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Concept
Focus (geometry)
Summary
In geometry, focuses or foci (ˈfəʊkaɪ; : focus) are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse.
Conic section#Eccentricity, focus and directrix Ellipse#FocusParabola#Position of the focusHyperbola#Directrix and focus and Confocal conic sections
An ellipse can be defined as the locus of points for which the sum of the distances to two given foci is constant.
A circle 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 the circle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of distances to the two foci.
A parabola is a limiting case of an ellipse in which one of the foci is a point at infinity.
A hyperbola can be defined as the locus of points for which the absolute value of the difference between the distances to two given foci is constant.
It is also possible to describe all conic sections in terms of a single focus and a single directrix, which is a given line 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 the eccentricity e. If 0 < e < 1 the conic is an ellipse, if e = 1 the conic is a parabola, and if e > 1 the conic is a hyperbola. If the distance to the focus is fixed and the directrix is a line 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.
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