## frustum.cc – Using the cone wall object to create a frustum

This code makes use of the conical wall object to carry out a Voronoi
tessellation for particles in a circular
frustum. A container
class is created using non-periodic boundary conditions, where the *z*
coordinate ranges from zero to one. On line 23, a `wall_cone`

class
is constructed, with the first three arguments specifying the apex position at
(0,0,2), and the next three coordinates specifying the axis direction along
(0,0,-1). The final argument sets the angle of the cone to be
`atan(0.5)`

. The intersection of this cone with the container is a
frustum, where the base at *z*=0 has radius 1, and the top at *z*=1
has radius 0.5.

In lines 28 to 32, a rectangular grid of particles is added to container,
only allowing those points that return true to the `point_inside()`

routine. The remainder of the code outputs the particle positions and Voronoi
cells in Gnuplot and POV-Ray formats.

The image above was rendered in POV-Ray using the header file
“frustum.pov”. The cells mesh together to form the frustum, although
some discrepancies are visible between them. This is due to the fact that the
`wall_cone`

class approximates the conical wall surface with a single
plane cut, in the same way as for the
cylinder example. In lines 44 to 48 the
accuracy of this approximation is tested, by comparing the sum of Voronoi
volumes, to the exact frustum volume:

`Exact frustum volume : 1.8326`

Voronoi cell volume : 1.85132

Difference : 0.0187281

Voronoi cell volume : 1.85132

Difference : 0.0187281

This is an error of 1.02%. Increasing the number of inserted particles or
improving the accuracy of the `wall_cone`

class would reduce
this.

## Movies

- Rotating movie of the cell construction: High quality QuickTime (13 MB) / Low quality AVI (14 MB)

## Code listing

1: // Frustum example code 2: // 3: // Author : Chris H. Rycroft (LBL / UC Berkeley) 4: // Email : chr@alum.mit.edu 5: // Date : August 30th 2011 6: 7: #include "voro++.hh" 8: using namespace voro; 9: 10: const double pi=3.1415926535897932384626433832795; 11: 12: int main() { 13: int i=0; 14: double x,y,z,evol,vvol; 15: 16: // Create a container with the geometry given above, and make it 17: // non-periodic in each of the three coordinates. Allocate space for 18: // eight particles within each computational block. 19: container con(-1.2,1.2,-1.2,1.2,0,1,14,14,7, 20: false,false,false,8); 21: 22: // Add a cylindrical wall to the container 23: wall_cone cone(0,0,2,0,0,-1,atan(0.5)); 24: con.add_wall(cone); 25: 26: // Place particles in a regular grid within the frustum, for points 27: // which are within the wall boundaries 28: for(z=0.1;z<1;z+=0.2) for(y=-0.85;y<1;y+=0.2) for(x=-0.95;x<1;x+=0.2) { 29: if (con.point_inside(x,y,z)) { 30: con.put(i,x,y,z);i++; 31: } 32: } 33: 34: // Output the particle positions and Voronoi cells in Gnuplot format 35: con.draw_particles("frustum_p.gnu"); 36: con.draw_cells_gnuplot("frustum_v.gnu"); 37: 38: // Output the particle positions and Voronoi cells in POV-Ray format 39: con.draw_particles_pov("frustum_p.pov"); 40: con.draw_cells_pov("frustum_v.pov"); 41: 42: // Compute the volume of the Voronoi cells and compare it to the 43: // exact frustum volume 44: evol=pi*1*(0.5*0.5+0.5*1+1*1)/3; 45: vvol=con.sum_cell_volumes(); 46: printf("Exact frustum volume : %g\n" 47: "Voronoi cell volume : %g\n" 48: "Difference : %g\n",evol,vvol,vvol-evol); 49: }