COMPUTATIONAL GEOMETRY (I)

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We now consider some basic problems of computational geometry that will be useful for our applications. We begin in 2 dimensions.

• Let S be a set of (closed) line segments. The problem we face is this: compute the set of pairs of segments that properly intersect.
• The computational paradigm here is the plane sweep. Assume a horizontal sweepline H(t) where t is the time and also the y-coordinate.
• Let S(t) be the set of line segments in S that is intersected by H(t). The elements in S(t) are naturally ordered by their x-coordinate: Write s < _t s¢ for this ordering.
• Events: This ordering can change in three ways.
  (1) When two segments s, s¢ intersect. Their relative ordering is switched.
  (2) A new segment enter the set S(t).
  (3) A segment exits from the set S(t).
• These events can be anticipated by using an event queue. Initially, the queue has all the events of types (2) and (3) put into it. We just have to show how to update the queue after each event.
• The algorithm is seen to take O(nlogn + K) where K is the number of intersections.

• Cell complex in R² is a set K of cells. The set is partitioned into the cells of dimension 0, 1 and 2: K = K⁰ ÈK¹ÈK². Dimension 0 and 1 cells are points and edges (open line segments). The rest are 2-cells, which are maximally connected connected components of R²- (K⁰ÈK¹).
• For cell complex, we only require that
  (1) endpoints of edges are 0-cells. If K also satisfy
  (2) 2-cells are simply connected, then we call K a subdivision.
• Let S be a set of non-crossing segments, where a segment may degenerate into an isolated point. we define n₀(S) to be the number of distinct endpoints in S. Let n₁(S)| is the number of non-degenerate segments, and n₂(S) be the number of 2-cells in the complement of ÈS. Finally, let b(S) be the number of connected components in ÈS. We have Euler's formula,
  

  n0(S) - n1(S) +n2(S) = 1 +b(S).

• It turns out that we could also define these numerical quantities for any undirected planar graph G. The only non-obvious part is whether n₂(G) is well-defined. This will be justified by Euler's formula, which continues to hold.

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