In geometry, a truncated projective plane (TPP), also known as a dual affine plane, is a special kind of a hypergraph or geometric configuration that is constructed in the following way. Take a finite projective plane. Remove one of the points (vertices) in the plane. Remove all lines (edges) containing that point. These objects have been studied in many different settings, often independent of one another, and so, many terminologies have been developed. Also, different areas tend to ask different types of questions about these objects and are interested in different aspects of the same objects. Example: the Pasch hypergraph Consider the Fano plane, which is the projective plane of order 2. It has 7 vertices {1,2,3,4,5,6,7} and 7 edges {123, 145, 167, 246, 257, 347, 356}. It can be truncated e.g. by removing the vertex 7 and the edges containing it. The remaining hypergraph is the TPP of order 2. It has 6 vertices {1,2,3,4,5,6} and 4 edges {123, 154, 624, 653}. It is a tripartite hypergraph with sides {1,6},{2,5},{3,4} (which are exactly the neighbors of the removed vertex 7). It is also called the Pasch hypergraph, due to its connection with Pasch's axiom. It is a 2-regular hypergraph (each vertex is in exactly two edges), and its maximum matching is of size 1 (every two of its edges intersect). Combinatorics of dual affine planes A finite projective plane of order has + 1 points on every line ( in the hypergraph description). There are total points and an equal number of lines. Each point is on + 1 lines. Every two distinct points lie on a unique line and every two distinct lines meet at a unique point. By removing a point and all the lines that pass through that point, the configuration that is left has points, 2 lines, each point is on lines and each line contains + 1 points. Each pair of distinct lines still meet at a unique point, but two distinct points are on at most one line. This dual affine plane is thus a configuration of type (). The points can be partitioned into + 1 sets of points apiece, where no two points in the same partition set are joined by a line. These sets are the analogs of classes of parallel lines in an affine plane, and some authors refer to the points in a partition piece as parallel points