Conic section

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting 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 it was sometimes considered 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 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 can be written in the form \(Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0.\) The geometric properties of the conic can be deduced from its equation.

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.

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 it 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.

Compare also spheric section (intersection of a plane with a sphere, producing a circle or point), and spherical conic (intersection of an elliptic cone with a concentric sphere).

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 e (called the eccentricity) 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.

The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.

If the angle between the surface of the cone and its axis is \(\beta\) and the angle between the cutting plane and the axis is \(\alpha,\) the eccentricity is \(\frac{\cos\alpha}{\cos\beta}.\)

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.

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 (ℓ).

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.

• \(\ \ell = p e\)
• \(\ c = a e\)
• \(\ p+c = \frac ae\)

Condensed: the full section is in Wikipedia.

After introducing Cartesian coordinates, the focus-directrix property can be used to produce the equations satisfied by the points of the conic section. By means of a change of coordinates (rotation and translation of axes) these equations can be put into standard forms. 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 c = a − b for an ellipse and c = a + b for a hyperbola. For a circle, c = 0 so a = b, with radius r = a = b. 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).

• Circle:

  \(x^2 + y^2 = a^2,\)

• Ellipse:

  \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\)

• Parabola:

  \(y^2 = 4ax, \text{ with } a > 0,\)

• Hyperbola:

  \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\)

• Rectangular hyperbola:

  \(xy = c^2.\)

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.

These standard forms can be written parametrically as,

• Circle:

  \((a \cos\theta, a \sin\theta),\)

• Ellipse:

  \((a \cos\theta, b \sin\theta),\)

• Parabola:

  \((at^2, 2at),\)

• Hyperbola:

  \((a\sec\theta, b\tan\theta), \text{ or } (\pm a\cosh \psi, b \sinh \psi)\),

• Rectangular hyperbola:

  \(\left(dt, \frac dt\right), \text{ where } d = \frac c{\sqrt{2}}.\)

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 most general equation is of the form

\(Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0,\)

with all coefficients real numbers and A, B, C not all zero.

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

\(r=\frac{l}{{1+e\cos\theta}},\)

where e is the eccentricity and l is the semi-latus rectum.

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.

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.

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.

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.

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 BC) as part of his solution of the Delian problem (Duplicating the cube). His work did not survive, not even the names he used for these curves, and is only known through secondary accounts. 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).

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 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.

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 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.

Pappus of Alexandria (died c. 350 AD) 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).

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

A century before the more famous work of Khayyam, Abu al-Jud used conics to solve quartic and cubic equations, although his solution did not deal with all the cases.

An instrument for drawing conic sections was first described in 1000 AD by Al-Kuhi.

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.

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. 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. De Witt invented the term 'directrix'.

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. 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.