Hyperbola

Pronunciation: /haɪˈpɜr.bə.lə/ Explain

Conic with hyperbolic section.
Figure 1: Conic section

A hyperbola is a conic section formed by intersecting a right circular conic surface and a plane that intersects both halves of the cone. The equations most often used for a hyperbola are:

((x-h)^2)/a-((y-k)^2)/b=1
where a is the semi-major axis parallel to the x-axis, b is the semi-minor axis parallel to the y-axis, and the point (h, k) is the center of the hyperbola, and
((y-k)^2)/b-((x-h)^2)/a=1
where a is the semi-minor axis parallel to the x-axis, b is the semi-major axis parallel to the y-axis, and the point (h, k) is the center of the hyperbola.

Graph of a Hyperbola

Click on the blue points and drag them to change the figure.

Why does the hyperbola disappear when a or b is zero?
Manipulative 1 - Hyperbola Created with GeoGebra.

A hyperbola can be defined as a set of points where the ratio of the distance from a fixed line called the directrix and the distance from the point is equal to the ratio of c to a. The equation for the directrices of a hyperbola with an east/west opening is

x=h+-a^2/c.
The equation for the directrices of a hyperbola with a north/south opening is
y=k+-b^2/c.

The foci for a hyperbola are the points at (h+-c,k) for a hyperbola with an east/west opening and (h,k+-c) for a hyperbola with a north/south opening.

In manipulative 1, click on the check box marked 'Show directrices'. Click on the blue point on the hyperbola and drag it. Notice that the ratio remains constant for a particular hyperbola.

Eccentricity of Hyperbolas

The eccentricity of a hyperbola can be considered as how far the hyperbola deviates from a circle. The larger the eccentricity, the flatter the hyperbolic curve. The formula for the eccentricity of a hyperbolic curve is

epsilon=square root(1+b^2/a^2)

Hyperbolas in Nature

A flashlight shining on a wall so the edge of the light forms a hyperbola.
Figure 2: Flashlight hyperbola. Click on the image to see a larger version.
A flashlight makes a cone of light. If a flashlight is held close to a wall and parallel to the wall, the plane of the wall 'cuts' the cone at in such a way that the 'edges' of the light form a hyperbola.

Drawing of a tube placed into liquid. The liquid is drawn up into the tube and forms a portion of a hyperbolic curve.
Figure 3: Capillary action.
A liquid in a small tube is drawn up the tube by a action known as capillary action. The liquid forms a hyperbolic curve. For more information see capillary action in The Internet Encyclopedia of Science.

The path of an object in space relative to a much larger object can be a hyperbola. The path of the object is a hyperbolic trajectory if the speed of the smaller object relative to the larger object is more than escape velocity. In actuality, very few hyperbolic trajectories are observed in nature. This is because an object on a hyperbolic trajectory 'breaks free' of the gravitational field of the larger object (in this case the sun). Because it breaks free, the object does not return close to the larger object so is rarely observed.

Hyperboloid

Hyperboloid of one sheet.
Figure 4: Hyperboloid of one sheet.

A hyperboloid is a 3-dimensional figure created by rotating a hyperbola about a line. Figures 4 and 5 show two types of hyperboloids.

Hyperboloid of two sheets.
Figure 5: Hyperboloid of two sheets.

References

  1. McAdams, David E.. All Math Words Dictionary, hyperbola. 2nd Classroom edition 20150108-4799968. pg 92. Life is a Story Problem LLC. January 8, 2015. Buy the book

More Information

  • McAdams, David E.. Conic Section. allmathwords.org. Life is a Story Problem LLC. 3/12/2009.

Cite this article as:

 McAdams, David E. Hyperbola. 4/22/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/h/hyperbola.html.

Image Credits

  • All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
  • Hyperbolic Conic Section. Unknown.
  • Hyperboloid of One Sheet: Svdmolen, https://en.wikipedia.org. Licensed under the terms of the GNU Free Documentation License, Version 1.2 or any later version. https://en.wikipedia.org/wiki/File:HyperboloidOfOneSheet.png.
  • Hyperboloid of Two Sheets: Svdmolen, https://en.wikipedia.org. Licensed under the terms of the GNU Free Documentation License, Version 1.2 or any later version.
  • Flashlight Making a Hyperbola: Dave Clauson. Used with permission. Do not copy.

Revision History

4/22/2019: Update equations and expressions to new format. (McAdams, David E.)
3/17/2019: Fixed broken link. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/16/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
2/8/2010: Added "References". (McAdams, David E.)
1/7/2010: Added "References". (McAdams, David E.)
12/12/2008: Added hyperboloid. (McAdams, David E.)
12/10/2008: Corrected equations. Added hyperbolas in nature (McAdams, David E.)
11/26/2008: Change equations to images. (McAdams, David E.)
8/8/2008: Added paragraph on eccentricity. Added https://www.allmathwords.org/conicsection.html to More Information. Added manipulative 1. Converted equation to hot_eqn (McAdams, David E.)
7/1/2008: Initial version. (McAdams, David E.)

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