# Manifolds (surfaces)

**Overview:**

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.

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.

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 dendtritic 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 dendtritic spine |

thylakoid | wB LB lB | Various, see (Paquay) | 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) |

**(Paquay)** Paquay and Kusters, Biophys. J., 110, 6, (2016).
preprint available at arXiv:1411.3019.