CMPUT 301 Convex Hulls in 2D

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CMPUT 301 Convex Hulls in 2D

• Lecturer Sherif Ghali
• Department of Computing Science
• University of Alberta

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Geometric Problems

• Voronoi Diagram

• Motion Planning

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Convexity of a region in 2D

• A region is convex iff pq is wholly inside the
  region for each p,q in the region.
• A convex hull is the smallest such convex set.

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The Convex Hull
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The Input and Output of aConvex Hull Computation
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Characterization of Points on the Convex Hull
two points p and q are on the convex hull if and
only if none of the other points lies on the left
side of pq
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Convex HullFirst Algorithm
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Implementation Details - time

• Left vs. right turn

• Connecting the hull

r1
r2
pqr1 is a left turn
q
pqr2 is a right turn
p
what is the time complexity?
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Handling Degeneracy
two points p and q are on the convex hull if and
only if none of the other points lies on the left
side of pq
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Reliability of floating point

• float a,b
• if (ab)

Conclusion SlowConvexHull can succumb to
degenerate inputs
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Incremental algorithms

• We know the solution for Si p1, p2, pi
• Insert pi1
• We compute the solution for Si1 p1, p2, pi
  ,pi1

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Types of Polygons

• Simple polygon edges intersect only at endpoints
• Convex polygon
• pq wholly inside
• turns are only right turns

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Upper and Lower Hull
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Incremental Insertion
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Lexicographic Sorting

• bool xcompare(Point a, Point b)
• return a.x lt b.x
• bool xycompare(Point a, Point b)
• if (a.x b.x)
• return a.y lt b.y
• else
• return a.x lt b.x

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Degeneracy

• Potential problems
• a hull point is mistakenly removed
• an interior point is not removed
• small dents in the hull
• Conclusion output not necessarily correct, but
  the program will not crash
• Perils of floating point

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Correctness

• Induction hypothesis assume the Hull Hi-1
• Hi-1 Hull(p1, p2, pi -1)
• is the upper hull
• Base case H2 Hull(p1, p2)
• is trivially correct
• Could a point be above the hull Hi
• Hi Hull(p1, p2, pi )
• after adding pi ?

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Time

• Lexicographic sorting O(n log n)
• A point is deleted at most once

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Degeneracies

• General position assumption
• Binary vs. ternary branching
• If affordable, use exact arithmetic (integers,
  rationals, algebraic numbers)
• Correctness of an algorithm is with respect to
  the definition

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Application Domains

• Computer Graphics
• Modeling and rendering, collision detection
• Robotics
• Motion planning, optimal robot arm placement
• Geographic Information Systems
• Maps, overlaying maps, terrains, nearness
• CAD/CAM
• prototyping, manufacturing, assembly
• Other
• molecular modeling, character/fingerprint
  recognition, database queries

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Reference
Chap. 1