Ingeometry,focuses orfoci (/ˈfoʊsaɪ/ or/ˈfoʊkaɪ/;sg.:focus) are special points with reference to which any of a variety ofcurves is constructed. For example, one or two foci can be used in definingconic sections, the four types of which are thecircle,ellipse,parabola, andhyperbola. In addition, two foci are used to define theCassini oval and theCartesian oval, and more than two foci are used in defining ann-ellipse.
Anellipse can be defined as thelocus of points for which the sum of the distances to two given foci is constant.
Acircle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as thecircle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of distances to the two foci.
Aparabola is a limiting case of an ellipse in which one of the foci is apoint at infinity.
Ahyperbola can be defined as the locus of points for which theabsolute value of the difference between the distances to two given foci is constant.
It is also possible to describe allconic sections in terms of a single focus and a singledirectrix, which is a givenline not containing the focus. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called theeccentricitye. If0 <e < 1 the conic is an ellipse, ife = 1 the conic is a parabola, and ife > 1 the conic is a hyperbola. If the distance to the focus is fixed and the directrix is aline at infinity, so the eccentricity is zero, then the conic is a circle.
It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle. The ellipse thus generated has its second focus at the center of the directrix circle, and the ellipse lies entirely within the circle.
For the parabola, the center of the directrix moves to the point at infinity (seeProjective geometry). The directrix "circle" becomes a curve with zero curvature, indistinguishable from a straight line. The two arms of the parabola become increasingly parallel as they extend, and "at infinity" become parallel; using the principles of projective geometry, the two parallels intersect at the point at infinity and the parabola becomes a closed curve (elliptical projection).
To generate a hyperbola, the radius of the directrix circle is chosen to be less than the distance between the center of this circle and the focus; thus, the focus is outside the directrix circle. The arms of the hyperbola approachasymptotic lines and the "right-hand" arm of one branch of a hyperbola meets the "left-hand" arm of the other branch of a hyperbola at the point at infinity; this is based on the principle that, in projective geometry, a single line meets itself at a point at infinity. The two branches of a hyperbola are thus the two (twisted) halves of a curve closed over infinity.
In projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others.
In thegravitationaltwo-body problem, the orbits of the two bodies about each other are described by two overlapping conic sections with one of the foci of one being coincident with one of the foci of the other at thecenter of mass (barycenter) of the two bodies.
Thus, for instance, theminor planetPluto's largestmoonCharon has an elliptical orbit which has one focus at the Pluto-Charon system's barycenter, which is a point that is in space between the two bodies; and Pluto also moves in an ellipse with one of its foci at that same barycenter between the bodies. Pluto's ellipse is entirely inside Charon's ellipse, as shown inthis animation of the system.
By comparison, the Earth'sMoon moves in an ellipse with one of its foci at the barycenter of the Moon and theEarth, this barycenter being within the Earth itself, while the Earth (more precisely, its center) moves in an ellipse with one focus at that same barycenter within the Earth. The barycenter is about three-quarters of the distance from Earth's center to its surface.
Moreover, the Pluto-Charon system moves in an ellipse around its barycenter with theSun, as does the Earth-Moon system (and every other planet-moon system or moonless planet in theSolar System). In both cases the barycenter is well within the body of the Sun.
Twobinary stars also move in ellipses sharing a focus at their barycenter; for an animation, seehere.
ACartesian oval is the set of points for each of which theweighted sum of the distances to two given foci is constant. If the weights are equal, the special case of an ellipse results.
ACassini oval is the set of points for each of which the product of the distances to two given foci is constant.
Ann-ellipse is the set of points all having the same sum of distances ton foci (then = 2 case being the conventional ellipse).
The concept of a focus can be generalized to arbitraryalgebraic curves. LetC be a curve of classm and letI andJ denote thecircular points at infinity. Draw them tangents toC through each ofI andJ. There are two sets ofm lines which will havem2 points of intersection, with exceptions in some cases due to singularities, etc. These points of intersection are the defined to be the foci ofC. In other words, a pointP is a focus if bothPI andPJ are tangent toC. WhenC is a real curve, only the intersections of conjugate pairs are real, so there arem in a real foci andm2 −m imaginary foci. WhenC is a conic, the real foci defined this way are exactly the foci which can be used in the geometric construction ofC.
LetP1,P2, …,Pm be given as foci of acurveC of classm. LetP be the product of the tangential equations of these points andQ the product of the tangential equations of the circular points at infinity. Then all the lines which are common tangents to bothP = 0 andQ = 0 are tangent toC. So, by theAF+BG theorem, the tangential equation ofC has the formHP +KQ = 0. SinceC has classm,H must be a constant andK but have degree less than or equal tom − 2. The caseH = 0 can be eliminated as degenerate, so the tangential equation ofC can be written asP +fQ = 0 wheref is an arbitrarypolynomial ofdegree2m.[1]
For example, letm = 2,P1 = (1, 0), andP2 = (−1, 0). The tangential equations are
soP =X2 − 1 = 0. The tangential equations for the circular points at infinity are
soQ =X2 +Y2. Therefore, the tangential equation for a conic with the given foci is
or
wherec is an arbitrary constant. In point coordinates this becomes