Lecture Notes in Computer Science Volume 2265, 2002, pp 407-421

Polar Coordinate Drawing of Planar Graphs with Good Angular Resolution

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Abstract

We present a novel way to draw planar graphs with good angular resolution. We introduce the polar coordinate representation and describe a family of algorithms which use polar representation. The main advantage of using a polar representation is that it allows us to exert independent control over grid size and bend positions. Polar coordinates allow us to specify different vertex resolution, bend-point resolution and edge separation. We first describe a standard (Cartesian) representation algorithm (CRA) which we then modify to obtain a polar representation algorithm (PRA). In both algorithms we are concerned with the following drawing criteria: angular resolution, bends per edge, vertex resolution, bend-point resolution, edge separation, and drawing area. The CRA algorithm achieves 1 bend per edge, unit vertex and bend resolution, $ \sqrt 2 /2$ edge separation, $ 5n \times \tfrac{{5n}} {2}$ drawing area and $ \tfrac{1} {{2d(\upsilon )}}$ angular resolution, where d(v) is the degree of vertex v. The PRA algorithm has an improved angular resolution of $ \tfrac{\pi } {{4d(\upsilon )}}$ , 1 bend per edge, and unit vertex resolution. For the PRA algorithm, the bend-point resolution and edge separation are parameters that can be modified to achieve different types of drawings and drawing areas. In particular, for the same parameters as the CRA algorithm (unit bend-point resolution and $ \sqrt 2 /2$ edge separation), the PRA algorithm creates a drawing of size $ 9n \times \tfrac{{9n}} {2}$ .