Finding Compact Coordinate

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