Hyperbola

A hyperbola is a conic section with two disconnected branches, each defined by a constant absolute difference in distances to two fixed points called foci.

Focus (geometry): A fixed point used to define a conic through distances to points on the curve. The two fixed points in the distance-difference definition are the hyperbola’s foci.
Conic section: A curve formed by intersecting a plane with a double cone. A hyperbola is one of the four principal conic sections.
Hyperbolic trajectory: An unbound orbit in which an object passes another body and escapes with positive excess speed. Its orbit has hyperbolic geometry when gravitational energy leaves the object unbound.
Hyperboloid: A three-dimensional quadric surface generated by rotating or extending a hyperbola. It extends hyperbolic geometry into a surface and is not itself a plane conic.
Menaechmus: An ancient Greek mathematician credited with discovering conic sections while studying the duplication of the cube. His work on sections of cones is among the earliest known origins of hyperbola study.
Eccentricity (mathematics): A dimensionless measure of a conic’s departure from circularity, defined through its focus and directrix. A hyperbola has eccentricity greater than one, distinguishing it from ellipses and parabolas.
Ellipse: A closed conic whose points have a constant sum of distances to two foci. Its focus definition uses a constant sum where a hyperbola uses a constant difference.
Multilateration: A positioning method that determines location from measured distances or distance differences to known points. A measured difference in distances places a transmitter on a hyperbola.
Rectangular coordinates: Coordinates measured along mutually perpendicular axes. The familiar horizontal and vertical hyperbola equations use this coordinate convention.
Apollonius of Perga: A Greek mathematician whose treatise Conics systematically developed the theory of conic sections. His terminology and systematic treatment established the classical study of hyperbolas.

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