The tessellation matrix is a special structure used to hold information on Voronoi polygons. Unlike the other matrices, tessellation matrices cannot be formally imported, but are rather generated internally within PASSaGE. They can be saved in the native PASSaGE file format for use at a later date, or exported as an ArcGIS shapefile.
Tessellation matrices are stored in a "wing-edged" data structure (Okabe et al. 2000). This is a somewhat complicated structure to explain and is described in more detail in the above reference. A tessellation contains three types of objects: vertices (representing points in space), edges (representing connections between pairs of vertices), and polygons (representing regions enclosed by a set of edges). One polygon is considered to be infinity and represents the entire area exterior to the points used to generate the tessellation. Each vertex, edge, and polygon is given a number, and the structure of the tessellation is stored by recording the following information. For each vertex, the matrix stores: (1) the number of one edge connected to the vertex, (2) the coordinates (x,y or x,y,z) of the vertex in space, and (3) whether the vertex represents a real location or a virtual location created out towards infinity. For each polygon, the matrix stores the number of one edge marking the border of the polygon. For each edge, the matrix stores (1) the numbers of the starting and ending vertices for the edge, (2) the numbers of the polygons found on the left and right hand sides of the edge, (3) the numbers of the edges found clockwise and counterclockwise around the starting vertex, and (4) the numbers of the edges found clockwise and counterclockwise around the ending vertex. This efficient storage allows any aspect of the tessellation (for example, the area or circumference of a specific polygon) to be easily reconstructed with minimum effort.