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 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.
Euclidean geometry
The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry.
Definition 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).
Eccentricity, focus and directrix
Alternatively, one can define a conic section purely in terms of plane geometry: it is the locus of all points whose distance to a fixed point (called the focus) is a constant multiple (called the eccentricity) of the distance from to a fixed line (called the directrix). For we obtain an ellipse, for a parabola, and for 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.Cohen, D., Precalculus: With Unit Circle Trigonometry (Stamford: Thomson Brooks/Cole, 2006), p. 844.
If the angle between the surface of the cone and its axis is and the angle between the cutting plane and the axis is the eccentricity is
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 ; while a hyperbola is the locus for which the difference of distances is . (Here 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 ; plus if the point is between the directrix and the latus rectum, minus otherwise.
Conic parameters
In addition to the eccentricity (), 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 () 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 () 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 (). When an ellipse or hyperbola are in standard position as in the equations below, with foci on the -axis and center at the origin, the vertices of the conic have coordinates and , with non-negative.
The minor axis is the shortest diameter of an ellipse, and its half-length is the semi-minor axis (), the same value as in the standard equation below. By analogy, for a hyperbola the parameter in the standard equation is also called the semi-minor axis.
The following relations hold:
For conics in standard position, these parameters have the following values, taking . conic section equation eccentricity () linear eccentricity () semi-latus rectum () focal parameter () circle ellipse parabola N/A hyperbola
Standard forms in Cartesian coordinates
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 -axis as principal axis and the origin (0,0) as center. The vertices are and the foci . Define by the equations for an ellipse and for a hyperbola. For a circle, so , with radius . For the parabola, the standard form has the focus on the -axis at the point and the directrix the line with equation . 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 is the principal axis. The foci then have coordinates and .
Circle Ellipse Parabola Hyperbola Rectangular hyperbola
The first four of these forms are symmetric about both the -axis and -axis (for the circle, ellipse and hyperbola), or about the -axis only (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines and .
These standard forms can be written parametrically as, Circle Ellipse Parabola Hyperbola Rectangular hyperbola
General Cartesian form 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
with all coefficients real numbers and not all zero.
Matrix notation
The above equation can be written in matrix notation as
The general equation can also be written as
This form is a specialization of the homogeneous form used in the more general setting of projective geometry (see below).
Discriminant The conic sections described by this equation can be classified in terms of the value , called the discriminant of the equation., Section 3.2, page 45 Thus, the discriminant is where is the matrix determinant
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 , the equation represents a rectangular hyperbola. In the notation used here, and are polynomial coefficients, in contrast to some sources that denote the semimajor and semiminor axes as and .
Invariants
The discriminant of the conic section's quadratic equation (or equivalently the determinant of the 2 × 2 matrix) and the quantity (the trace of the 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes,Pettofrezzo, Anthony, Matrices and Transformations, Dover Publ., 1966, p. 110. as is the determinant of the 3 × 3 matrix above. The constant term and the sum are invariant under rotation only.
Eccentricity in terms of coefficients When the conic section is written algebraically as
the eccentricity can be written as a function of the coefficients of the quadratic equation.Ayoub, Ayoub B., "The eccentricity of a conic section", The College Mathematics Journal 34(2), March 2003, 116–121. If 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 if the determinant of the 3 × 3 matrix above is negative and if that determinant is positive.
It can also be shownSpain, B., Analytical Conics (Mineola, NY: Dover, 2007). Originally published in 1957 by Pergamon. that the eccentricity is a positive solution of the equation
where again This has precisely one positive solution—the eccentricity— in the case of a parabola or ellipse, while in the case of a hyperbola it has two positive solutions, one of which is the eccentricity.
Conversion to canonical form
In the case of an ellipse or hyperbola, the equation
can be converted to canonical form in transformed variables asAyoub, A. B., "The central conic sections revisited", Mathematics Magazine 66(5), 1993, 322–325.
or equivalently
where and are the eigenvalues of the matrix — that is, the solutions of the equation
— and is the determinant of the 3 × 3 matrix above, and is again the determinant of the 2 × 2 matrix. In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form.
Polar coordinates
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 -axis, is given by the equation
where is the eccentricity and is the semi-latus rectum.
As above, for , the graph is a circle, for the graph is an ellipse, for a parabola, and for 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.
Properties