In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a circular conical surface) with a plane. A conic section is therefore a restriction of a quadric surface to the plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
Types of conics
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. 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.
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 focus-directrix 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 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 .
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 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 semi-minor axis.
Note that A and B are just polynomial coefficients, not the lengths of semi-major/minor axis as defined in the following sections.
In matrix notation the equation above becomes:
Such forms will be symmetrical about the x-axis and for the circle, ellipse and hyperbola symmetrical about the y-axis.
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 A1 = A2, it is an rectangular hyperbola if δ<0 and A1 = -A2. It can be proven that in the complex projective plane CP 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.
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 x-axis, is given by the equation
where e is the eccentricity and l is the semi-latus 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 can be associated with a conic section.
For every conic section, there exist a fixed point F, a fixed line L and a non-negative number e such that the conic section consists of all points whose distance to F equals e times their distance to L. e is called the eccentricity of the conic section.
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 relation p = l / e holds.
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 two-body problem.
Intersecting two conics
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.
Published - July 2009
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