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====== FieldSymmetry ======

The FieldSymmetry class represents the symmetry group of FlowFields. Specifically, the periodic in 
x,z and polynomial in y expansions inherent in FlowFields allow the follow symmetries

<latex> $ \begin{align*}
  [u,v,w](x,y,z) &\rightarrow [-u,-v,-w](x,y,z) \\
  [u,v,w](x,y,z) &\rightarrow [-u, v, w](-x,y,z) \\
  [u,v,w](x,y,z) &\rightarrow [ u, -v, w](x,-y,z) \\
  [u,v,w](x,y,z) &\rightarrow [ u, v, -w](x, y,-z) \\
  [u,v,w](x,y,z) &\rightarrow [ u, v, w](x+\ell_x, y, z+\ell_z) 
\end{align*} $ </latex>

Let G be the group generated by these symmetries. Specific choice of boundary conditions at the walls 
might constrain the symmetry group of a specific flow to a subgroup G. But for generality,the FieldSymmetry 
class can represent any symmetry in G. The FieldSymmetry class parameterizes symmetries σ ⊂ G symmetries 
as follows. Any element of the symmetry group is defined by six parameters

<latex> $ \begin{align*}
  \sigma &= (s_x, s_y, s_x, a_x, a_z, s)\\
  sx, sy, sz, s &= \pm 1\\
  ax, az &\in [-0.5, 0.5)
\end{align*} $ </latex>

with the action of σ on a velocity field u as

<latex>
  \sigma [u, v, w](x,y,z) = s (s_x u, s_y v, s_z w)(s_x x + a_x L_x, s_y y, s_z z + a_z L_z)
</latex>

In C++ code, elements of the symmetry group can be defined as follows

<code c++>
  FieldSymmetry sigma0(sx, sy, sz, ax, az, s);

  FieldSymmetry sigma1(sx, sy, sz, ax, az);  // The s argument s defaults to 1

  FieldSymmetry sigma2(sx, sy, sz);          // s defaults to 1; ax,az to 0

  FieldSymmetry tau(ax, az);                 // Pure translation: s,sx,sy,sz default to 1

  FieldSymmetry identity;                    // The identity: s defaults to 1; ax,az to 0; sx,sy,sz to 1
</code>

where sx,sy,sz are of type int and ax, az are of type Real. FieldSymmetries act on each
other and velocity fields as follows

<code c++>
  FieldSymmetry sigma4 = sigma2 * sigma1;     // group multiplication

  sigma4 *= sigma3;                           // sigma4 now equals sigma3*sigma2*sigma1 ---note order!

  FlowField u("u");                           // read velocity field u from disk

  FlowField v = sigma1 * u;                   // Make new field v = sigma1 u

  v *= sigma2;                                // v now equals sigma2 * sigma1 * u
</code>

FieldSymmetries can be saved to / read from disk...

<code c++>
  sigma1.save("sigma1");                      // saves to ASCII file sigma1.asc

  FieldSymmetry sigma7("sigma1");             // make a new symmetry element sigma7 == sigma1
</code>

...compared to each other...

<code c++>
  if (sigma1 != identity) 
    ...
  else if (sigma1 == sigma7)
    ...
</code>

For a complete description of the FieldSymmetry class, see the header file fieldsymmetry.h.
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