On the Complexity of the Union of Geometric Objects

Given a family C of regions bounded by simple closed curves in the plane, the complexity of their union is denned as the number of points along the boundary of ?C, which belong to more than one curve Similarly, one can define the complexity of the union of 3-dimensional bodies, as the number of points on the boundary of the union, belonging to the surfaces of at least three distinct members of the family. We survey some upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These problems play a central role in the design and analysis of many geometric algorithms arising in motion planning and computer graphics.

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