<P> In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set . A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows . The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone . (The other conic sections are the parabola and the ellipse . A circle is a special case of an ellipse .) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola . </P> <P> Hyperbolas arise in many ways: </P> <Ul> <Li> as the curve representing the function f (x) = 1 / x (\ displaystyle f (x) = 1 / x) in the Cartesian plane, </Li> <Li> as the path followed by the shadow of the tip of a sundial, </Li> <Li> as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spacecraft during a gravity assisted swing - by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, </Li> <Li> as the path of a single - apparition comet (one travelling too fast ever to return to the solar system), </Li> <Li> as the scattering trajectory of a subatomic particle (acted on by repulsive instead of attractive forces but the principle is the same), </Li> <Li> in radio navigation, when the difference between distances to two points, but not the distances themselves, can be determined, </Li> </Ul> <Li> as the curve representing the function f (x) = 1 / x (\ displaystyle f (x) = 1 / x) in the Cartesian plane, </Li>

The hyperbola is used as a mathematical model in which of the following areas
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