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# smoothSubfan -- computes the subfan of all smooth cones

- Usage:
    `F1 = smoothSubfan F`
- Inputs:
  - F, an instance of the type Fan
- Outputs:
  - F1, an instance of the type Fan

## Description

For a given Fan F the function computes the subfan F1 of all smooth cones.

Let's consider the fan consisting of the following three dimensional cone and all of its faces:

    i1 : C = posHull  matrix {{1,-1,0},{1,1,0},{1,1,1}}

    o1 = {ambient dimension => 3           }
          dimension of lineality space => 0
          dimension of the cone => 3
          number of facets => 3
          number of rays => 3

    o1 : Cone

    i2 : F = fan C

    o2 = {ambient dimension => 3         }
          number of generating cones => 1
          number of rays => 3
          top dimension of the cones => 3

    o2 : Fan

This cone is not smooth, therefore also the fan is not. But if we remove the interior and one of the two dimensional faces the resulting subfan is smooth.

    i3 : F1 = smoothSubfan F

    o3 = {ambient dimension => 3         }
          number of generating cones => 2
          number of rays => 3
          top dimension of the cones => 2

    o3 : Fan

    i4 : apply(maxCones F1, rays)

    o4 = {| 0 1 |, | 0 -1 |}
          | 0 1 |  | 0 1  |
          | 1 1 |  | 1 1  |

    o4 : List

## Ways to use smoothSubfan:

- `smoothSubfan(Fan)`

## For the programmer

The object smoothSubfan is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldPolyhedra.m2:6677:0.