Free-Form Surface Partition in 3-D

  • Danny Z. Chen
  • Ewa Misiołek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)


We study the problem of partitioning a spherical representation S of a free-form surface F in 3-D, which is to partition a 3-D sphere S into two hemispheres such that a representative normal vector for each hemisphere optimizes a given global objective function. This problem arises in important practical applications, particularly surface machining in manufacturing. We model the spherical surface partition problem as processing multiple off-line sequences of insertions/deletions of convex polygons alternated with certain point queries on the common intersection of the polygons. Our algorithm combines nontrivial data structures, geometric observations, and algorithmic techniques. It takes \(O(\min\{m^2n \log \log m + \frac{m^3 \log^2(mn) \log^2(\log m)}{\log^3 m}, m^3\log^2n+mn\})\) time, where m is the number of polygons, of size O(n) each, in one off-line sequence (generally, m ≤ n). This is a significant improvement over the previous best-known O(m 2 n 2) time algorithm. As a by-product, our algorithm can process O(n) insertions/deletions of convex polygons (of size O(n) each) and queries on their common intersections in O(n 2 loglogn) time, improving over the “standard” O(n 2 logn) time solution for off-line maintenance of O(n 2) insertions/deletions of points and queries. Our techniques may be useful in solving other problems.


Convex Polygon Binary Search Partial Intersection Point Query Common Intersection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Danny Z. Chen
    • 1
  • Ewa Misiołek
    • 2
  1. 1.Dept. of Computer Science and EngineeringUniv. of Notre DameNotre DameUSA
  2. 2.Mathematics DepartmentSaint Mary’s CollegeNotre DameUSA

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