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单词 ConicSection
释义

conic section


Definitions

In EuclideanPlanetmathPlanetmath 3-space, a conic sectionMathworldPlanetmath, or simply a conic, is theintersectionMathworldPlanetmath of a plane with a right circular double cone.

But a conic can also defined, in several equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ways,without using an enveloping 3-space.

In the Euclidean planeMathworldPlanetmath, let d be a line and F a point not ond. Let ϵ be a positive real number.For an arbitrary point P, write |Pd| for the perpendicularMathworldPlanetmathPlanetmathPlanetmath(or shortest) distanceMathworldPlanetmathPlanetmath from P to the line d.The set of all points P such that |PF|=ϵ|Pd| is a conicwith eccentricity ϵ, focus F, anddirectrix d.

An ellipseMathworldPlanetmath, parabola, or hyperbolaMathworldPlanetmath has eccentricity <1, =1, or>1 respectively.For a parabola, the focus and directrix are unique.Any ellipse other than a circle, or any hyperbola, may be defined byeither of two focus-directrix pairs; the eccentricity is the same for both.

The definition in terms of a focus and a directrix leaves out the caseof a circle; still, the circle can be thought of as a limiting case:eccentricity zero, directrix at infinityMathworldPlanetmath, and two coincident foci.

The chord through the given focus, parallelMathworldPlanetmathPlanetmath to the directrix,is called the latus rectum; its length is traditionallydenoted by 2l.

Given a conic σ which is the intersection ofa circular cone C with a plane π, and given a focus F of σ,there is a unique sphere tangent to π at F and tangent also to Cat all points of a circle.That sphere is called the Dandelin sphere for F.(Consider a spherical ball resting on a table.Suppose that a point source of light, at some point above the tableand outside the ball, shines on the ball.The margin of the shadow of the ball is a conic,the ball is one of the Dandelin spheres of that conic,and the ball meets the table at the focus corresponding to that sphere.)

Degenerate conics; coordinatesMathworldPlanetmathPlanetmath in 2 or 3 dimensionsMathworldPlanetmathPlanetmathPlanetmath

The intersection of a plane with a cone may consist of asingle point, or a line, or a pair of lines.Whether we should regard these sets as conics is a matterof convention, but in general they are not so regarded.

In the Euclidean plane with the usual Cartesian coordinatesMathworldPlanetmath,a conic is the set of solutions of an equation of the form

P(x,y)=0

where P is a polynomial of the second degree over .For a degenerate conic, P has discriminantMathworldPlanetmath zero.

In three dimensions, if a conic is defined as the intersectionof the cone

z2=x2+y2

with a plane

αx+βy+γz=δ

then, assuming γ0, we can eliminate z to get apolynomial for the curve in terms of x and y only; alinear change of variables will then give Cartesian coordinates,within the plane, for the given conic.If γ=0 we can eliminate x or y instead, with the sameresult.

Conics and physics

Kepler revolutionized astronomy by describing simple mathematical laws for planetary motion, which Newton then derived from his laws of motion using calculus, which he invented for the purpose (although several other people invented calculus at more or less the same time). One of Kepler’s laws was that planets move on ellipses with the Sun at one focus. More generally, objects under the influence of the Sun’s gravity (and no other forces) move in conic sections with the Sun at one focus. Comets moving on parabolae or hyperbolae do not return; comets moving on elongated ellipses return.

To work with conic sections in such an astronomical context, it is very useful to have a description in terms of polar coordinatesMathworldPlanetmath centered at one focus. Measuring the angle θ from the closest approach to the focus, an ellipse with semi-major axis a can be described by

r(θ)=a(1-ϵ2)1-ϵcosθ.

A parabola or hyperbola can also be described in an essentially similarMathworldPlanetmathPlanetmath way, although they do not have a semi-major axis. If we call the length of the latus rectum 2l, then any conic can be described as

r(θ)=l1-ϵcosθ.

Observe that as ϵ goes to zero we approach a circle; as ϵ goes to 1 we approach a parabola; and for ϵ>1 we obtain a hyperbola. (Normally one does not consider the case ϵ<0, as this simply amounts to choosing a different place to measure angle).

Conics in a projective planeMathworldPlanetmath

Conic sections can be defined in a projective plane, even though,in such a plane, there is no notion of angle nor any notion of distance.In fact there are several equivalent ways to define it.The following elegant definitionwas discovered by von Staudt:A conic is the set of self-conjugate points of a hyperbolic polarityMathworldPlanetmathPlanetmath.In a little more detail, the polarity is a pair of bijectionsMathworldPlanetmath

f:PL  g:LP

where P is the set of points of the plane, L is the set of lines,f maps collinear points to concurrent lines, and g maps concurrent linesto collinear points.The set of fixed pointsPlanetmathPlanetmath of gf is a conic, and f(x) is the tangentto the given conic at the given point x.

A projective conic has no focus, directrix, or eccentricity, for in aprojective plane there is no notion of distance (nor angle).Indeed all projective conics are alike; there is no distinction between a parabola and a hyperbola, for example.

If we visualize the projective plane as the set of all lines in three-space, then a conic is precisely a cone. An embeddingMathworldPlanetmathPlanetmath of the plane into the projective plane, in this context, is just the selection of a plane not passing through the origin. Then we identify a projective “point” (a line through the origin) with its intersection with this plane. Points whose line does not intersect this plane are considered points at infinity. With this interpretationMathworldPlanetmathPlanetmath, it is clear why all conic sections are essentially the same in the projective plane: if we choose a plane cutting the cone perpendicularly, we obtain a circle; if we tip the plane, we get an ellipse; as we tip it further, at a certain point there is exactly one point at infinity on the cone, giving a parabola; finally we get two branches, connectedPlanetmathPlanetmathPlanetmath by two points at infinity, that is, a hyperbola. The only difference is how we choose to embed the plane in the projective plane.

Titleconic section
Canonical nameConicSection
Date of creation2013-03-22 13:08:45
Last modified on2013-03-22 13:08:45
Ownerdrini (3)
Last modified bydrini (3)
Numerical id18
Authordrini (3)
Entry typeDefinition
Classificationmsc 51N20
Synonymconic
Related topicHyperbola2
Related topicParabola2
Related topicSqueezingMathbbRn
Related topicAnalyticGeometry
Related topicAsymptoteOfLamesCubic
Related topicQuadraticCurves
Related topicIntersectionOfQuadraticSurfaceAndPlane
Related topicSimplestCommonEquationOfConics
Related topicPropertiesOfEllipse
Definesellipse
Definesparabola
Defineshyperbola
Definesfocus
Defineseccentricity
Definesdirectrix
Defineslatus rectum
DefinesDandelin sphere
Definesdegenerate conic
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