Computational Geometry 2D Convex Hulls

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Computational Geometry2D Convex Hulls
Joseph S. B. Mitchell Stony Brook University
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Convexity

• Set X is convex if p,q?X ? pq? X
• Point p? X is an extreme point if there exists a
  line (hyperplane) through p such that all other
  points of X lie strictly to one side
• Fact If XS is a finite set of points in 2D,
  then CH(X) is a convex polygon whose vertices
  (extreme points) are points of S.

Extreme points in red
r
p
q
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Fundamental Problem 2D Convex Hulls
More generally CH(polygons)

• Input n points S (p1, p2, , pn)
• Output A boundary representation, e.g., ordered
  list of vertices (extreme points), of the convex
  hull, CH(S), of S (convex polygon)

p2
p7
p4
p8
p3
p1
p6
p5
p9
Output (9,6,4,2,7,8,5)
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Equivalent Definitions of Convex Hull, CH(X)

• all convex combinations of d1 points of X
  Caratheodorys Thm (in any dimension d)
• Set-theoretic smallest convex set containing X.
  
• In 2D min-area (or min-perimeter) enclosing
  convex body containing X
• In 2D

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2D Convex Hull Algorithms

• O(n4) simple, brute force (but finite!)
• O(n3) still simple, brute force
• O(nh) simple, output-sensitive
• h output size ( vertices)
• O(n log n) worst-case optimal (as fcn of n)
• O(n log h) ultimate time bound (as fcn of
  n,h)
• Randomized, expected O(n log n)
• Lower bound ?(n log n)

Simple, elegant
y x2
(xi ,xi2 )
From SORTING
Note Even if the output of CH is not required to
be an ordered list of vertices (e.g., just the
of vertices), ?(n log n) holds
xi
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Primitive Computation

• Left tests sign of a cross product
  (determinant), which determines the orientation
  of 3 points
• Time O(1) (constant)

c
b
Left( a, b, c ) TRUE ? ab ? ac gt 0
c is left of ab
a
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SuperStupidCH O(n4)

• Fact If s ??pqr, then s is not a vertex of CH(S)
• ? p
• ? q ? p
• ? r ? p,q
• ? s ? p,q,r If s ? ?pqr then mark s as
  NON-vertex
• Output Vertices of CH(S)
• Can sort (O(n log n) ) to get ordered

q
s
p
O(n3)
r
O(n)
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StupidCH O(n3)

• Fact If all points of S lie strictly to one side
  of the line pq or lie in between p and q, then pq
  is an edge of CH(S).
• ? p
• ? q ? p
• ? r ? p,q If r ? red then mark pq as NON-edge
  (ccw)
• Output Edges of CH(S)
• Can sort (O(n log n) ) to get ordered

O(n2)
O(n)
Caution!! Numerical errors require care to
avoid crash/infinite loop!
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O(nh) Gift-Wrapping
Jarvis March

• Idea Use one edge to help find the next edge.
• Output Vertices of CH(S)
• Demo applet of Jarvis march

O(n) per step h steps Total O(nh)
r
q
p
Key observation Output-sensitive!