# Pérez &Aacute;guila, Ricardo - Capítulo 6. Extreme Edges in 4D-OPP&acute;- 4D Orthogonal Polytopes

Pérez &Aacute;guila, Ricardo
- Capítulo 6. Extreme
Edges in 4D-OPP&acute;-
4D Orthogonal Polytopes
-- Licenciatura en Ingeniería
en Sistemas Computacionales. - Departamento de Ingeniería en
Sistemas Computacionales. - Escuela de Ingeniería, - Universidad de las Américas
Puebla.

## Teaser

Chapter 6 Extreme Edges in 4D-OPP’s 6.1 The 3D Extreme Vertices [Aguilera, 98] defines a brink or extended-edge as the maximal uninterrupted segment, built out of a sequence of collinear and contiguous two-manifold edges of a 3DOPP with the following properties:  Non-manifold edges do not belong to brinks.  Every two-manifold edge belongs to a brink, whereas every brink consists of m edges (m  1), and contains m  1 vertices.  Two of the vertices of type V3, V4N1 or V6N1 (section 5.4.1) are at either extreme of the brink (extreme or ending vertices).
These vertices have in common that they are the only ones that have exactly three incident two-manifold and perpendicular edges, regardless of the number of incident non-manifold edges, therefore those vertices mark the end of brinks in all three orthogonal directions.  The m  1 vertices of type V4, V4N2, V5N or V6 are the only common point of two collinear edges of a same brink (interior vertices).  Due to all six incident edges of a V6N2 vertex are non-manifold edges, none of them belongs to a brink, thus this vertex does not belong to any brink. 76 Although the above properties for brinks were defined for 3D-OPP’s, [Aguilera, 98] points that also can be defined the properties for brinks in 1D-OPP’s and 2D-OPP’s as follows:  For 1D-OPP’s the only elements are vertices and edges.
If a vertex has one incident edge, it is an Extreme Vertex.
For instance, edges and brinks are in this case equivalent.  For 2D-OPP’s there are only two types of vertices (section 5.3.1), the vertex with two incident manifold edges (V2) and the vertex with four incident manifold edges (V4N). In a 2D-OPP’s brink, vertices of type V2 are the Extreme Vertices since each of those vertices has two incident manifold edges perpendicular to each other, while vertices of type V4N are the inner ones because each one is the only common point of two collinear edges of a brink, so they can not be ending verti...