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Subsections

exponential and logarithm

Syntax

Quaternion   exp()const;
Quaternion   Log()const;
Quaternion   power(const Real t)const;

Description

A unit quaternion can be represented by $q = cos(\theta) + u
sin(\theta)$. Euler's identity for complex numbers generalizes to quaternions $exp(u\theta) = cos(\theta) + u
sin(\theta)$, where $exp(x)$ is replace by $exp(u\theta)$ and $uu$ is replace by $-1$. With this identity we obtain the exponential of the quaternion $q =
(0,\theta v)$, where $q$ is not necessary a unit quaternion. It is then possible to define the logarithm and the power of a unit quaternion [2].
$\displaystyle Log(q)$ $\textstyle =$ $\displaystyle Log\big(\cos(\theta) + u \sin(\theta)\big) =
Log\big(exp(u\theta)\big) = u\theta$ (2.20)
$\displaystyle q^t$ $\textstyle =$ $\displaystyle cos(t\theta) + u sin(t\theta)$ (2.21)

$Log(q)$ is not necessary a unit quaternion even if $q$ is a unit quaternion.

Return Value

Quaternion for exp, Log



Richard Gourdeau 2004-07-06