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<div class="section" id="manifolds-surfaces">
<h1>Manifolds (surfaces)</h1>
<p><strong>Overview:</strong></p>
<p>This doc page is not about a LAMMPS input script command, but about
manifolds, which are generalized surfaces, as defined and used by the
USER-MANIFOLD package, to track particle motion on the manifolds. See
the src/USER-MANIFOLD/README file for more details about the package
and its commands.</p>
<p>Below is a list of currently supported manifolds by the USER-MANIFOLD
package, their parameters and a short description of them. The
parameters listed here are in the same order as they should be passed
to the relevant fixes.</p>
<table border="1" class="docutils">
<colgroup>
<col width="5%" />
<col width="5%" />
<col width="45%" />
<col width="44%" />
<col width="1%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td><em>manifold</em></td>
<td><em>parameters</em></td>
<td><em>equation</em></td>
<td><em>description</em></td>
<td>&nbsp;</td>
</tr>
<tr class="row-even"><td>cylinder</td>
<td>R</td>
<td>x^2 + y^2 - R^2 = 0</td>
<td>Cylinder along z-axis, axis going through (0,0,0)</td>
<td>&nbsp;</td>
</tr>
<tr class="row-odd"><td>cylinder_dent</td>
<td>R l a</td>
<td>x^2 + y^2 - r(z)^2 = 0, r(x) = R if | z | &gt; l, r(z) = R - a*(1 + cos(z/l))/2 otherwise</td>
<td>A cylinder with a dent around z = 0</td>
<td>&nbsp;</td>
</tr>
<tr class="row-even"><td>dumbbell</td>
<td>a A B c</td>
<td>-( x^2 + y^2 ) * (a^2 - z^2/c^2) * ( 1 + (A*sin(B*z^2))^4) = 0</td>
<td>A dumbbell</td>
<td>&nbsp;</td>
</tr>
<tr class="row-odd"><td>ellipsoid</td>
<td>a b c</td>
<td>(x/a)^2 + (y/b)^2 + (z/c)^2 = 0</td>
<td>An ellipsoid</td>
<td>&nbsp;</td>
</tr>
<tr class="row-even"><td>plane</td>
<td>a b c x0 y0 z0</td>
<td>a*(x-x0) + b*(y-y0) + c*(z-z0) = 0</td>
<td>A plane with normal (a,b,c) going through point (x0,y0,z0)</td>
<td>&nbsp;</td>
</tr>
<tr class="row-odd"><td>plane_wiggle</td>
<td>a w</td>
<td>z - a*sin(w*x) = 0</td>
<td>A plane with a sinusoidal modulation on z along x.</td>
<td>&nbsp;</td>
</tr>
<tr class="row-even"><td>sphere</td>
<td>R</td>
<td>x^2 + y^2 + z^2 - R^2 = 0</td>
<td>A sphere of radius R</td>
<td>&nbsp;</td>
</tr>
<tr class="row-odd"><td>supersphere</td>
<td>R q</td>
<td>| x |^q + | y |^q + | z |^q - R^q = 0</td>
<td>A supersphere of hyperradius R</td>
<td>&nbsp;</td>
</tr>
<tr class="row-even"><td>spine</td>
<td>a, A, B, B2, c</td>
<td>-(x^2 + y^2)*(a^2 - z^2/f(z)^2)*(1 + (A*sin(g(z)*z^2))^4), f(z) = c if z &gt; 0, 1 otherwise; g(z) = B if z &gt; 0, B2 otherwise</td>
<td>An approximation to a dendtritic spine</td>
<td>&nbsp;</td>
</tr>
<tr class="row-odd"><td>spine_two</td>
<td>a, A, B, B2, c</td>
<td>-(x^2 + y^2)*(a^2 - z^2/f(z)^2)*(1 + (A*sin(g(z)*z^2))^2), f(z) = c if z &gt; 0, 1 otherwise; g(z) = B if z &gt; 0, B2 otherwise</td>
<td>Another approximation to a dendtritic spine</td>
<td>&nbsp;</td>
</tr>
<tr class="row-even"><td>thylakoid</td>
<td>wB LB lB</td>
<td>Various, see <a class="reference internal" href="#paquay1"><span class="std std-ref">(Paquay)</span></a></td>
<td>A model grana thylakoid consisting of two block-like compartments connected by a bridge of width wB, length LB and taper length lB</td>
<td>&nbsp;</td>
</tr>
<tr class="row-odd"><td>torus</td>
<td>R r</td>
<td>(R - sqrt( x^2 + y^2 ) )^2 + z^2 - r^2</td>
<td>A torus with large radius R and small radius r, centered on (0,0,0)</td>
<td>&nbsp;</td>
</tr>
</tbody>
</table>
<p id="paquay1"><strong>(Paquay)</strong> Paquay and Kusters, Biophys. J., 110, 6, (2016).
preprint available at <a class="reference external" href="http://arxiv.org/abs/1411.3019/">arXiv:1411.3019</a>.</p>
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