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Falling cat. Optimal control.
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\begin{document}
\input epsf
\title{Optimal rotations of deformable bodies \\ and orbits in magnetic fields}%
\author{ J.E. Avron,
O. Gat, O. Kenneth and U. Sivan}
\address{Department of Physics, Technion, Haifa 32000, Israel
}%
\email{avron@physics.technion.ac.il}

\date{\today}%
\pacs{02.40.-k,07.10.Cm,83.50.-v}
% ----------------------------------------------------------------

% ----------------------------------------------------------------

\begin{abstract}
Deformations can induce rotation with zero angular momentum where
dissipation is a natural ``cost function''. This gives rise to an
optimization problem of finding the most effective rotation with
zero angular momentum. For certain plastic and viscous media in
two dimensions the optimal path is the orbit of a charged particle
on a surface of constant negative curvature with magnetic field
whose total flux is half a quantum unit.
\end{abstract}
\maketitle

Rotations with zero angular momentum are intriguing. The most
celebrated phenomenon of this kind is the rotation of a falling
cat. A mechanical model \cite{ref:kane} replacing the cat by two
rigid bodies that can rotate relative to each other, has been
extensively studied, see \cite{ref:montgomery,ref:marsden} and
references therein. Here we address rotations with zero angular
momentum under linear deformations. Our motivation comes from
nano-mechanics: Imagine an elastic or plastic material with its
own energy source, and ask what is the most efficient way of
turning it through an appropriate sequence of autonomous
deformations without external torque.

Deformations can generate rotations because order matters: A cycle
of deformations will, in general, result in a rotation. The ratio
between the rotation and the (infinitesimal) area of the controls
can be interpreted as curvature \cite{ref:novikov,ref:shapere}.
Consequently, small cycles
%of the controls
are ineffective since a cycle of length $\eps$ in the controls
leads to a rotation of order $\eps^2$. The search for optimal
paths forces one to mind deformations that are not small.

The problem we address has three parts. The first part is to
determine the rotation for a given path of the controls. We solve
this problem for general linear deformations. In two dimensions
this leads to curvature on the space of the controls which is
exponentially localized. If one thinks of the curvature as a
magnetic field, then the total magnetic flux is that of half a
unit of quantum flux. The second part is to set up a model for the
cost function. We choose the cost function to be a measure of
dissipation and focus on two settings, one where the dissipation
is rate independent, as is the case in certain plastics, and the
other where it is rate dependent as in liquids. Both cost
functions lead to the same metric on the space of deformations.
The third part is to pose and solve the problem of finding the
path of minimal dissipation for a given rotation. In two
dimensions and for either model of dissipation, the problem maps
to finding the shortest path that starts at a given point and
encloses a given amount magnetic flux. Optimal paths tend to
linger near the circle in the space of controls where the ratio of
eigenvalues of the quadrupole moment of the body is
$(\phi+\sqrt{\phi^2-1})^2$. $\phi=(1+\sqrt{5})/2$ is the golden
ratio.

Deformations generate rotations {\em because} angular momentum is
conserved \cite{ref:shapere}. Consider a collection of point
masses $m_\alpha$ at positions $x_\alpha$. Internal forces may
deform the body, but there are no external forces. Suppose that
the center of mass of the body is at rest at the origin and that
the body has zero angular momentum. The total angular momentum,
$L_{ij}$, must then stay zero for all times.

A linear deformation is represented by a matrix $M$ that sends
$x\to M x$.  The $i,j$ component of the angular momentum is
\begin{equation}L_{ij}
=\Tr\,(\dot M Q M^t\, \ell_{ij}),%\nonumber
\end{equation}
where $Q$ is the quadrupole moment of the body
\begin{equation}\label{eq:qmoment}
Q_{ij}=\sum_\alpha m_\alpha (x_\alpha)_i(x_\alpha)_j
\end{equation}  and  $\ell_{ij}, \ i<j$ are the $n\choose 2$
generators of rotations in $n$ dimensions, {\it i.e.}
$(x,\ell_{ij}y)=x_iy_j-x_jy_i$. Since $\ell_{ij}$ span the
anti-symmetric matrices, the set of  $n\choose 2$ equations
$L_{ij}=0$  imply that the matrix $(dM)QM^t$ is symmetric.
%%%%%%%%%%%%%%%%%%%%

Two immediate consequences of this symmetry are:
\begin{itemize}
\item Isotropic bodies:  With $Q=1$, and $M$ close to the identity,
the symmetry of $(dM)QM^t$ reduces to $dM$ being symmetric: the
linear transformation must be a strain \cite{ref:landau}.
%%%%%%%%%%%%%%%%%%%%%
\item Pointers: Pointers are  bodies with large aspect ratios,
such as needles and discs. In the limit of infinite aspect ratio,
$Q$ may be identified with a projection where $\dim Q$ is the
dimension of the pointer. With $M$ near the identity, the symmetry
of $ (dM) Q$ implies that $ (1-Q)(dM) Q=(1-Q) Q (dM)^t = 0. $
Since $Q$ does not acquire a component in the normal direction,
$1-Q$, under $dM$ a pointer keeps its orientation.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%

We now derive the fundamental relation between the response
(rotation) and the controls (deformations). To this end we use the polar
decomposition $M=RS$ with $R$ a rotation and $S$ a positive matrix.
Assuming $S$ positive is a choice of a gauge which makes the
representation unique with $S=\sqrt{M^tM}$. The symmetry of
$(dM)QM^t$ gives
 \begin{equation}\label{eq:CAR0}\{
\A,SQS\} =SQ(dS)-(dS)QS, \quad \A=R^{-1}\dbar R.\end{equation}
Eq.~(\ref{eq:CAR0})  determines the  differential rotation, $\A$,
in terms of the variation of the controls $dS$.  The symbol
$\dbar$  stresses that the differential will not, in general,
integrate to a function on the space of deformations.
Geometrically, the differential rotation is the connection 1-form
which fixes a notion of parallel transport.

Eq.~(\ref{eq:CAR0}) can be interpreted in terms of a variational
principle: The motion is such that the kinetic energy is minimal
for a given deformation. To see this, let $M=1$ and
$\dot M =\dot R +\dot S $
with $\dot R$ antisymmetric ({\it i.e.} a rotation) and $\dot S$
symmetric ({\it i.e.} a strain). The kinetic energy is
\begin{equation}\label{eq:kinetic}
E=\frac 1 2 \Tr\,\big( \dot M Q\dot M^t\big) =\frac 1
2\,\Tr\,\big( (\dot R+\dot S) Q (-\dot R+\dot S)\big)\; .
\end{equation}
Minimizing with respect to $\dot R$ gives
\begin{equation}
0=\delta E= \frac 1 2 \Tr\,\big( \delta \dot R \big(-\{Q,\dot
R\}+[ Q,\dot S]\big)\; .
\end{equation}
The trace is of a product of antisymmetric matrices, and its
vanishing for an arbitrary antisymmetric $\delta \dot R$ implies
$\{\dot R,Q\}=[ Q,\dot S]$ which is Eq.~(\ref{eq:CAR0}) for $M=1$.

One readily sees that if $\A$ is the solution of Eq.~(\ref{eq:CAR0})
 given $S$ and $Q$, then it is also a solution for $\lambda
S$ and $Q$ for $\lambda$ a scalar valued function. Hence scaling
does not drive rotations and we may restrict ourselves to volume
(or area) preserving deformations with $\det S=1$.

Since any $Q$ is obtainable by a linear deformation of the
identity, we may assume without loss of generality that $Q=1$.
Eq.~(\ref{eq:CAR0}) reduces to
 \begin{equation}\label{eq:CAR}
\{\A,S^2\}=[S,dS]
 \; .
\end{equation}
 Eq.~(\ref{eq:CAR})  is
conveniently solved in a basis where $S$ is diagonal. Let $s_j$
denote the eigenvalues of $S$ then
\begin{equation}\label{eq:connection}
\A_{ij}=\frac {s_i-s_j} {s_i^2+s_j^2}\,(dS)_{ij} \; .
\end{equation}
Normally, the curvature $F$ is more interesting than the
connection. It is defined by $F=d\A+\A\wedge \A$. Calculation
gives
\begin{equation}\label{eq:curvature}
F_{ij}=2\sum_k \frac{s_i\,s_j\, (s_i+s_k)(s_j+s_k)}{(s_i^2+s_j^2)
(s_j^2+s_k^2)(s_i^2+s_k^2)}\, (dS)_{ik}\wedge (dS)_{kj} \; .
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The situation is particularly simple in two dimensions. We use the
Pauli matrices $\sigma_x, \sigma_z, i\sigma_y$ and the identity as
basis for the real $2\times 2$ matrices. Positive unimodular
matrices correspond to points on the hyperboloid $t^2-x^2-z^2=1,\;
t>0$ shown in Fig.~\ref{fig:hyper} where
\begin{equation}\label{eq:2-dS}
S^2(t,x,z)=t+ x\sigma_x+ z\sigma_z\,.
\end{equation}
Natural coordinates are  $(\tau,\theta)$
\begin{equation}\label{eq:coordinates}
(t,x,z)=(\cosh\tau,\sinh\tau\,\cos\theta,\sinh\tau\,\sin\theta)\;,
\end{equation}
with $S=\cosh (\tau/2)+
\sinh(\tau/2)(\sigma_x\cos\theta+\sigma_z\sin\theta)$. Let
$R^t\dbar R=-i\,\sigma_y\,\dbar\varphi$.
Eqs.~(\ref{eq:connection}) and (\ref{eq:curvature}) give
\begin{equation}\label{eq:CAR2}
\dbar \varphi=a(\tau) \sinh\tau\, d\theta,\quad
a(\tau)=\frac{\cosh\tau-1}{\sinh 2\tau},
\end{equation}
and
\begin{equation}\label{eq:curv2}
 F(\tau)= f(\tau) \sinh\tau\, d\tau\wedge d\theta, \quad
 f(\tau)=\frac{1}{2\cosh^2 \tau}.
\end{equation}

\begin{figure}[h]
\includegraphics[width=4cm]{hyper.eps}
\caption{Positive, area preserving linear transformations may be
identified with the hyperboloid of revolution. The induced metric
obtained by embedding the hyperboloid in Minkowski space gives it
the structure of the Lobachevsky plane with constant negative
curvature.}\label{fig:hyper}
\end{figure}

The total curvature is easily computed from Eq.~(\ref{eq:CAR2})
and one finds $\int F =\pi$. This, together with the positivity of
$f$, implies that in any single closed cycle ({\it i.e.} one
without self intersections), the angle of rotation is at most
$\pi$. When interpreted as magnetic flux, $\pi$ corresponds to
half a unit of quantum flux \cite{remark}.

%%%%%%%%%%%%%%%%%%%%

The cost function must include some measure of dissipation. For,
without dissipation energy is a function of the controls and no
change in energy is associated with a closed loop. We consider two
models of dissipation in isotropic media. Both lead to the same
metric on the space of controls, namely
\begin{equation}\label{eq:inv-metric} (d\ell)^2=
\,\Tr\left(S^{-2}\,d(S^2)\otimes S^{-2}d(S^2)\right)\ .
\end{equation}
The metric is
distinguished by symmetry: It is invariant under congruence, $g\to
AgA^t$, for an arbitrary invertible matrix $A$.

Consider a medium with viscosity tensor $\eta$. The power due to
dissipation is
\begin{equation}\label{eq:viscosity-dissipation}
P=  \frac 1 2\,\sum \eta^{ijkl}\dot u_{ij}\dot u_{kl}\; ,
\end{equation}
where $u$ is the strain tensor and $\dot u$ the strain rate. $S$
is related to $u$ by  $2u=S^2-1$. To see this recall that the
strain is {defined} as the change in the distance between
two neighboring points caused by a deformation. If the deformation
is described by a metric $g$ then $g=1+2u$, where
$1$ is the metric associated with the undeformed reference
system~\cite{ref:landau}.
When considering {\em linear} deformation described by a symmetric
matrix $S$, the resulting metric is $g=S^2$ (regarding the
covariant components of $g$ as the elements of a positive matrix).
This establishes the claimed connection between $u$ and $S$.

The space of (symmetric) 4-th rank isotropic tensors
is two dimensional and spanned by the two viscosity coefficients
$\eta$ and $\eta'$
 \cite{ref:landau}
\begin{equation}\label{eq:viscosity-scalar}
\eta^{ijkl}=\eta g^{ik}g^{jl}+\eta' g^{ij}g^{kl}\ .
\end{equation}
We therefore find that in an isotropic medium
\begin{equation}\label{eq:dissipation}
P= 2\eta\, Tr \left(g^{-1} \dot g g^{-1}\dot g\right)   +2\eta'\,
Tr\,\left( g^{-1}\dot g\right)Tr\,\left( g^{-1}\dot g\right)
\end{equation}
%%%%%
For volume preserving transformations the term multiplying $\eta'$
vanishes and one is left with the first term alone. By choosing
the unit of time appropriately one can take  $\eta=1$. This leads
to the metric of Eq.~(\ref{eq:inv-metric}).

The dissipation in certain plastic materials can be rate
independent. This is the continuum mechanics analog of the
dissipation due to friction when one body slides on another
\cite{ref:ori}. In plastics, rate independence  is a consequence
of the L\'evy-Mise constitutive relation: $s\,
\delta\lambda=\delta u$, where $s$ is the stress and $\delta
\lambda$ a scalar valued function \cite{ref:hill}. The
constitutive relation is formally the same as for fluids
\cite{ref:hill}, and by isotropy, the dissipation must be a
function of $d\ell$ of Eq.~(\ref{eq:inv-metric}). If the material
is memoryless, dissipation is additive with respect to
concatenating paths and must be proportional to $d\ell$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In two dimensions, with the parametrization of
Eq.~(\ref{eq:2-dS}), the metric Eq.~(\ref{eq:inv-metric}) gives
\begin{equation}\label{eq:metric-2d}
(d\ell)^2=
(dx)^2+(dz)^2-(dt)^2=(d\tau)^2+\sinh^2\tau(d\theta)^2\,.
\end{equation}
This is the metric on the hyperboloid induced from Minkowski
space. It gives the hyperboloid the geometry of the pseudo-sphere
i.e. it makes it into a surface of constant negative curvature
$-1$ \cite{ref:novikov}.

The metric enables us to assign a scalar to the curvature 2-form
$F$ of Eq.~(\ref{eq:curv2}) as the ratio between $F$ and the area
form of the pseudo-sphere, $ \sinh\tau\, d\tau\wedge d\theta$.
This ratio is $f(\tau)$ and is plotted in Fig. \ref{fig:curv},
along with $a(\tau)$ which serves as the $\theta$ component of a
vector potential for $f$. The curvature is everywhere positive and
it is concentrated  near the origin, $\tau=0$. It decays
exponentially with $\tau$. This means that large deformations are
ineffective. We already know that small deformations are
ineffective. This bring us to the optimization problem.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\hbox{
\includegraphics[width=4.2cm]{curvature.eps}
\includegraphics[width=4.2cm]{connection.eps}
} \caption{The scalar curvature, $f(\tau)$, which can be
interpreted as a magnetic field (left) and the $\theta$ component
of the vector potential $a(\tau)$. $f$ is exponentially localized
near the origin of the control space $(\tau,\theta)$, while $a$
has a maximum on the circle $\cosh\tau=\phi$, $\phi$ the golden
ratio. $a^2/2$ appears also as a potential in the effective
1-dimensional dynamics of the optimization problem.}
\label{fig:curv}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The control problem is to find a closed path $\gamma$ in the space
of deformations, starting at $S_0=\sqrt Q_0$, ($Q_0$ is the
initial quadrupole),  which rotates the quadrupole by $\Phi$
radians, with minimal dissipation. If the dissipation is rate
dependent, one adds a constraint   that the time of traversal is
$1$.

For viscous media $\gamma=\big(\tau(t),\theta(t)\big)$ is then the
solution of the variational problem
\begin{equation}\label{eq:variation}
{\delta}\int_0^1\left( \frac{ \dot\tau^2 +\dot\theta^2
\sinh^2\tau}2\,-\lambda a(\tau)\sinh(\tau)\dot\theta\right) dt=0
\end{equation}
where $\lambda$ is a Lagrange multiplier and
$\gamma(0)=\gamma(1)=S_0=S(\tau_0,\theta_0)$. This is the
(variation of the) action of a classical particle with charge
$\lambda$ and unit mass moving on the hyperbolic plane (i.e. the
pseudo-sphere) in the presence of a magnetic field $f(\tau)$ given
in Eq.~(\ref{eq:curv2}).

Since motion in a magnetic field conserves kinetic energy the
particle moves at constant speed and the dissipation, $\frac 1 2
|\gamma|^2$, depends only on the length of the path. The
variational problem can therefore be cast in purely geometric
terms: Find the shortest closed path starting at a given point
which encloses a given amount of magnetic flux. The shortest path
is evidently also the solution in the case that the dissipation is
rate independent.
%%%%%%%%%


%%%%%%%%%%%%%%%%%%
Consider the family of isospectral deformations $\tau=const$ which
keep the the eigenvalues of $S$ (or $Q$) constant while rotating
its eigenvectors. We call these ``stirring'', see
Fig.~\ref{fig:chain}.
 One can not stir an isotropic
body, since its eigenvectors don't have well defined directions.

Among the stirring cycles there is an optimal one which maximizes
the rotation per unit length. From
Eqs.~(\ref{eq:CAR2},\ref{eq:metric-2d}) the flux to length ratio
for stirring cycles is $a(\tau)$ of Eq.~(\ref{eq:CAR2}). The
function $a$ takes its maximum at
$\cosh\tau_s=\phi,\tau_s\approx1.061$, see Fig.~\ref{fig:curv}.
Every cycle of the controls rotates by
$\Phi_s=(2-\phi)\pi\approx.382\pi$ radians, somewhat less than a
quarter turn.

To make full use of the optimal stirring cycle the initial
conditions must be right. This is the case for a quadrupole with
$Q=S(2\tau_s,\theta)$, $\theta$ arbitrary. With other initial
conditions and for large angles of rotations, the optimal paths
approach the optimal stirring cycle, linger near it, eventually
returning to the initial point. This is shown in
Fig.~\ref{fig:pi-cat} for a half turn of an isotropic body.

\begin{figure}[h]
\hbox{\includegraphics[width=4cm]{cat1.eps}
\includegraphics[width=4cm]{cats5rh.eps}}
\caption{A reference shape (left), assumed to have $Q=1$, and four
instances from the optimal stirring cycle (right).
The configurations on the right are ordered clockwise with
increasing $\theta$, starting with the top left. Both the first and the
last correspond to $\theta=0$ and are therefore related by a pure
rotation.}
\label{fig:chain}
\end{figure}


\begin{figure}[h]
\includegraphics[width=5cm]{pi_catv3.eps}
\caption{The optimal path in the space of shapes
for a rotation by $\pi$ of a body with initial
quadrupole moment $Q=1$. Polar coordinates $(\tau,\theta)$ are
used to parametrize the plane. The
path reaches exponentially close to the optimal stirring cycle at $\cosh\tau=\phi$,
and winds around it twice before returning to the initial configuration.
The optimal stirring cycle is not distinguishable from the envelope of the orbit
in the scale of this figure.}
\label{fig:pi-cat}
\end{figure}

Since the magnetic field $F(\tau)$ is rotationally invariant, the
angular momentum
\begin{equation}\label{eq:angular-mom}
J=\sinh\tau\left(\dot\theta\sinh\tau - \lambda a(\tau)\right)
\end{equation}
is conserved. Conservation of energy gives
\begin{equation}\label{eq:radial}
0\le\dot\tau^2 =2E- \left(\frac J
{\sinh\tau}+\lambda{a(\tau)}\right)^2,
\end{equation}
which reduces the problem to quadrature. The three conditions,
$\tau(1)=\tau(0)$, $\theta(1)\equiv\theta(0)\,{\rm mod}\ 2\pi$,
and the constraint of enclosed flux $\Phi$, determine the three
parameters $E$, $J$ and $\lambda$.

The initial condition $\tau(0)=0$ is special: By Eq.~(\ref{eq:angular-mom})
the angular momentum $J$ is forced to have the value 0. In turn, there is
also one less condition to satisfy, as the value of $\theta$ when
$\tau=0$ is meaningless. It then follows from Eqs.~(\ref{eq:angular-mom})
and~(\ref{eq:radial}) that the optimal orbit, $\tau(\theta)$, depends
only on the ratio $E/\lambda^2$. Rescaling time properly, we can achieve
$\lambda=1$ by relaxing the constraint that the time to complete
a cycle should be 1.

The key equation controlling the dynamics is Eq.~(\ref{eq:radial}), which
describes effective one-dimensional motion in the potential $a^2(\tau)/2$.
As shown above, $a(\tau)$ has a maximum at $\tau_s$. Therefore, closed orbits
which correspond to optimal paths have energy values $E<a(\tau_s)^2/2$. The
motion is quite simple: Trajectories leave the origin with a positive $\dot\tau$,
reach the turning point $\tau_t=a^{-1}(\sqrt{2E})$, and return symmetrically to
the origin, thereby completing a cycle. The flux accumulated during a cycle
is
\begin{equation}
\Phi=\int_0^{t_{\hbox{\tiny cycle}}}a(\tau)\sinh(\tau)\dot\theta dt\ ,
\end{equation}
where $\dot\theta$ is obtained from Eq.~(\ref{eq:angular-mom}).

Increasingly longer orbits are obtained when $E$ approaches the
separatrix energy $a(\tau_s)^2/2$. The orbit corresponding to the
unstable equilibrium point at $\tau=\tau_s$ is the optimal
stirring cycle, which is a $J=0$ orbit. The flux accumulated
during a complete turn in parameter space, where $\theta$
increases by $2\pi$, is hence bounded by $\Phi_s$. Large values
$\Phi$ require many turns during which the orbit approaches
exponentially the optimal stirring cycle, see
Fig.~\ref{fig:pi-cat}. A circular disc of playdough would
therefore rotate by $\pi$, with no angular momentum, in about
three cycles.





%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\bf Acknowledgment:} We thank Amos Ori for pointing out that
dissipation in plastic materials is geometric. This work is
supported by the Technion fund for promotion of research and by
the EU grant HPRN-CT-2002-00277.
\begin{thebibliography}{99}

\bibitem {ref:kane} T.R. Kane and M.P.Scher, Int. J. Solid
Structures {\bf 5}, 663-670 (1969).
\bibitem{ref:montgomery} R. Montgomery, Fields Istitute Comm. {\bf
1}, 193 (1993); M. Fecko, J. Math. Phys. {\bf36}, 6709 (1995).

\bibitem {ref:marsden} J. Marsden, Motion control
and geometry, Proceeding of a Symposium, National Academy of
Science 2003

\bibitem{ref:novikov} B.A. Dubrovin, A.T. Fomenko, S.P. Novikov ;
{\it Modern geometry--methods and applications } translated by
Robert G. Burns, Springer (1992)
\bibitem{ref:shapere} F. Wilczek and A. Shapere, {\it
Geometric Phases in Physics} , World Scientific, Singapore,
(1989).
%\bibitem{ref:berry} M. V. Berry, Proc. Roy. Soc. (London),
%{\bf 392}, 45 (1984)
\bibitem{ref:landau} L.D. Landau and I.M.  Lifshitz, {\it Theory
of elasticity}, Pergamon
\bibitem{remark}
It is noteworthy that the total curvature of the hyperboloid is
$\pi$ and not an integer multiple of $2\pi$ as one would expect
from Gauss Bonnet Chern theorem. The reason is the restriction to
the hyperboloid with $t>0$. If both signs of $t$ are taken, the
two hyperboloids can be glued at infinity to give a sphere with
total curvature $2\pi$.

\bibitem{ref:hill} R. Hill, {\it The mathematical theory of
Plasticity}, Oxford, (1950).

\bibitem{ref:ori}We are indebted to Amos Ori for this observation.
\end{thebibliography}
\end{document}
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	5 -1 roll
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/Mimage {
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1.000000 -1.000000 scale
0.000000 0.000000 translate
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1.000000 -1.000000 scale
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