Quaternions: Difference between revisions
imported>Ragnar Schroder (→Related topics: formatting) |
imported>Jitse Niesen (expand on connection with rotation, general clean up) |
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== Definition & basic operations == | == Definition & basic operations == | ||
The quaternions, <math>\mathbb{H}</math>, | The quaternions, <math>\mathbb{H}</math>, form a four-dimensional normed division algebra over the [[Real number|real numbers]].<br/><br/> | ||
:<math>\mathbb{H}=\left\lbrace a+\mathit{i}b+\mathit{j}c+\mathit{k}d | :<math>\mathbb{H}=\left\lbrace a+\mathit{i}b+\mathit{j}c+\mathit{k}d \mid a,b,c,d\in\mathbb{R}\right\rbrace</math> | ||
:<math>\mathit{i}^2=\mathit{j}^2=\mathit{k}^2=\mathit{ijk}=-1</math> | :<math>\mathit{i}^2=\mathit{j}^2=\mathit{k}^2=\mathit{ijk}=-1 \,</math> | ||
== Properties == | == Properties == | ||
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== Applications == | == Applications == | ||
Quaternions can be used to model the three-dimensional rotation group. | Quaternions can be used to model the three-dimensional rotation group. First, every 3-dimensional vector <math>(x,y,z)</math> is associated with the quaternion <math>ix + jy + kz</math>. A [[rotation]] over an angle <math>\alpha</math> around the axis defined by the unit vector <math>u</math> is then represented by the unit quaternion | ||
:<math>\cos \left( \frac{\alpha}{2} \right) + u \sin \left( \frac{\alpha}{2} \right). </math> | |||
In other words, the image of a vector under this rotation is the same as the product of the quaternion representing the rotation with the quaternion representing the vector. | |||
The set of such unit quaternions form a [[group]] under quaternion multiplication. | The set of such unit quaternions form a [[group (mathematics)|group]] under quaternion multiplication. | ||
==See also== | ==See also== | ||
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== References == | == References == | ||
* | *Henry Baker. [http://home.pipeline.com/~hbaker1/QuaternionRefs.txt Quaternion references]. Electronic document. | ||
*Simon Altmann (2005). ''Rotations, Quaternions, and Double Groups''. Dover Publications. ISBN 978-0486445182. (First edition appeared in 1977). | |||
* | |||
==External links== | ==External links== | ||
*[http://mathworld.wolfram.com/Quaternion.html MathWorld | *[http://mathworld.wolfram.com/Quaternion.html Quaternion] at MathWorld | ||
Revision as of 05:46, 26 November 2007
Quaternions are a non-commutative extension of the complex numbers. They were first described by Sir William Rowan Hamilton in 1843. He famously inscribed their defining equation on Broom Bridge in Dublin when walking with his wife on 16 October 1843. They have many possible applications, including in computer graphics, but have during their history proved comparatively unpopular, with vectors being preferred instead.
Definition & basic operations
The quaternions, , form a four-dimensional normed division algebra over the real numbers.
Properties
Applications
Quaternions can be used to model the three-dimensional rotation group. First, every 3-dimensional vector is associated with the quaternion . A rotation over an angle around the axis defined by the unit vector is then represented by the unit quaternion
In other words, the image of a vector under this rotation is the same as the product of the quaternion representing the rotation with the quaternion representing the vector.
The set of such unit quaternions form a group under quaternion multiplication.
See also
Related topics
References
- Henry Baker. Quaternion references. Electronic document.
- Simon Altmann (2005). Rotations, Quaternions, and Double Groups. Dover Publications. ISBN 978-0486445182. (First edition appeared in 1977).
External links
- Quaternion at MathWorld