On finite-precision representations of geometric objects
Journal of Computer and System Sciences
Coordinate representation of order types requires exponential storage
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
A rational rotation method for robust geometric algorithms
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
I3D '01 Proceedings of the 2001 symposium on Interactive 3D graphics
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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.