Conic section

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

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

In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a projective plane, the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.

The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry.

The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone.

A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes). It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic.

There are three types of conics: the ellipse, parabola, and hyperbola. The circle is a special kind of ellipse, although historically Apollonius considered as a fourth type. Ellipses arise when the intersection of the 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, this means the cutting plane is perpendicular to the axis. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola: the plane intersects both halves of the cone, producing two separate unbounded curves.

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

A circle is a limiting case and is not defined by a focus and directrix in the Euclidean plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but its directrix can only be taken as the line at infinity in the projective plane.[2]

The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.[3]: 844 

A proof that the above curves defined by the focus-directrix property are the same as those obtained by planes intersecting a cone is facilitated by the use of Dandelin spheres.[5]

Alternatively, an ellipse can be defined in terms of two focus points, as the locus of points for which the sum of the distances to the two foci is 2a; while a hyperbola is the locus for which the difference of distances is 2a. (Here a is the semi-major axis defined below.) A parabola may also be defined in terms of its focus and latus rectum line (parallel to the directrix and passing through the focus): it is the locus of points whose distance to the focus plus or minus the distance to the line is equal to 2a; plus if the point is between the directrix and the latus rectum, minus otherwise.

In addition to the eccentricity (e), foci, and directrix, various geometric features and lengths are associated with a conic section.

The principal axis is the line joining the foci of an ellipse or hyperbola, and its midpoint is the curve's center. A parabola has no center.

The linear eccentricity (c) is the distance between the center and a focus.

The latus rectum is the chord parallel to the directrix and passing through a focus; its half-length is the semi-latus rectum ().

The focal parameter (p) is the distance from a focus to the corresponding directrix.

The major axis is the chord between the two vertices: the longest chord of an ellipse, the shortest chord between the branches of a hyperbola. Its half-length is the semi-major axis (a). When an ellipse or hyperbola are in standard position as in the equations below, with foci on the x-axis and center at the origin, the vertices of the conic have coordinates (−a, 0) and (a, 0), with a non-negative.

The minor axis is the shortest diameter of an ellipse, and its half-length is the semi-minor axis (b), the same value b as in the standard equation below. By analogy, for a hyperbola the parameter b in the standard equation is also called the semi-minor axis.

After introducing Cartesian coordinates, the focus-directrix property can be used to produce the equations satisfied by the points of the conic section.[7] By means of a change of coordinates (rotation and translation of axes) these equations can be put into standard forms.[8] For ellipses and hyperbolas a standard form has the x-axis as principal axis and the origin (0,0) as center. The vertices are a, 0) and the foci c, 0). Define b by the equations c2 = a2b2 for an ellipse and c2 = a2 + b2 for a hyperbola. For a circle, c = 0 so a2 = b2. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In standard form the parabola will always pass through the origin.

For a rectangular or equilateral hyperbola, one whose asymptotes are perpendicular, there is an alternative standard form in which the asymptotes are the coordinate axes and the line x = y is the principal axis. The foci then have coordinates (c, c) and (−c, −c).[9]

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

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

This form is a specialization of the homogeneous form used in the more general setting of projective geometry (see below).

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

The discriminant B2 – 4AC of the conic section's quadratic equation (or equivalently the determinant ACB2/4 of the 2 × 2 matrix) and the quantity A + C (the trace of the 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes,[14][15][16] as is the determinant of the 3 × 3 matrix above.[17]: pp. 60–62  The constant term F and the sum D2 + E2 are invariant under rotation only.[17]: pp. 60–62 

the eccentricity can be written as a function of the coefficients of the quadratic equation.[18] If 4AC = B2 the conic is a parabola and its eccentricity equals 1 (provided it is non-degenerate). Otherwise, assuming the equation represents either a non-degenerate hyperbola or ellipse, the eccentricity is given by

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

It can also be shown[17]: p. 89  that the eccentricity is a positive solution of the equation

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

As above, for e = 0, the graph is a circle, for 0 < e < 1 the graph is an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.

The polar form of the equation of a conic is often used in dynamics; for instance, determining the orbits of objects revolving about the Sun. [20]

Just as two (distinct) points determine a line, five points determine a conic. Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true in both the Euclidean plane and its extension, the real projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion.

Four points in the plane in general linear position determine a unique conic passing through the first three points and having the fourth point as its center. Thus knowing the center is equivalent to knowing two points on the conic for the purpose of determining the curve.[21]

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.[22]

Any point in the plane is on either zero, one or two tangent lines of a conic. A point on just one tangent line is on the conic. A point on no tangent line is said to be an interior point (or inner point) of the conic, while a point on two tangent lines is an exterior point (or outer point).

All the conic sections share a reflection property that can be stated as: All mirrors in the shape of a non-degenerate conic section reflect light coming from or going toward one focus toward or away from the other focus. In the case of the parabola, the second focus needs to be thought of as infinitely far away, so that the light rays going toward or coming from the second focus are parallel.[23][24]

Pascal's theorem concerns the collinearity of three points that are constructed from a set of six points on any non-degenerate conic. The theorem also holds for degenerate conics consisting of two lines, but in that case it is known as Pappus's theorem.

Non-degenerate conic sections are always "smooth". This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence.

It is believed that the first definition of a conic section was given by Menaechmus (died 320 BCE) as part of his solution of the Delian problem (Duplicating the cube).[b][25] His work did not survive, not even the names he used for these curves, and is only known through secondary accounts.[26] The definition used at that time differs from the one commonly used today. Cones were constructed by rotating a right triangle about one of its legs so the hypotenuse generates the surface of the cone (such a line is called a generatrix). Three types of cones were determined by their vertex angles (measured by twice the angle formed by the hypotenuse and the leg being rotated about in the right triangle). The conic section was then determined by intersecting one of these cones with a plane drawn perpendicular to a generatrix. The type of the conic is determined by the type of cone, that is, by the angle formed at the vertex of the cone: If the angle is acute then the conic is an ellipse; if the angle is right then the conic is a parabola; and if the angle is obtuse then the conic is a hyperbola (but only one branch of the curve).[27]

Euclid (fl. 300 BCE) is said to have written four books on conics but these were lost as well.[28] Archimedes (died c. 212 BCE) is known to have studied conics, having determined the area bounded by a parabola and a chord in Quadrature of the Parabola. His main interest was in terms of measuring areas and volumes of figures related to the conics and part of this work survives in his book on the solids of revolution of conics, On Conoids and Spheroids.[29]

The greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga (died c. 190 BCE), whose eight-volume Conic Sections or Conics summarized and greatly extended existing knowledge.[30] Apollonius's study of the properties of these curves made it possible to show that any plane cutting a fixed double cone (two napped), regardless of its angle, will produce a conic according to the earlier definition, leading to the definition commonly used today. Circles, not constructible by the earlier method, are also obtainable in this way. This may account for why Apollonius considered circles a fourth type of conic section, a distinction that is no longer made. Apollonius used the names 'ellipse', 'parabola' and 'hyperbola' for these curves, borrowing the terminology from earlier Pythagorean work on areas.[31]

Pappus of Alexandria (died c. 350 CE) is credited with expounding on the importance of the concept of a conic's focus, and detailing the related concept of a directrix, including the case of the parabola (which is lacking in Apollonius's known works).[32]

An instrument for drawing conic sections was first described in 1000 CE by the Islamic mathematician Al-Kuhi.[33]: 30 [34]

Apollonius's work was translated into Arabic, and much of his work only survives through the Arabic version. Persians found applications of the theory, most notably the Persian mathematician and poet Omar Khayyám,[35] who found a geometrical method of solving cubic equations using conic sections.[36][37]

Johannes Kepler extended the theory of conics through the "principle of continuity", a precursor to the concept of limits. Kepler first used the term 'foci' in 1604.[38]

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

René Descartes and Pierre Fermat both applied their newly discovered analytic geometry to the study of conics. This had the effect of reducing the geometrical problems of conics to problems in algebra. However, it was John Wallis in his 1655 treatise Tractatus de sectionibus conicis who first defined the conic sections as instances of equations of second degree.[39] Written earlier, but published later, Jan de Witt's Elementa Curvarum Linearum starts with Kepler's kinematic construction of the conics and then develops the algebraic equations. This work, which uses Fermat's methodology and Descartes' notation has been described as the first textbook on the subject.[40] De Witt invented the term 'directrix'.[40]

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem.

The reflective properties of the conic sections are used in the design of searchlights, radio-telescopes and some optical telescopes.[41] A searchlight uses a parabolic mirror as the reflector, with a bulb at the focus; and a similar construction is used for a parabolic microphone. The 4.2 meter Herschel optical telescope on La Palma, in the Canary islands, uses a primary parabolic mirror to reflect light towards a secondary hyperbolic mirror, which reflects it again to a focus behind the first mirror.

The conic sections have some very similar properties in the Euclidean plane and the reasons for this become clearer when the conics are viewed from the perspective of a larger geometry. The Euclidean plane may be embedded in the real projective plane and the conics may be considered as objects in this projective geometry. One way to do this is to introduce homogeneous coordinates and define a conic to be the set of points whose coordinates satisfy an irreducible quadratic equation in three variables (or equivalently, the zeros of an irreducible quadratic form). More technically, the set of points that are zeros of a quadratic form (in any number of variables) is called a quadric, and the irreducible quadrics in a two dimensional projective space (that is, having three variables) are traditionally called conics.

The Euclidean plane R2 is embedded in the real projective plane by adjoining a line at infinity (and its corresponding points at infinity) so that all the lines of a parallel class meet on this line. On the other hand, starting with the real projective plane, a Euclidean plane is obtained by distinguishing some line as the line at infinity and removing it and all its points.

In a projective space over any division ring, but in particular over either the real or complex numbers, all non-degenerate conics are equivalent, and thus in projective geometry one simply speaks of "a conic" without specifying a type. That is, there is a projective transformation that will map any non-degenerate conic to any other non-degenerate conic.[42]

The three types of conic sections will reappear in the affine plane obtained by choosing a line of the projective space to be the line at infinity. The three types are then determined by how this line at infinity intersects the conic in the projective space. In the corresponding affine space, one obtains an ellipse if the conic does not intersect the line at infinity, a parabola if the conic intersects the line at infinity in one double point corresponding to the axis, and a hyperbola if the conic intersects the line at infinity in two points corresponding to the asymptotes.[43]

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

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

Metrical concepts of Euclidean geometry (concepts concerned with measuring lengths and angles) can not be immediately extended to the real projective plane.[d] They must be redefined (and generalized) in this new geometry. This can be done for arbitrary projective planes, but to obtain the real projective plane as the extended Euclidean plane, some specific choices have to be made.[44]

Fix an arbitrary line in a projective plane that shall be referred to as the absolute line. Select two distinct points on the absolute line and refer to them as absolute points. Several metrical concepts can be defined with reference to these choices. For instance, given a line containing the points A and B, the midpoint of line segment AB is defined as the point C which is the projective harmonic conjugate of the point of intersection of AB and the absolute line, with respect to A and B.

A conic in a projective plane that contains the two absolute points is called a circle. Since five points determine a conic, a circle (which may be degenerate) is determined by three points. To obtain the extended Euclidean plane, the absolute line is chosen to be the line at infinity of the Euclidean plane and the absolute points are two special points on that line called the circular points at infinity. Lines containing two points with real coordinates do not pass through the circular points at infinity, so in the Euclidean plane a circle, under this definition, is determined by three points that are not collinear.[45]: 72 

It has been mentioned that circles in the Euclidean plane can not be defined by the focus-directrix property. However, if one were to consider the line at infinity as the directrix, then by taking the eccentricity to be e = 0 a circle will have the focus-directrix property, but it is still not defined by that property.[46] 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.

A synthetic (coordinate-free) approach to defining the conic sections in a projective plane was given by Jakob Steiner in 1867.

By the Principle of Duality in a projective plane, the dual of each point is a line, and the dual of a locus of points (a set of points satisfying some condition) is called an envelope of lines. Using Steiner's definition of a conic (this locus of points will now be referred to as a point conic) as the meet of corresponding rays of two related pencils, it is easy to dualize and obtain the corresponding envelope consisting of the joins of corresponding points of two related ranges (points on a line) on different bases (the lines the points are on). Such an envelope is called a line conic (or dual conic).

In the real projective plane, a point conic has the property that every line meets it in two points (which may coincide, or may be complex) and any set of points with this property is a point conic. It follows dually that a line conic has two of its lines through every point and any envelope of lines with this property is a line conic. At every point of a point conic there is a unique tangent line, and dually, on every line of a line conic there is a unique point called a point of contact. An important theorem states that the tangent lines of a point conic form a line conic, and dually, the points of contact of a line conic form a point conic.[52]: 48–49 

Karl Georg Christian von Staudt defined a conic as the point set given by all the absolute points of a polarity that has absolute points. Von Staudt introduced this definition in Geometrie der Lage (1847) as part of his attempt to remove all metrical concepts from projective geometry.

A polarity, π, of a projective plane, P, is an involutory (i.e., of order two) bijection between the points and the lines of P that preserves the incidence relation. Thus, a polarity relates a point Q with a line q and, following Gergonne, q is called the polar of Q and Q the pole of q.[53] An absolute point (line) of a polarity is one which is incident with its polar (pole).[e]

A von Staudt conic in the real projective plane is equivalent to a Steiner conic.[54]

No continuous arc of a conic can be constructed with straightedge and compass. However, there are several straightedge-and-compass constructions for any number of individual points on an arc.

One of them is based on the converse of Pascal's theorem, namely, Specifically, given five points, A, B, C, D, E and a line passing through E, say EG, a point F that lies on this line and is on the conic determined by the five points can be constructed. Let AB meet DE in L, BC meet EG in M and let CD meet LM at N. Then AN meets EG at the required point F.[55]: 52–53  By varying the line through E,as many additional points on the conic as desired can be constructed.

if the points of intersection of opposite sides of a hexagon are collinear, then the six vertices lie on a conic.

Another method, based on Steiner's construction and which is useful in engineering applications, is the parallelogram method, where a conic is constructed point by point by means of connecting certain equally spaced points on a horizontal line and a vertical line.[56] Specifically, to construct the ellipse with equation x2/a2 + y2/b2 = 1, first construct the rectangle ABCD with vertices A(a, 0), B(a, 2b), C(−a, 2b) and D(−a, 0). Divide the side BC into n equal segments and use parallel projection, with respect to the diagonal AC, to form equal segments on side AB (the lengths of these segments will be b/a times the length of the segments on BC). On the side BC label the left-hand endpoints of the segments with A1 to An starting at B and going towards C. On the side AB label the upper endpoints D1 to Dn starting at A and going towards B. The points of intersection, AAiDDi for 1 ≤ in will be points of the ellipse between A and P(0, b). The labeling associates the lines of the pencil through A with the lines of the pencil through D projectively but not perspectively. The sought for conic is obtained by this construction since three points A, D and P and two tangents (the vertical lines at A and D) uniquely determine the conic. If another diameter (and its conjugate diameter) are used instead of the major and minor axes of the ellipse, a parallelogram that is not a rectangle is used in the construction, giving the name of the method. The association of lines of the pencils can be extended to obtain other points on the ellipse. The constructions for hyperbolas[57] and parabolas[58] are similar.

Yet another general method uses the polarity property to construct the tangent envelope of a conic (a line conic).[59]

Further unification occurs in the complex projective plane CP2: the non-degenerate conics cannot be distinguished from one another, since any can be taken to any other by a projective linear transformation.

It can be proven that in CP2, two conic sections have four points in common (if one accounts for multiplicity), so there are between 1 and 4 intersection points. The intersection possibilities are: four distinct points, two singular points and one double point, two double points, one singular point and one with multiplicity 3, one point with multiplicity 4. If any intersection point has multiplicity > 1, the two curves are said to be tangent. If there is an intersection point of multiplicity at least 3, the two curves are said to be osculating. If there is only one intersection point, which has multiplicity 4, the two curves are said to be superosculating.[60]

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

What should be considered as a degenerate case of a conic depends on the definition being used and the geometric setting for the conic section. There are some authors who define a conic as a two-dimensional nondegenerate quadric. With this terminology there are no degenerate conics (only degenerate quadrics), but we shall use the more traditional terminology and avoid that definition.

In the Euclidean plane, using the geometric definition, a degenerate case arises when the cutting plane passes through the apex of the cone. The degenerate conic is either: a point, when the plane intersects the cone only at the apex; a straight line, when the plane is tangent to the cone (it contains exactly one generator of the cone); or a pair of intersecting lines (two generators of the cone).[61] These correspond respectively to the limiting forms of an ellipse, parabola, and a hyperbola.

In the real projective plane, since parallel lines meet at a point on the line at infinity, the parallel line case of the Euclidean plane can be viewed as intersecting lines. However, as the point of intersection is the apex of the cone, the cone itself degenerates to a cylinder, i.e. with the apex at infinity. Other sections in this case are called cylindric sections.[63] The non-degenerate cylindrical sections are ellipses (or circles).

When viewed from the perspective of the complex projective plane, the degenerate cases of a real quadric (i.e., the quadratic equation has real coefficients) can all be considered as a pair of lines, possibly coinciding. The empty set may be the line at infinity considered as a double line, a (real) point is the intersection of two complex conjugate lines and the other cases as previously mentioned.

To distinguish the degenerate cases from the non-degenerate cases (including the empty set with the latter) using matrix notation, let β be the determinant of the 3 × 3 matrix of the conic section—that is, β = (AC B2/4)F + BEDCD2AE2/4; and let α = B2 − 4AC be the discriminant. Then the conic section is non-degenerate if and only if β ≠ 0. If β = 0 we have a point when α < 0, two parallel lines (possibly coinciding) when α = 0, or two intersecting lines when α > 0.[64]

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.[65]: 64  The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.[66]: 127 

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 3 × 3 symmetric matrix which depends on six parameters.

The procedure to locate the intersection points follows these steps, where the conics are represented by matrices:[67]

Conics may be defined over other fields (that is, in other pappian geometries). However, some care must be used when the field has characteristic 2, as some formulas can not be used. For example, the matrix representations used above require division by 2.

A generalization of a non-degenerate conic in a projective plane is an oval. An oval is a point set that has the following properties, which are held by conics: 1) any line intersects an oval in none, one or two points, 2) at any point of the oval there exists a unique tangent line.

Generalizing the focus properties of conics to the case where there are more than two foci produces sets called generalized conics.

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.

In this interactive SVG, move left and right over the SVG image to rotate the double cone