Probing a set of hyperplanes by lines and related problems

  • Yasukazu Aoki
  • Hiroshi Imai
  • Keiko Imai
  • David Rappaport
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 709)


Suppose that for a set H of n unknown hyperplanes in the Euclidean d-dimensional space, a line probe is available which reports the set of intersection points of a query line with the hyperplanes. Under this model, this paper investigates the complexity to find a generic line for H and further to determine the hyperplanes in H. This problem arises in factoring the u-resultant to solve systems of polynomials (e.g., Renegar [13]). We prove that d+1 line probes are sufficient to determine H. Algorithmically, the time complexity to find a generic line and reconstruct H from O(dn) probed points of intersection is important. It is shown that a generic line can be computed in O(dn log n) time after d line probes, and by an additional d line probes, all the hyperplanes in H are reconstructed in O(dn log n) time. This result can be extended to the d-dimensional complex space. Also, concerning the factorization of the u-resultant using the partial derivatives on a generic line, we touch upon reducing the time complexity to compute the partial derivatives of the u-resultant represented as the determinant of a matrix.


Grid Point Time Complexity Generic Direction Generic Point Dual Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Yasukazu Aoki
    • 1
  • Hiroshi Imai
    • 1
  • Keiko Imai
    • 2
  • David Rappaport
    • 3
  1. 1.Department of Information ScienceUniversity of TokyoTokyoJapan
  2. 2.Department of Information and System EngineeringChuo UniversityTokyoJapan
  3. 3.Department of Computing and Information ScienceQueen's UniversityKingstonCanada

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