Finding compact coordinate representations for polygons and polyhedra

A standard technique in solid modeling is to represent planes (or lines) by explicit equations and to represent vertices and edges implicitly by means of combinatorial information. Numerical problems that arise from using floating-point arithmetic to implement operations on solids can be avoided by using exact arithmetic. Since the execution time of exact arithmetic operators increases with the number of bits required to represent the operands, it is important to avoid increasing the number of bits required to represent the plane (or line) equation coefficients. Set operations on solids do not increase the number of bits required. However, rotating a solid greatly increases the number of bits required, thus adversely affecting efficiency. One proposed solution to this problem is to round the coefficients of each plane (or line) equation without altering the combinatorial information. We show that such rounding is NP-complete.