Hyperbola.

Hyperbola. Hyperbola, two-dimensional curve with two mirror-image, U-shaped branches that do not touch and face in opposite directions. The arms of each branch separate as they recede. The difference between the distances from any point on a hyperbola to two fixed points, called the foci, is a constant. Each branch contains one of the foci in its interior area; the line joining the foci intersects each branch in a point called a vertex. The line through the vertices and the foci is called the transverse axis. The line perpendicular to the transverse axis and passing through the point midway between the vertices is the conjugate axis. The two axes meet at the center of the hyperbola. The hyperbola is symmetric with respect to each axis, meaning that the portion that lies to one side of the line is a mirror image of the portion on the other side. A hyperbola has two asymptotes passing through the center. An asymptote of a curve is a straight line that gets closer and closer to the curve but never quite touches it as both the curve and the asymptote extend to infinity. A hyperbola that has asymptotes that are perpendicular to each other is called a rectangular or equilateral hyperbola. A hyperbola can be drawn by graphing its equation on xy axes. The equation of a hyperbola with a center at (0, 0)--the point where the x and y axes meet--is while the more general equation of a hyperbola centered at any point (h, k) is , . The variable a represents the distance from the hyperbola's center to either vertex. The variable b is one-half the distance between the two asymptotes along a line tangent to a vertex (a line that just barely touches the hyperbola at the vertex). The hyperbola is a member of a family of curves called conic sections. Conic sections are formed by a plane that cuts through a double-ended right circular cone (two cones held tip to tip). The angle at which the plane passes through the cone determines the shape of the curve it produces. Hyperbolas are produced when the plane passes through both ends of the cone. Hyperbolas have useful mathematical and scientific properties. In particular, the angle between two lines joining a point on the hyperbola to the two foci is bisected (divided exactly in half) by the tangent to the hyperbola at that point. In astronomy, some orbits are hyperbolic in shape. For example, certain comets--those that have a high enough velocity and a large enough mass so that they cannot be captured by the Sun's gravitational field--move in hyperbolic orbits. The modern navigational system called loran also uses hyperbolas. Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.