bipyramid -- computes the bipyramid over a polyhedron

Synopsis

• Usage:

  Q = bipyramid P

• Inputs:
  • P, a convex polyhedron
• Outputs:
  • Q, a convex polyhedron

Description

The bipyramid over a Polyhedron in n-space is constructed by embedding the Polyhedron into (n+1)-space, computing the barycentre of the vertices, which is a point in the relative interior, and taking the convex hull of the embedded Polyhedron and the barycentre x {+/- 1}.

As an example, we construct the octahedron as the bipyramid over the square (see hypercube).

i1 : P = hypercube 2

o1 = {ambient dimension => 2           }
      dimension of lineality space => 0
      dimension of polyhedron => 2
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o1 : Polyhedron

i2 : Q = bipyramid P

o2 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 8
      number of rays => 0
      number of vertices => 6

o2 : Polyhedron

i3 : vertices Q

o3 = | -1 1  -1 1 0  0 |
     | -1 -1 1  1 0  0 |
     | 0  0  0  0 -1 1 |

              3        6
o3 : Matrix QQ  <--- QQ