Manifolds (surfaces)
====================

**Overview:**

This page is not about a LAMMPS input script command, but about
manifolds, which are generalized surfaces, as defined and used by the
MANIFOLD package, to track particle motion on the manifolds.  See
the src/MANIFOLD/README file for more details about the package
and its commands.

Below is a list of currently supported manifolds by the 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.

+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| *manifold*     | *parameters*   | *equation*                                                                                                                             | *description*                                                                                                                      |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| cylinder       | R              | x\^2 + y\^2 - R\^2 = 0                                                                                                                 | Cylinder along z-axis, axis going through (0,0,0)                                                                                  |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| cylinder_dent  | R l a          | x\^2 + y\^2 - r(z)\^2 = 0, r(x) = R if \| z \| > l, r(z) = R - a\*(1 + cos(z/l))/2 otherwise                                           | A cylinder with a dent around z = 0                                                                                                |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| dumbbell       | a A B c        | -( x\^2 + y\^2 ) + (a\^2 - z\^2/c\^2) \* ( 1 + (A\*sin(B\*z\^2))\^4) = 0                                                               | A dumbbell                                                                                                                         |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| ellipsoid      | a  b c         | (x/a)\^2 + (y/b)\^2 + (z/c)\^2 = 0                                                                                                     | An ellipsoid                                                                                                                       |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| gaussian_bump  | A l rc1 rc2    | if( x < rc1) -z + A \* exp( -x\^2 / (2 l\^2) ); else if( x < rc2 ) -z + a + b\*x + c\*x\^2 + d\*x\^3; else z                           | A Gaussian bump at x = y = 0, smoothly tapered to a flat plane z = 0.                                                              |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| plane          | a b c x0 y0 z0 | a\*(x-x0) + b\*(y-y0) + c\*(z-z0) = 0                                                                                                  | A plane with normal (a,b,c) going through point (x0,y0,z0)                                                                         |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| plane_wiggle   | a w            | z - a\*sin(w\*x) = 0                                                                                                                   | A plane with a sinusoidal modulation on z along x.                                                                                 |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| sphere         | R              | x\^2 + y\^2 + z\^2 - R\^2 = 0                                                                                                          | A sphere of radius R                                                                                                               |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| supersphere    | R q            | \| x \|\^q + \| y \|\^q + \| z \|\^q - R\^q = 0                                                                                        | A supersphere of hyperradius R                                                                                                     |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| spine          | a, A, B, B2, c | -(x\^2 + y\^2) + (a\^2 - z\^2/f(z)\^2)\*(1 + (A\*sin(g(z)\*z\^2))\^4), f(z) = c if z > 0, 1 otherwise; g(z) = B if z > 0, B2 otherwise | An approximation to a dendritic spine                                                                                              |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| spine_two      | a, A, B, B2, c | -(x\^2 + y\^2) + (a\^2 - z\^2/f(z)\^2)\*(1 + (A\*sin(g(z)\*z\^2))\^2), f(z) = c if z > 0, 1 otherwise; g(z) = B if z > 0, B2 otherwise | Another approximation to a dendritic spine                                                                                         |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| thylakoid      | wB LB lB       | Various, see :ref:`(Paquay) <Paquay1>`                                                                                                 | A model grana thylakoid consisting of two block-like compartments connected by a bridge of width wB, length LB and taper length lB |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+
| torus          | R r            | (R - sqrt( x\^2 + y\^2 ) )\^2 + z\^2 - r\^2                                                                                            | A torus with large radius R and small radius r, centered on (0,0,0)                                                                |
+----------------+----------------+----------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------+

.. _Paquay1:

**(Paquay)** Paquay and Kusters, Biophys. J., 110, 6, (2016).
preprint available at `arXiv:1411.3019 <https://arxiv.org/abs/1411.3019/>`_.