Editing Quaternion (section)

==Functions of a quaternion variable==
{{Main|Quaternionic analysis}}
[[File:Quaternion Julia x=-0,75 y=-0,14.jpg|thumb|The Julia sets and Mandelbrot sets can be extended to the Quaternions, but they must use cross sections to be rendered visually in 3&nbsp;dimensions. This Julia set is cross sectioned at the {{mvar|x y}} plane.]]
Like functions of a [[complex variable]], functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the [[Mandelbrot set]] and [[Julia set]]s into 4-dimensional space.<ref>{{cite web |title=[no title cited] |website=bridgesmathart.org |series=archive |url=http://archive.bridgesmathart.org/2010/bridges2010-247.pdf |access-date=19 August 2018}}</ref>

===Exponential, logarithm, and power functions===
Given a quaternion,

<math display=block> q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k = a + \mathbf{v}, </math>

the exponential is computed as<ref name=Särkkä2007>{{cite web |url=http://www.lce.hut.fi/~ssarkka/pub/quat.pdf |website=Lce.hut.fi |title=Notes on Quaternions |first=Simo |last=Särkkä |date=June 28, 2007 |archive-url=https://web.archive.org/web/20170705123142/http://www.lce.hut.fi/~ssarkka/pub/quat.pdf |archive-date=5 July 2017}}</ref>

<math display=block>
\exp(q)
= \sum_{n=0}^\infty \frac{q^n}{n!}
= e^{a} \left(\cos \|\mathbf{v}\| + \frac{\mathbf{v}}{\|\mathbf{v}\|} \sin \|\mathbf{v}\|\right),
</math>

and the logarithm is<ref name=Särkkä2007/>

<math display=block>\ln(q) = \ln \|q\| + \frac{\mathbf{v}}{\|\mathbf{v}\|} \arccos \frac{a}{\|q\|}.</math>

It follows that the polar decomposition of a quaternion may be written

<math display=block>q=\|q\|e^{\hat{n}\varphi} = \|q\| \left(\cos(\varphi) + \hat{n} \sin(\varphi)\right),</math>

where the [[angle]] <math>\varphi</math>{{efn|name="θ"|Books on applied mathematics, such as Corke (2017)<ref>{{cite book |title=Robotics, Vision, and Control – Fundamental Algorithms in MATLAB |last=Corke |first=Peter |publisher=[[Springer Publishing|Springer]] |year=2017 |isbn=978-3-319-54413-7}}</ref> often use different notation with {{math|''φ'' :{{=}} {{sfrac|1|2}}''θ''}} — that is, [[change of variables|another variable]] {{math|''θ'' {{=}} 2''φ''}}.}}

<math display=block>a = \| q \| \cos( \varphi )</math>

and the unit vector <math>\hat{n}</math> is defined by:

<math display=block>\mathbf{v} = \hat{n} \|\mathbf{v}\|= \hat{n}\|q\|\sin(\varphi).</math>

Any unit quaternion may be expressed in polar form as:

<math display=block>q=\exp{(\hat{n}\varphi)}.</math>

The [[Power (mathematics)|power]] of a quaternion raised to an arbitrary (real) exponent {{mvar|x}} is given by:

<math display=block>q^x = \|q\|^x e^{\hat{n} x \varphi} = \|q\|^x \left(\cos(x\varphi) + \hat{n}\,\sin(x\varphi)\right).</math>

===Geodesic norm===
The [[Great-circle distance|geodesic distance]] {{nowrap|{{math|''d''<sub>g</sub>(''p'', ''q'')}}}} between unit quaternions {{mvar|p}} and {{mvar|q}} is defined as:<ref name=Geodesic>{{cite journal |last1=Park |first1=F.C. |last2=Ravani |first2=Bahram |title=Smooth invariant interpolation of rotations |journal=ACM Transactions on Graphics |date=1997 |volume=16 |number=3 |pages=277–295 |doi=10.1145/256157.256160|s2cid=6192031 |doi-access=free }}</ref>

<math display=block>d_\text{g}(p, q) = \lVert \ln(p^{-1} q) \rVert.</math>

and amounts to the absolute value of half the angle subtended by {{mvar|p}} and {{mvar|q}} along a [[Arc (geometry)|great arc]] of the {{math|S<sup>3</sup>}} sphere.
This angle can also be computed from the quaternion [[dot product]] without the logarithm as:

<math display=block>d_\text{g}(p, q) = \arccos(2(p \cdot q)^2 - 1).</math>