  At one of the two vertices erect a perpendicular (talus rectum) of a certain length (which is determined below), and join the extremity of this line to the other vertex. At any point on the latus transversum erect an ordinate. Then the square of the ordinate intercepted between the diameter and the curve is equal to the rectangle contained by the portion of the diameter between the first vertex and the foot of the ordinate, and the segment of the ordinate intercepted between the diameter and the line joining the extremity of the latus rectum to the second vertex. This property is true for all conics, and it served as the basis of most of the constructions and propositions given by Apollonius. The conics are distinguished by the ratio between the latus rectum (which was originally called the latus erectum, and now often referred to as the parameter) and the segment of the ordinate intercepted between the diameter and the line joining the second vertex with the extremity of the latus rectum. When the cutting plane is inclined to the base of the cone at an angle less than that made by the sides of the cone, the latus rectum is greater than the intercept on the ordinate, and we obtain the ellipse; if the plane is inclined at an equal angle as the side, the latus rectum equals the intercept, and we obtain the parabola; if the inclination of the plane be greater than that of the side, we obtain the hyperbola. In modern notation, if we denote the ordinate by y, the distance of the foot of the ordinate from the vertex (the abscissa) by x, and the latus rectum by p, these relations may be expressed as 31 2 <px for the ellipse, 31 2 = px for the parabola, and y 2 > for the hyperbola. Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the parabola, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle. The word parabola was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.
We may now summarize the contents of the Conics of Apollonius. The first book deals with the generation of the three conics; the second with the asymptotes, axes and diameters; the third with various metrical relations between transversals, chords, tangents, asymptotes, &c.; the fourth with the theory of the pole and polar, including the harmonic division of a straight line, and with systems of two conics, which he shows to intersect in not more than four points; he also investigates conics having single and double contact. The fifth book contains properties of normals and their envelopes, thus embracing the germs of the theory of evolutes, and also maxima and minima problems, such as to draw the longest and shortest lines from a given point to a conic; the sixth book is concerned with the similarity of conics; the seventh with complementary chords and conjugate diameters; the eighth book, according to the restoration of Edmund Halley, continues the subject of the preceding book. His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a conic, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a parabola. He also considered the two branches of a hyperbola, calling the second branch the "opposite" hyperbola, and shows the relation which existed between many metrical properties of the ellipse and hyperbola. The focus of the parabola was discovered by Pappus, who also introduced the notion of the directrix.
The Conics of Apollonius was translated into Arabic by Tobit ben Korra in the 9th century, and this edition was followed by Halley in 1710. Although the Arabs were in full possession of the store of knowledge of the geometry of conics which the Greeks had accumulated, they did little to increase it; the only advance made consisted in the application of describing intersecting conics so as to solve algebraic equations. The great pioneer in this field was Omar Khayyam, who flourished in the 11 th century. These discoveries were unknown in western Europe for many centuries, and were reinvented and developed by many European mathematicians. In 1522 there was published an original work on conics by Johann Werner of Nuremburg. This work, the earliest published in Christian Europe, treats the conic sections in relation to the original cone, the procedure differing from that of the Greek geometers. Werner was followed by Franciscus Maurolycus of Messina, who adopted the same method, and added considerably to the discoveries of Apollonius. Claude Mydorge (15851647), a French geometer and friend of Descartes, published a work De sectionibus conicis in which he greatly simplified the cumbrous proofs of Apollonius, whose method of treatment he followed.
Johann Kepler (15711630) made many important discoveries in the geometry of conics. Of supreme importance is the fertile conception of the planets revolving about the sun in elliptic orbits. On this is based the great structure of celestial mechanics and the theory of universal gravitation; and in the elucidation of problems more directly concerned with astronomy, Kepler, Sir Isaac Newton and others discovered many properties of the conic sections (see Mechanics). Kepler's greatest contribution to geometry lies in his formulation of the "principle of continuity" which enabled him to show that a parabola has a "caecus (or blind) focus" at infinity, and that all lines through this focus are parallel (see Geometrical Continuity). This assumption (which differentiates ancient from modern geometry) has been developed into one of the most potent methods of geometrical investigation (see Geometry: Projective). We may also notice Kepler's approximate value for the circumference of an ellipse (if the semiaxes be a and b, the approximate circumference is ir(a+b)).
An important generalization of the conic sections was developed about the beginning of the 17th century by Girard Desargues and Blaise Pascal. Since all conics derived from a circular cone appear circular when viewed from the apex, they conceived the treatment of the conic sections as projections of a circle. From this conception all the properties of conics can be deduced. Desargues has a special claim to fame on account of his beautiful theorem on the involution of a quadrangle inscribed in a conic. Pascal discovered a striking property of a hexagon inscribed in a conic (the hexagrammum mysticum); from this theorem Pascal is said to have deduced over 400 corollaries, including most of the results obtained by earlier geometers. This subject is mathematically discussed in the article Geometry: Projective. While Desargues and Pascal were founding modern synthetic geometry, Rene Descartes was developing the algebraic representation of geometric relations. The subject of analytical geometry which he virtually created enabled him to view the conic sections as algebraic equations of the second degree, the form of the section depending solely on the coefficients. This method rivals in elegance all other methods; problems are investigated by purely algebraic means, and generalizations discovered which elevate the method to a position of paramount importance. John Wallis, in addition to translating the Conics of Apollonius, published in 1655 an original work entitled De sectionibus conicis nova methodo expositis, in which he treated the curves by the Cartesian method, and derived their properties from the definition in piano, completely ignoring the connexion between the conic sections and a cone. The analytical method was also followed by G. F. A. de l'Hopital in his Traite analytique des sections coniques (1707). A mathematical investigation of the conics by this method is given in the article Geometry: Analytical. Philippe de la Hire, a pupil of Desargues, wrote several works on the conic sections, of which the most important is his Sectiones Conicae (1685). His treatment is synthetic, and he follows his tutor and Pascal in deducing the properties of conics by projection from a circle.
A method of generating conics essentially the same as our modern method of homographic pencils was discussed by Jan de Witt in his Elementa linearum curvarum (1650); but he treated the curves by the Cartesian method, and not synthetically.
Similar methods were devised by Sir Isaac Newton and Colin Maclaurin. In Newton's method, two angles of constant magnitude are caused to revolve about their vertices which are fixed in position, in such a manner that the intersection of two limbs moves along a fixed straight line; then the two remaining limbs envelop a conic. Maclaurin's method, published in his Geometria organica (1719), is based on the proposition that the locus of the vertex of a triangle, the sides of which pass through three fixed points, and the base angles move along two fixed lines, is a conic section. Both Newton's and Maclaurin's methods have been developed by Michel Chasles. In modern times the study of the conic sections has proceeded along the lines which we have indicated; for further details reference should be made to the article Geometry.
For the ancient geometry of conic sections, especially of Apollonius, reference should be made to T. L. Heath's Apollonius of Perga (1886); more general accounts are given in James Gow, A Short History of Greek Mathematics (1884), and in H. G. Zeuthen, Die Lehre von dem Kegelschnitten in Alterthum (1886). Michel Chasles in his Apercu historique sur l'origine et le de'veloppement des methodes en geometrie (1837, a third edition was published in 1889), gives a valuable account of both the ancient and modern geometry of conics; a German translation with the title Geschichte der Geometrie was published in 1839 by L. A. Sohncke. A copious list of early works on conic sections is given in Fred. W. A. Murhard, Bibliotheca mathematics (Leipzig, 1798). The history is also treated in general historical treatises (see Mathematics).
Geometrical constructions are treated in T. H. Eagles, Constructive Geometry of Plane Curves (1886); geometric investigations primarily based on the relation of the conic sections to a cone are given in Hugo Hamilton's De Sectionibus Conicis (1758); this method of treatment has been largely replaced by considering the curves from their definition in piano, and then passing to their derivation from the cone and cylinder. This method is followed in most modern works. Of such textbooks there is an everincreasing number; here we may notice W. H. Besant, Geometrical Conic Sections; C. Smith, Geometrical Conics; W. H. Drew, Geometrical Treatise on Conic Sections. Reference may also be made to C. Taylor, An Introduction to Ancient and Modern Geometry of Conics (1881).
See also list of works under GEOMETRY: Analytical and Projective.
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