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Trapezoidal Maps in Computational Geometry

Explore the concept of trapezoidal maps, a planar subdivision with distinct abscissas, offering efficient algorithms for point location queries and incremental construction. Learn about the Randomized Incremental and Querying Algorithms, plus Intersection properties.

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Trapezoidal Maps in Computational Geometry

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  1. Trapezoidal Maps Shmuel Wimer Bar Ilan Univ., School of Engineering

  2. Trapezoidal Map Planar subdivision Abscissas are all distinct n segments 6n+4 vertices at most 3n+1 trapezoids at most

  3. Trapezoidal map can be constructed in O(nlogn) time by a scan-line algorithm.

  4. Randomized Incremental Algorithm

  5. x-node y-node trapezoid Inner nodes have degree 2

  6. Querying a point location Does q lie to the left or to the right ? Does q lie above or below?

  7. Randomized Construction Algorithm

  8. New segment insertion

  9. Assuming that a point is contained in Δ, the sub tree replacing its leaf is sufficient to determine whether the point is in A, B, C or D. The information attached to new trapezoids is their left and right neighbor trapezoids, top and bottom segment and points defining their left and right vertical segment. If the information in Δ is properly stored, above info can be determined in a constant time from si and Δ. If pi=leftPoint(Δ) and / or qi=rightPoint(Δ), Δis divided into two or three trapezoids and sub-tree replacement is simpler.

  10. Intersection with more than one trapezoid

  11. ■

  12. Given a set of segments, nothing is guaranteed on the maximal run time, which can be quadratic. Considering all possible problems of n segments, what is the expected maximal query time? O(logn)

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