Finding compact coordinate representations for polygons and polyhedra

Practical solid modeling systems are plagued by numerical problems that arise from using floating-point arithmetic. For example, polyhedral solids are often represented by a combination of geometric and combinatorial information. The geometric information may consist of explicit plane equations, with floating-point coefficients; the combinatorial information may consist of face, edge, and vertex adjacencies and orientations, with edges defined by face-face adjacencies and vertices by edge-edge adjacencies. Problems arise when numerical roundoff error in geometric operations causes the geometric information to become inconsistent with the combinatorial information. These problems can be avoided by using exact arithmetic instead of floating-point arithmetic. However, some operations, such as rotation, increase the number of bits required to represent the plane equation coefficients. Since the execution time of exact arithmetic operators increases with the number of bits in the operands, the increased number of bits in the plane equation coefficients can cause performance problems. One proposed solution to this performance problem is to round the plane equation coefficients without altering the combinatorial information. We show that such rounding is NP-complete.