## Kill Vertex

The Kill Vertex tool looks like this  and is located on the Modify Tab of the Toolbar. Click the Kill Vertex button. Geodesica prompts you to pick a primal vertex. The picked vertex, its edges and surrounding faces are all deleted. A single primal face replaces the deleted vertex. The dual face is also deleted and its surrounding faces converge at the center of the new primal face. The dual is regenerated automatically.

Primal before Kill Vertex

After Kill Vertex: a hexagonal face

Primal + dual before Kill Vertex

Primal + dual after Kill Vertex

### Illegal operations

If the lattice is to satisfy Euler’s formula for convex polyhedra V - E + F = 2, then the following conditions must be met:

1. All faces are bounded by a single ring of edges, and there are no holes in the faces.
2. The lattice has no holes ‘through’ it.
3. Each edge adjoins exactly two faces and is terminated by a vertex at each end.
4. At least three edges meet at a vertex.
[Mortenson]

However, during the modelling process, conditions 1 and 3 may be temporarily excluded if the next step in the modelling phase corrects the situation. Geodesica allows vertices subtended by only two edges, providing that V - E + F = 2.

If Kill Vertex detects an illegal operation, it will stop and warn the user.

#### Example 1

In fig. 2., the yellow vertex was picked for deletion. But the dialog explains that the vertex with only two edges (labelled ‘D’) must be deleted first. If the yellow vertex was deleted first, an unconnected edge would result at‘D’.

Fig. 2

#### Example 2

Another illegal operation is the creation of “island” facets. In fig. 3., the yellow vertex is the only link left between the pentagonal island of facets and the surrounding lattice. If you try to delete the yellow vertex, the application will halt with an error dialog. (Version 0-0-1-7z)

Fig. 3

Situations like the above can be rectified by using the Make Edge Tool.

#### Keeping the manifold closed

In fig. 4., Kill Vertex has eaten away at the sphere until it looks like a broken egg. But the sphere is still closed. The jagged rim marks the boundary of the top “face” with 23 non-coplanar vertices. This face is equivalent to a topological disk. The dual, which is projected to the envelope, shows this to be the case.

Fig. 4

TIP: Non-coplanar cells may be selected using the menu Select->Non coplanar cells. These can be turned into trifans using the menu Modify->Trifan selected cells. This is documented here.