Conic section: Difference between revisions
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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]]. |
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]]. |
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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 |
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. |
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==Polar coordinates== |
==Polar coordinates== |
Revision as of 17:15, 8 January 2007
![]() | It has been suggested that Latus rectum be merged into this article. (Discuss) Proposed since November 2006. |
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In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a 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
Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an open curve and the plane is not parallel to a generator line of the cone, the figure is a hyperbola. (In this case the plane will intersect both halves of the cone, producing two separate curves, though often one is ignored.)
Degenerate cases
The degenerate cases, where the plane passes through the apex of the cone, resulting in an intersection figure of a point, a straight line, or a pair of intersecting lines, are often excluded from the list of conic sections.
(There are two more degenerate cases. They require the cone itself to be degenerate; where the angle of the cone to its axis is either 90 or 0 degrees. When the angle is 90 degrees, the interior of the cone encompasses all of three-dimensional space and the exterior of the cone is the plane passing through the apex and orthogonal to the cone's axis. That same plane may be chosen as the intersector, yielding the plane, included in its entirety. On the other hand, when the cone's angle to the axis is zero and the plane is parallel to (but not encompassing) the cone's axis, the intersection is null.
Algebraically, those are obtained by setting all parameters equal to 0 (giving the entire plane) or setting c not equal to zero while all other parameters do equal zero, giving the empty set.)
Conics as point loci
Each type of conic can be defined as the locus of all points P with a certain property:
- Circle: , where C is a fixed point (the center), and r is a fixed constant (the radius).
- Parabola: , where F is a fixed point (the focus), and L is a fixed line (the directrix) not containing F.
- Ellipse: , where A,B are distinct fixed points (the foci), and is a fixed constant (the major diameter)
- Hyperbola: , where A,B are distinct fixed points (the foci), and is a fixed constant.
In projective geometry, each of the conics can be defined as the locus of all points equidistant from a fixed point (the focus) and a fixed curve (the directrix); see this detailed description. All conics have identical properties in projective space, and every theorem in projective geometry true of one conic section is true of all the others.
Eccentricity
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Eccentricity.png/280px-Eccentricity.png)
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 , 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 e = 0 and one imagines 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.
Cartesian coordinates
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
- with , , not all zero.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conic_sections_2.png/450px-Conic_sections_2.png)
then:
- if , the equation represents an ellipse (unless the conic is degenerate, for example );
- if and , the equation represents a circle;
- if , the equation represents a parabola;
- if , the equation represents a hyperbola;
- if we also have , the equation represents a rectangular hyperbola.
Note that A and B are just polynomial coefficients, not the lengths of semi-major/minor axis as defined in the previous sections.
Through change of coordinates these equations can be put in standard forms:
- Circle:
- Ellipse:
- Parabola:
- Hyperbola:
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,
- Circle: ,
- Ellipse: ,
- Parabola: ,
- Hyperbola: or .
Homogeneous coordinates
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 degenered, this means that the conic section is in fact a union of two straight lines. A conic section that intersects itself is always degenered, however not all degenered 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 CP2 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 exist at least one intersection point with multiplicity > 1, then the two conic sections are said to be tangent. If there be only one intersection point, which has multiplicity 4, the two conic sections are said to be oscillating.
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.
Polar coordinates
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Elps-slr.svg/300px-Elps-slr.svg.png)
The semi-latus rectum of a conic section, usually denoted l, is the distance from the single focus, or one of the two foci, to the conic section itself, measured along a line perpendicular to the major axis. It is related to the semi-major axis a, and the semi-minor axis b, by the formula , or .
In polar coordinates, a conic section with one focus at the origin and, if any, the other on the positive x-axis, is given by the equation
- .
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.
Properties
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.
Applications
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.
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.
Dandelin spheres
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.
See also
- Focus (geometry), an overview of properties of conic sections related to the foci
- Quadrics, the higher-dimensional analogs of conics
- Matrix representation of conic sections
- Quadratic function
- Rotation of Axes