Rotation vector and operator
================================

Appendix 1 gives preliminary results concerning antisymmetric matrices of order 3.




Vector-rotation
----------------

Suppose that, in the system of general axes :math:`P\text{}{e}_{1}\text{}{e}_{2}\text{}{e}_{3}` [:ref:`fig 4.1-a <fig 4.1-a>`], the point :math:`M\text{'}` is deduced from :math:`M` by the rotation of angle :math:`\alpha` around the axis passing through :math:`P` and by the unit vector :math:`u`. Let's say:

:math:`\Psi =\alpha u`

:math:`\Psi` is called the **rotation-vector** to go from :math:`M` to :math:`M\text{'}`.

According to the Euler-Rodrigues formula [:ref:`bib6 <bib6>`] p. 186 and [:ref:`bib7 <bib7>`]:

:math:`\overrightarrow{\mathrm{PM}\text{'}}=\overrightarrow{\mathrm{PM}}+\text{sin}\alpha (u\wedge \overrightarrow{\mathrm{PM}})+(1-\mathrm{cos}\alpha )\left[u\wedge \left[u\wedge \overrightarrow{\mathrm{PM}}\right]\right]`.


.. image:: images/10000000000001050000015CEC799D7C4F27596D.png
    :width: 2.2402in
    :height: 2.7591in


.. _RefImage_10000000000001050000015CEC799D7C4F27596D.png:

**Figure 4.1-a: Representation of a finite rotation**

In general, the rotation vector of the product of two rotations **isn't** the geometric sum of the component rotation vectors. The 2D case is a particularly simple exception: the vectors-rotation, perpendicular to the plane, are added algebraically.




Rotation operator
---------------------

Given [:ref:`éq An1-3 <éq An1-3>`], the previous equation is written as:

:math:`\overrightarrow{\mathrm{PM}\text{'}}=\left[1+\mathrm{sin}\alpha \stackrel{ˆ}{u}+(1-\mathrm{cos}\alpha ){\stackrel{ˆ}{u}}^{2}\right]\overrightarrow{\mathrm{PM}}`

The expression in square brackets defines the **rotation operator** :math:`R` to go from :math:`\mathrm{PM}` to :math:`\mathrm{PM}\text{'}`:

 :math:`R=1+\text{sin}\alpha \stackrel{ˆ}{u}+(1-\text{cos}\alpha ){\stackrel{ˆ}{u}}^{2}\text{.}` **eq 4.2-1**

The following four numbers are called "Euler parameters" of rotation:

 :math:`\begin{array}{cc}{e}_{0}=\text{cos}\frac{\alpha }{2}& {e}_{1}=\text{sin}\frac{\alpha }{2}{u}_{1}\\ {e}_{2}=\text{sin}\frac{\alpha }{2}{u}_{2}& {e}_{3}=\text{sin}\frac{\alpha }{2}{u}_{3}\end{array}` **eq 4.2-2**

Of course we have:

:math:`{e}_{0}^{2}+{e}_{1}^{2}+{e}_{2}^{2}+{e}_{3}^{2}=1\text{.}`

Let's say:

:math:`e=\left\{\begin{array}{c}{e}_{1}\\ {e}_{2}\\ {e}_{3}\end{array}\right\}=\text{sin}\frac{\alpha }{2}u\text{.}`

Using the [:ref:`éq A1-4 <éq A1-4>`] relationship, you can easily put the expression [:ref:`éq 4.2-1 <éq 4.2-1>`] for :math:`R` in the form:

 :math:`R=(2{e}_{0}^{2}-1)1+2({ee}^{T}+{e}_{0}\stackrel{ˆ}{e})\text{.}` **eq 4.2-3**

On the other hand, replacing :math:`\text{sin}\alpha` and :math:`\text{cos}\alpha`, to the second member of [:ref:`éq 4.2-1 <éq 4.2-1>`], by their developments in entire series, he comes:

:math:`\begin{array}{cc}R=1& +\left[\alpha -\frac{{\alpha }^{3}}{3!}+\frac{{\alpha }^{5}}{5!}+\text{...}+{(-1)}^{p-1}\frac{{\alpha }^{\mathrm{2p}-1}}{(\mathrm{2p}-1)!}+\text{...}\right]\stackrel{ˆ}{u}\\ & +\left[\frac{{\alpha }^{2}}{2!}-\frac{{\alpha }^{4}}{4!}+\frac{{\alpha }^{6}}{6!}+\text{...}+{(-1)}^{p-1}\frac{{\alpha }^{\mathrm{2p}}}{(\mathrm{2p})!}\right]{\stackrel{ˆ}{u}}^{2}\end{array}`

or, using [:ref:`éq A1-5 <éq A1-5>`] and [:ref:`éq A1-6 <éq A1-6>`], the **exponential form** of the rotation operator:

 :math:`\begin{array}{c}R=1+\alpha \stackrel{ˆ}{u}+\frac{{(\alpha \stackrel{ˆ}{u})}^{2}}{2!}+\text{...}+\frac{{(\alpha \stackrel{ˆ}{u})}^{p}}{p!}+\text{...}\\ =\text{exp}(\alpha \stackrel{ˆ}{u})=\text{exp}(\stackrel{ˆ}{\Psi })\text{.}\end{array}` **eq 4.2-4**

It appears on [:ref:`éq 4.2-4 <éq 4.2-4>`] that when :math:`\parallel \Psi \parallel \to 0`,

 :math:`R\approx 1+\Psi \wedge \text{.}` **eq 4.2-5**

:math:`R=\text{exp}(\alpha \stackrel{ˆ}{u})` is obviously not calculated by the development [:ref:`éq 4.2-4 <éq 4.2-4>`], but by the expression [:ref:`éq 4.2-1 <éq 4.2-1>`].

Since :math:`{\stackrel{ˆ}{u}}^{T}=-\stackrel{ˆ}{u}`, transposing all the terms on the second member of [:ref:`éq 4.2-4 <éq 4.2-4>`] gives:

 :math:`{\left[\text{exp}(\stackrel{ˆ}{\Psi })\right]}^{T}=\text{exp}(-\stackrel{ˆ}{\Psi })`, i.e.: **eq 4.2-6**

:math:`{R}^{T}={R}^{-1}`

and:

 :math:`{\mathrm{RR}}^{T}=1` **eq 4.2-7**

The rotation operators, orthogonal according to the equation [:ref:`éq 4.2-7 <éq 4.2-7>`], form a group in relation to the multiplication operation - non-commutative in 3D - called **Lie group** and designated by :math:`\text{SO}(3)` (Special Orthogonal group).