Integer Approximation to the Intersection of Three Planes with Planar Constraints
The intersection point of three planes specified with rational coordinates is approximated by a point with integer-valued coordinates so that the integer point is constrained within the pyramid specified by three half-spaces of the planes. An O(N log N) algorithm guarantees that the obtained integer point is closest to the apex of the pyramid. An O(log N) algorithm, not yet proven to guarantee the closest point, yields promising results.
KeywordsLine Intersection Pyramid
Unable to display preview. Download preview PDF.
- D. Dobkin and D. Silver. Recipes for geometry and numerical analysis - part I: An empirical study. In Proceedings of the ACM Symposium on Computational Geometry, pages 93 7 105. Champaign-Urbana, Illinois, June 1988.Google Scholar
- D. H. Greene and F. F. Yao. Finite resolution computational geometry. In Proceedings of Twenty-seventh Annual IEEE-FOCS Conference, pages 143 — 152. Toronto, Canada, October 1986.Google Scholar
- L. Guibas, D. Salesin, and J. Stolfi. Epsilon-geometry: Building robust algorithms from imprecise computations. In Proceedings of the ACM Symposium on Computational Geometry, pages 208–217. Saarbrücken, Germany, June 1989.Google Scholar
- D. E. Knuth. The Art of Computer Programming - Seminumerical Algorithms (Vol. II). Addison Wesley, 1981.Google Scholar
- J. C. Lagarias. The computational complexity of simultaneous Diophantine approximation problems. In Proceedings of Twenty-third Annual IEEE-FOCS Conference, pages 32–39, 1982.Google Scholar
- A. K. Lenstra. Polynomial factorization by root approximation. In J. Fitch, editor, Lecture Notes on Computer Science: Int. Symp. Symbolic and Algebraic Computations, volume 174, pages 272–276. Springer-Verlag, 1984.Google Scholar
- L. R. Nackman M. S. Karasick, D. Lieber. Efficient Delaunay triangulation using rational arithmetic. IBM Research Report, RC 14455, IBM Research Division, Yorktown Heights, New York, March 1989.Google Scholar
- D. W. Matula and P. Kornerup. Foundations of Finite Precision Rational Arithmetic. In Computing, Suppl 2, pages 85–111. Springer-Verlag, 1980.Google Scholar
- S. Mehta, M. Mukherjee, and G. Nagy. Constrained integer approximation to 2-D line intersections. In Proceedings Second Canadian Conf. on Computational Geometry, pages 302–305. Ottawa, Canada, August 1990.Google Scholar
- V. J. Milenkovic. Verifiable implementations of geometric algorithms using finite precision arithmetic. In D. Kapur and J. L. Mundy, editors, Geometric Reasoning, pages 377–401. Cambridge, Massachusetts, MIT Press, 1989.Google Scholar
- M. Segal and C. H. Sequin. Consistent calculations for solid modeling. In Proceedings of the ACM Symposium on Computational Geometry, pages 29 — 38. Baltimore, Maryland, June 1985.Google Scholar
- J. Vignes. New methods for evaluating the validity of mathematical computations. In Mathematics and Computers in Simulation, volume 20, pages 227–249, 1978.Google Scholar