2dRotations.tex

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% Copyright © 2017 Peeter Joot.  All Rights Reserved.
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\index{rotation}
Plotting \cref{eqn:2dMultiplication:180}, as in
\cref{fig:rotationOfe1:rotationOfe1Fig1},
 shows that multiplication by \( i \) rotates the \R{2} basis vectors by \( \pm \pi/2 \) radians,
with the
rotation direction dependent on the order of multiplication.
\pmathImageTwoFigures
{../figures/GAelectrodynamics/\subfigdir/}
{rotationOfe1Fig1}
{rotationOfe2Fig1}
{Multiplication by \( \Be_1 \Be_2 \).}{fig:rotationOfe1:rotationOfe1Fig1}{scale=0.5}
{orientedAreas.nb}

Multiplying a polar vector representation
\begin{equation}\label{eqn:2dRotations:280}
   \Bx = \rho \lr{ \Be_1 \cos\theta + \Be_2 \sin\theta },
\end{equation}
by \( i \) shows that a \( \pi/2 \) rotation is induced.
\index{pseudoscalar}

Multiplying the vector from the right by \( i \) gives
\begin{equation}\label{eqn:2dRotations:300}
\begin{aligned}
\Bx i
&= \Bx \Be_1 \Be_2 \\
&= \rho \lr{ \Be_1 \cos\theta + \Be_2 \sin\theta } \Be_1 \Be_2 \\
&= \rho \lr{ \Be_2 \cos\theta - \Be_1 \sin\theta },
\end{aligned}
\end{equation}
a counterclockwise rotation of \( \pi/2 \) radians, and
multiplying the vector by \( i \) from the left gives
\begin{equation}\label{eqn:2dRotations:3}
\begin{aligned}
i \Bx
&= \Be_1 \Be_2 \Bx \\
&= \rho \Be_1 \Be_2 \lr{ \Be_1 \cos\theta + \Be_2 \sin\theta } \Be_1 \Be_2 \\
&= \rho \lr{ -\Be_2 \cos\theta + \Be_1 \sin\theta },
\end{aligned}
\end{equation}
a clockwise rotation by \( \pi/2 \) radians
(\cref{problem:2dRotations:1}).

The transformed vector \( \Bx' = \Bx \Be_1 \Be_2 = -\Be_1 \Be_2 \Bx \,(= \Bx i = -i \Bx) \) has been rotated in the direction that takes \( \Be_1 \) to \( \Be_2 \), as illustrated
in \cref{fig:rotationOfV:rotationOfVFig1}.
\pmathImageFigure{../figures/GAelectrodynamics/\subfigdir/}{rotationOfVFig1}{\( \pi/2\) rotation in the plane using pseudoscalar multiplication.}{fig:rotationOfV:rotationOfVFig1}{0.3}{orientedAreas.nb}

In complex number theory the complex exponential \( e^{i\theta} \) can be used as a rotation operator.
Geometric algebra puts this rotation operator into the vector algebra toolbox, by utilizing
Euler's formula
\index{Euler's formula}
\begin{equation}\label{eqn:2dRotations:1140}
e^{i\theta} = \cos\theta + i \sin\theta,
\end{equation}
valid for this pseudoscalar imaginary representation too (\cref{problem:2dRotations:Euler}).
\index{complex exponential}
By writing \( \Be_2 = \Be_1 \Be_1 \Be_2 \),
a complex exponential can be factored directly out of the polar vector representation \cref{eqn:2dRotations:280}
\begin{equation}\label{eqn:2dRotations:940}
\begin{aligned}
\Bx
&= \rho \lr{ \Be_1 \cos\theta + \Be_2 \sin\theta } \\
&= \rho \lr{ \Be_1 \cos\theta + (\Be_1 \Be_1) \Be_2 \sin\theta } \\
&= \rho \Be_1 \lr{ \cos\theta + \Be_1 \Be_2 \sin\theta } \\
&= \rho \Be_1 \lr{ \cos\theta + i \sin\theta } \\
&= \rho \Be_1 e^{i\theta}.
\end{aligned}
\end{equation}

We end up with a complex exponential multivector factor on the right.
Alternatively, since \( \Be_2 = \Be_2 \Be_1 \Be_1 \), a complex exponential can be factored out on the left
\begin{equation}\label{eqn:2dRotations:960}
\begin{aligned}
\Bx &= \rho \lr{ \Be_1 \cos\theta + \Be_2 \sin\theta } \\
&= \rho \lr{ \Be_1 \cos\theta + \Be_2 (\Be_1 \Be_1) \sin\theta } \\
&= \rho \lr{ \cos\theta - \Be_1 \Be_2 \sin\theta } \Be_1 \\
&= \rho \lr{ \cos\theta - i \sin\theta } \Be_1 \\
&= \rho e^{-i\theta} \Be_1.
\end{aligned}
\end{equation}

Left and right exponential expressions have now been found for the polar representation
\begin{equation}\label{eqn:2dRotations:1120}
\rho \lr{ \Be_1 \cos\theta + \Be_2 \sin\theta }
= \rho e^{-i\theta} \Be_1 = \rho \Be_1 e^{i\theta}.
\end{equation}

This is essentially a recipe for rotation of a vector in the x-y plane.
Such rotations are illustrated in \cref{fig:rotationOfX:rotationOfXFig1}.
\pmathImageFigure{../figures/GAelectrodynamics/\subfigdir/}{rotationOfXFig1}{Rotation in a plane.}{fig:rotationOfX:rotationOfXFig1}{0.3}{orientedAreas.nb}

This generalizes to rotations of \R{N} vectors constrained to a plane.
Given orthonormal vectors \( \Bu, \Bv \) and any vector in the plane of these two vectors (\( \Bx \in \Span\setlr{\Bu,\Bv} \)), this vector is rotated \( \theta \) radians in the direction of rotation that takes \( \Bu \) to \( \Bv \) by
\begin{equation}\label{eqn:2dRotations:1160}
\Bx' = \Bx e^{ \Bu \Bv \theta } = e^{-\Bu \Bv \theta} \Bx.
\end{equation}

The sense of rotation for the rotation \( e^{ \Bu \Bv \theta} \) is opposite that of \( e^{\Bv \Bu \theta} \), which provides a first hint that bivectors can be characterized as having an orientation, somewhat akin to thinking of a vector as having a head and a tail.

\makeexample{Velocity and acceleration in polar coordinates.}{example:2dRotations:1180}{
Complex exponential representations of rotations work very nicely for describing vectors in polar coordinates.
A radial vector can be written as
\begin{equation}\label{eqn:2dRotations:1200}
\Br = r \rcap,
\end{equation}
as illustrated in \cref{fig:radialVectorCylindrical:radialVectorCylindricalFig1}.
The polar representation of the radial and azimuthal unit vector are simply
\pmathImageFigure{../figures/GAelectrodynamics/\subfigdir/}{radialVectorCylindricalFig1}{Radial vector in polar coordinates.}{fig:radialVectorCylindrical:radialVectorCylindricalFig1}{0.2}{radialVectorCylindricalFig1.nb}
\begin{equation}\label{eqn:2dRotations:1220}
\begin{aligned}
\rcap &= \Be_1 e^{i\theta} =
\Be_1 \lr{ \cos\theta + \Be_1 \Be_2 \sin\theta } = \Be_1 \cos\theta + \Be_2 \sin\theta \\
\thetacap &= \Be_2 e^{i\theta} =
\Be_2 \lr{ \cos\theta + \Be_1 \Be_2 \sin\theta } = \Be_2 \cos\theta - \Be_1 \sin\theta,
\end{aligned}
\end{equation}
where \( i = \Be_{12} \) is the unit bivector for the x-y plane.  We can easily show that these unit vectors are orthogonal
\begin{equation}\label{eqn:2dRotations:1340}
\begin{aligned}
\rcap \thetacap
&= \lr{ \Be_1 e^{i \theta}} \lr{ e^{-i\theta} \Be_2} \\
&= \Be_1 \cancel{e^{i \theta} e^{-i\theta}} \Be_2 \\
&= \Be_1 \Be_2.
\end{aligned}
\end{equation}
By \cref{thm:multiplication:anticommutationNormal}, since the product of \( \rcap \thetacap \) is a bivector,
\( \rcap \) is orthogonal to \( \thetacap \).

We can find the
velocity and acceleration by taking time derivatives
\begin{equation}\label{eqn:2dRotations:1240}
\begin{aligned}
\Bv &= r' \rcap + r \rcap' \\
\Ba &= r'' \rcap + 2 r' \rcap' + r \rcap'',
\end{aligned}
\end{equation}
but to make these more meaningful want to evaluate the \( \rcap, \thetacap \) derivatives explicitly.  Those are
\begin{equation}\label{eqn:2dRotations:1260}
\begin{aligned}
\rcap' &= \lr{ \Be_1 e^{i \theta} }' =
\mathLabelBox[ labelstyle={yshift=1.2em}, linestyle={} ]
{
\Be_1 i
}
{
\(\Be_1 (\Be_1 \Be_2) = (\Be_1 \Be_1) \Be_2 \)
}
 e^{i\theta} \theta' = \Be_2 e^{i\theta} \theta' = \thetacap \omega \\
\thetacap' &= \lr{ \Be_2 e^{i \theta} }' =
\mathLabelBox[ labelstyle={below of=m\themathLableNode, below of=m\themathLableNode} ]
{
\Be_2 i
}
{
\(\Be_2 \Be_1 \Be_2
=
(-\Be_1 \Be_2) \Be_2\)
}
 e^{i\theta} \theta' = -\Be_1 e^{i\theta} \theta' = -\rcap \omega,
\end{aligned}
\end{equation}
where \( \omega = d\theta/dt \), and primes denote time derivatives.  The velocity and acceleration vectors can now be written explicitly in terms of radial and azimuthal components.  The velocity is
\begin{equation}\label{eqn:2dRotations:1280}
\Bv = r' \rcap + r \omega \thetacap,
\end{equation}
and the acceleration is
\begin{equation}\label{eqn:2dRotations:1300}
\begin{aligned}
\Ba
&= r'' \rcap + 2 r' \omega \thetacap + r (\omega \thetacap)' \\
&= r'' \rcap + 2 r' \omega \thetacap + r \omega' \thetacap - r \omega^2 \rcap,
\end{aligned}
\end{equation}
or
\begin{equation}\label{eqn:2dRotations:1320}
\Ba
= \rcap \lr{ r'' - r \omega^2 }
+ \inv{r} \thetacap \lr{ r^2 \omega }'.
\end{equation}

Using \cref{eqn:2dRotations:1220}, we also have the option of factoring out the rotation operation from the position vector or any of its derivatives
\begin{equation}\label{eqn:2dRotations:1360}
\begin{aligned}
\Br &= \lr{ r \Be_1 } e^{i \theta } \\
\Bv &= \lr{ r' \Be_1 + r \omega \Be_2 } e^{i \theta } \\
\Ba &= \lr{ \lr{ r'' - r \omega^2 } \Be_1 + \inv{r} \lr{ r^2 \omega }' \Be_2 } e^{i\theta}.
\end{aligned}
\end{equation}

In particular,
for uniform circular motion, each of the position, velocity and acceleration vectors can be represented by a vector that is fixed in space, subsequently rotated by an angle \( \theta \).
} % example
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