PM Quadtree

A PM (Polygonal Map) Quadtree describes a collection of quadtrees that can store points and edges. A q-edge refers to a subset of an edge formed by the partitions of the PM Quadtree.

PM1 Quadtree

A PM1 Quadtree follows the following rules:

1. At most, one vertex can lie in a region represented by a quadtree leaf node.
2. If a quadtree leaf node's region contains a vertex, then it can contain no q-edge that does not include that vertex.
3. If a quadtree leaf node's region contains no vertices, then it can contain, at most, one q-edge.
4. Each region's quadtree leaf node is maximal.

Here is an example.

PM2 Quadtree

A PM2 Quadtree follows the following rules:

1. At most, one vertex can lie in a region represented by a quadtree leaf node.
2. If a quadtree leaf node's region contains a vertex, then it can contain no q-edge that does not include that vertex.
3. If a quadtree leaf node's region contains no vertices, then it can contain only q-edges that meet at a common vertex exterior to the region.
4. Each region's quadtree leaf node is maximal.

Here is an example.

PM3 Quadtree

A PM3 Quadtree follows the following rules:

1. At most, one vertex can lie in a region represented by a quadtree leaf node.
2. Each region's quadtree leaf node is maximal.

Here is an example.

Applications

PM Quadtrees are useful in determining the region in which a point lies, the determination of the boundaries of all regions lying within a given distance of a point, and overlaying two maps.

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