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====== symmetryop ======

''symmetryop'': apply a symmetry operation to a given field

<code>
options :
  -sn       --sn       <int>       default == 0     apply s1, s2, or s3 symmetry (0 => use sx,sy,sz,ax,az,a)
  -sx       --x-sign                                change u,x sign
  -sy       --y-sign                                change v,y sign
  -sz       --z-sign                                change w,z sign
  -ax       --axshift  <real>      default == 0     translate x by ax*Lx
  -az       --azshift  <real>      default == 0     translate z by az*Lz
  -anti     --anti                                  antisymmetry instead of symmetry
  -v        --verbose                               print out some diagnostics
  <flowfield>  (trailing arg 2)                     input field
  <flowfield>  (trailing arg 1)                     output field
</code>

The [[:docs:math:symmetry|symmetries of 3D fields]] on x,z periodic, y bounded domains with 
Dirichlet or Neumann conditions at the walls can be parameterized 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>

where s,sx,sy,sz, and are +/-1 and ax,az are real. The options to symmetryop set the values of 
these parameters, and the resulting symmetry is applied to the input field and saved to disk as 
the output field. The -sx,-sy,-sz options work as flags: using -sx sets sx = -1; otherwise sx = 1.
Likewise the -anti option sets s = -1.

For convenience the -sn option allows application of a few predefined symmetries that
figure prominently in our research, namely

<latex> $ \begin{align*}
s_1 [u,v,w](u,v,w) &= [u, v, -w](x + L_x/2 , \, y, \, -z) \\
s_2 [u,v,w](u,v,w) &= [-u, -v, w](-x + L_x/2 , \, -y, \, z+L_z/2) \\
s_3 [u,v,w](u,v,w) &= [-u, -v, -w](-x , \, -y, \, -z+L_z/2)
\end{align*} $ </latex>

====== Usage examples ======

===== Translation =====

Translate the field g.ff by Lx/2 in x and save it in h.ff
<code>
  symmetryop -ax 0.5 g h
</code>

===== General =====

Apply s1 = [u,v,w] %%->%% [-u, -v, w](-x + L_x/2 , -y,  z+L_z/2) to g.ff and save it in h.ff

<code>
  symmetryop -sx -sy -ax 0.5 -az 0.5 g h
</code>

===== Convenience =====

Apply s1 to g.ff and save it in h.ff, using the convenience -sn option

<code>
  symmetryop -sn 1  g h
</code>
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