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In mathematics, a hyperbola is a type of conic section.

• Geometrically, it is defined as the intersection between a cone and a plane which cuts through both halves of the cone.

• Analytically, it is defined as the set of all points for which the difference in the distance to two fixed points (called the foci) is constant.

For a simple geometric proof that the two characterizations above are equivalent to each other, see Dandelin spheres.

• It can also be defined as the locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant larger than 1. This constant is the eccentricity of the hyperbola. These foci lie on the transverse axis and their midpoint is called the center.

A hyperbola comprises two disconnected curves called its arms which separate the foci. At large distances from the foci the hyperbola begins to approximate two lines, known as asymptoteSee also Asymptotic analysis but contrast asymptotic curve''. An asymptote to a curve is a straight line that the curve approaches in such a manner that it becomes as close as one might wish to the line by going far enough along the line. A "real-life" exs.

A hyperbola has the property that a rayRay has several meanings. For the concept in geometry, see Ray (geometry) For the fish, see Ray (fish) For the 2004 movie biography of legendary musician Ray Charles, see Ray (movie) For the city in Iran, see Ray, Iran For the small settlement in North Da originating at one of the foci is reflectedIn mathematics, reflection (also spelt reflexion refers to an involutive automorphism of a space which leaves invariant a subspace of codimension 1. This means that a two-dimensional n dimensional) space is flipped around a one-dimensional n-1 dimensional in such a way as to appear to have originated at the other focus.

A special case of the hyperbola is the equilateral or rectangular hyperbola, in which the asymptotes intersect at right angleThis article is about angles in geometry. For other articles, see Angle (disambiguation An angle (from the Lat. angulus a corner, a diminutive, of which the primitive form, angus does not occur in Latin; cognate are the Lat. angere, to compress into a bens. The rectangular hyperbola with the coordinate axes as its asymptotes is given by the equation xy=c, where c is a constant.

Just as the sine and cosine functions give a parametric equationMultivariate calculus In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. Abstractly, a relation is given in the form of an equation, and it is shown also to be the image of a func for the ellipseThe Ellipse is also an elliptical street immediately in front of the White House. In mathematics, an ellipse is a figure corresponding to a circle which has been stretched in one direction. This is an example of a conic section and can be defined as the l, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.

A body that has sufficient energy to escape the gravitational field of a massive body moves in a hyperbolic trajectory with the massive body at one of the foci.