Hyperbola
A hyperbola is a type of conic section, which is formed by the intersection of a double cone and a plane that cuts through both halves. It can be defined in various ways, including as a locus of points where the absolute difference in distances to two fixed points, known as foci, remains constant. Hyperbolas have two distinct branches and two foci, and they approach but never intersect their asymptotes, which are straight lines that guide their shape. The hyperbola can also be represented in parametric equations involving hyperbolic functions, which are analogous to trigonometric functions.
In practical applications, hyperbolas describe the trajectories of objects moving rapidly enough to escape gravitational fields, such as spacecraft. The Long Range Navigation (LORAN) system employs hyperbolic geometry to determine positions based on the timing of radio pulses emitted from multiple stations. Additionally, hyperbolic functions play a significant role in various fields, including physics and engineering, where they model phenomena such as the shape of hanging cables and the motion of objects in resistive media. Understanding hyperbolas and their properties is essential in both theoretical and applied mathematics.
Subject Terms
Hyperbola
A hyperbola is one of the four conic sections: hyperbola, parabola, ellipse, and circle. These are called conic because they are the result of cutting a double-cone with a plane.
Overview
There are five definitions of a hyperbola. All five are equivalent:
1) The intersection of a double-cone and a plane which intersects both halves of the double-cone.
2) Given two points (each is called a focus, plural foci), the locus of all points for which the absolute value of the difference between their distances to each of the two foci is the same.
3) The curve in the xy coordinate plane defined by
4) The curve defined by the parametric equations x = a cosh t and y = b sinh t, where sinh and cosh are the hyperbolic functions
5) The locus of all points whose distance from a fixed point is ε times its distance from a fixed line, with ε > 1.
In the above definitions, ε is called the eccentricity, 2a is the length of the transverse axis, 2b is the length of the conjugate axis, 2c is the distance between the two foci. The eccentricity ε = c/a.
Parts of the hyperbola
The hyperbola has two branches and two foci. The two crossing lines in the above figure are the asymptotes. The hyperbola approaches, but never reaches, its asymptotes.
A hyperbola centered on the origin can be rotated through an angle θ by substituting x cos θ – y sin θ for x, and x sin θ + y cos θ for y. This is equivalent to multiplying the vector (x, y) by the angle θ rotation matrix.
The hyperbola x2 – y2 = 1 (that is, a hyperbola with a = b = 1) can be rotated counterclockwise through an angle of π/4 (45 degrees). The result is xy = 1 or y = 1/x, which is the familiar formula for a hyperbola oriented 45 degrees to the x and y axes and lying entirely in the first and third quadrants.
Uses of the hyperbola
An object moving around a source of gravity (for example, the sun or the earth) will follow a path that is one of the conic sections. If the object is moving faster than the escape velocity of the gravity source, its path will be a hyperbola. Similarly, a charged object moving around another charged object will follow a conic path, and if it’s moving fast enough to escape, its path will be a hyperbola.
The LORAN (Long Range Navigation) system uses the second definition (see the Overview, above). A primary station sends out a radio pulse. When a secondary station hears the pulse, it sends out a pulse of its own. A ship or airplane is on the unique hyperbola determined by the time between the two pulses. By having three secondary stations, three such hyperbolas are generated, and they intersect at a unique point.
The hyperbolic functions are related to trigonometric functions by the following identities, where i is the square root of –1:
The hyperbolic functions have numerous uses in physics and engineering. For example, a cable hanging under its own weight assumes the form of a catenary, with the formula y = a cosh x. If a body is moving through a medium in which the resistance is proportional to the square of the velocity (air resistance has this property), the position and speed of the body are given by equations involving sinh and cosh:
Bibliography
Coxeter, H. S. M. Introduction to Geometry. New York: Wiley, 1989.
McKellar, Danica. Girls Get Curves. New York: Penguin, 2012.
Posamentier, Alfred S, and Robert L. Bannister. Geometry, Its Elements and Structure. Mineola, NY: Dover, 2014.
Stankowski, James F., ed. Geometry and Trigonometry. New York: Rosen, 2015.
Swokowski, Earl W., Jeffery A. Cole. Algebra and Trigonometry with Analytic Geometry. Belmont CA: Cengage, 2011.