CHAPTER 33 Computational Geometry

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CHAPTER 33Computational Geometry

• Is the branch of computer science that studies
  algorithms for solving geometric problems.
• Has applications in many fields, including
• computer graphics
• robotics,
• VLSI design
• computer aided design
• statistics
• Deals with geometric objects such as points,
  line segments, polygons, etc.
• Some typical problems considered
• whether intersections occur in a set of lines.
• finding vertices of a convex hull for points.
• whether a line can be drawn separating two sets
  of points.
• whether one point is visible from a second
  point, given some polygons that may block
  visibility.
• optimal location of fire towers to view a
  region.
• closest or most distant pair of points.
• whether a point is inside or outside a polygon.

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Cross products
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Cross products (cont.)
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Intersection of two line segments
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Intersection of two line segments (cont.)
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Determining whether any pair of segments
intersect.
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The Algorithm ANY-SEGMENTS-INTERSECTION(S)

• To maintain a total ordering T on the sweep line
  as it moves, we need the following operations
• INSERT(T,s) insert segment s into T
• DELETE(T,s) delete segment s from T
• ABOVE(T,s) return the segment from T
  immediately above segment s
• BELOW(T,s) return the segment from T
  immediately below segment s
• Red-black trees can provide a balanced search
  tree which can support above operations in O(log
  n) time (see Ch. 13 in CLRS)

• T? emptyset
• Sort segment endpoints from left to right. Break
  ties by putting point with lower y-coordinate
  first.
• For each point p in the sorted list (in order)
• if p is left endpoint of segment s then
• INSERT(T,s)
• if ABOVE(T,s) exists and
  it intersects s or
• BELOW(T,s) exists and it intersects s
  then return TRUE
• if p is the right endpoint of segment
  then
• if both ABOVE(T,s) and
  BELOW(T,s) exist and
• ABOVE(T,s) intersects BELOW(T,s)
  then return TRUE
• DELETE(T,s)
• return FALSE
  

Animated demo http//www.lupinho.de/gishur/html/Sw
eeps.htmlsegment
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Comments and Correctness
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Correctness and Runtime

• In either case, the order of T is correct before
  q is processed.
• Only two cases are possible.
• Either a or b is inserted into T at z and the
  other segment is above or below it. (In this
  case, intersection is detected by line (6) .)
• Segments a and b are already in T and a segment c
  between them is deleted at z. (In this case, an
  intersection is detected by line (8).)
  

• In either case the intersection of a and b at p
  is detected, so procedure cannot return a FALSE,
  as supposed.

a
p
c
q
b
x
z

• Running Time
• sort on line (2) takes O(n logn) time, using
  merge or heap sort.
• since there are 2n events (endpoints) where
  sweep line stops, the loop iterates at most 2n
  times.
• each iteration of loop takes O(log n) time,
  since red-black tree operations take O(log n)
  time.
• Total time is O(n logn).

READ Ch. 33.1 and Ch 33.2 in CLRS.
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Convex Hull Algorithms
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The Problem and Approaches
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Gift Wrapping or The Jarvis March Algorithm
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Complexity of Jarvis March
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Graham Scan Algorithm
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Example and Runtime Computation
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Correctness of Graham Scan
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Correctness of Graham Scan (cont.)
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Closest Pair of Points
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Divide and Conquer Steps
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Combine Step and Runtime
d
d
L
d
d
Possible coincident points, one in P and one in
P.
L
Runtime

• The initial preprocessing step costs O(n logn)
• The recurrence relation for the recursive
  portion is T(n),
• where
• Its solution is T(n) O(n logn). This gives also
  the total running time.

READ Ch. 33.3 and 33.4 in CLRS.