Convex Hull in Two Dimensions

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Convex Hull in Two Dimensions

• Jyun-Ming Chen
• Refs deBerg et al. (Chap. 1)
• ORourke (Chap. 3)

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Outline

• Definitions of CH
• Algorithms for CH
• Analysis of algorithms

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Convex Hull
??(Convex Hull)

• Definition of convexity

• Convex hull of a point set S, CH(S)
• Smallest convex set containing S
• Intersection of all convex sets containing S

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Convex Hull

• Why computing CH?
• Many reasons e.g., faster collision detection
• How to compute CH?
• Good solutions come from
• A thorough understanding of the geometric
  properties of the problem
• Proper application of algorithmic techniques and
  data structures

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General Steps

• Sketch your problem in general position
• Observe the desired output and reason about the
  possible steps to get to the goal
• Check for degeneracy
• Time/resource complexity
• General case
• Best/worst cases

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Computing CH

• of a finite set P of n points in 2D
• Geometric intuition Rubberband analogy

• Representation
• Input point set
• Output CH(P) CW-ordered point set

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Observation

• Def convex edges

• Idea
• Search through all pairs of points for convex
  edges
• Form a loop

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Naïve Algorithm
You can add a break here if you like
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Analysis of Naïve Algorithm

• Assumption points are in general position
• (usually this is done first to simplify the code)
• (consider the degenerate cases later)
• How to compute lie to the left of directed
  line?
• Complexity analysis O(n3)
• (n2n)/2 pairs, each with n2 other points to
  examine
• High complexity due to
• simply translate geometric insight into an
  algorithm in a brute-force manner

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Considering Degeneracy

• Collinearity
• Modify the test to all points to the right, or
  on the line
• Does this solve the problem?!

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Degenerate Case (cont)

• Consider numerical inaccuracy
• Two possibilities

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Jarvis March (Gift Wrapping)

• A modification of naïve algorithm
• Start at some extreme point, which is guaranteed
  to be on the hull.
• At each step, test each of the points, and find
  the one which makes the smallest right-hand turn.
  That point has to be the next one on the hull.
• The final CW loop is the CH.

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Jarvis March (cont)

• Complexity
• Crude count O(n2)
• Output sensitive
• O(nh), h of segments on the hull

Worst case
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A Better Algorithm

• Sort by x-coord
• The extreme points must be on the CH
• Goal in 2 linked lists
• Upper hull p1 to pn
• Lower hull pn to p1

• Idea
• In CW manner, always making right turns
• If fail to turn right, delete previous point
  until the turn is correct.

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Step-by-step
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Algorithm (cont)

• This is a variation of Graham Scan
• Proof of correctness p.8
• Complexity analysis O(n log n)
• Lexicographical sorting of points O(n log n)
• Computation of lower hull O(n)
• Consider the for and while

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Consider Degeneracy Again

• Effects on sorting
• Lexico. ordering

• Effects on making turns

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Other Version of Graham Scan

1. Find an extreme point. This point will be the
   pivot, is guaranteed to be on the hull, and is
   chosen to be the point with smallest y
   coordinate.
2. Sort the points in order of increasing angle
   about the pivot. We end up with a star-shaped
   polygon (one in which one special point, in this
   case the pivot, can "see" the whole polygon).
3. Build the hull, by marching around the
   star-shaped poly, adding edges when we make a
   left turn, and back-tracking when we make a right
   turn.

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Graham Scan (ver2)

• Also handles collinearity

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Incremental Algorithm

• Suitable for dynamic CGeom and 3D hull
• If pn?CHn-1,
• CHnCHn-1
• Else
• Find two tangents, update CHn

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Incremental Algorithm

• Input a set of points P in 2D
• Output a list L containing the points of CH(P)
  in CCW order
• 1. Take first three points of P to form a
  triangle
• 2. For i 4 to n
• If (pi is in CHi-1) do nothing
• Else
• FindTangent (pi, CHi-1)?t1, t2
• Replace the items t2 t1 in L by t2, pi, t1

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FindTangent (p, CH)

• (Go through all points in CH circularly)
• For each point pi
• If XOR (p is left_or_on (pi-1,pi), p is
  left_or_on(pi,pi1))
• Mark pi as the tangent point
• (There will be two tangent points)
• Determine t1, t2

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Incremental Algorithm (cont)

• Finding Tangents
• Analysis

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Quick Hull
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Quick Hull (cont)

• Time complexity
• Find extreme point c O(n)
• Cost of recursive call
• T(n) O(n) T(A) T(B)
• Best case A B n/2
• T(n) 2T(n/2) O(n) T(n) O(n log n)
• Worst case A 1, B n-1
• T(n) T(n-1) O(n) T(n) O(n2)

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Divide and Conquer Algorithm

• Computing 3D hull
• Graham scan (probably not ?angle sort)
• Preparata and Hong (1977)

a-1 a1 at left side of ab b-1 b1 at left
side of ab
T(n) 2T(n/2) O(n) T(n) O(n log n)
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Extend to 3D
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Floating Point Inaccuracies

• Handled by
• Interval arithmetic
• Exact (rational) arithmetic

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Interval arithmetic

• Ref http//www.eng.mu.edu/corlissg/VC02/READ_ME.h
  tml

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Exact Math

• Rational arithmetic (never floating point)
• In6 1/12 - 7/9 31/36
• Out6 - 1/6
• The Exact Arithmetic Competition Level 0 Tests
• Ref XR

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Exercise

• From this applet, develop a Graham scan algorithm
  that returns a CW-ordered convex hull
• Express the algorithm in pseudocode. Make sure it
  works for general position and degenerate cases
  (shown right)
• Run your example on test cases (general position
  and degeneracies)

• Explain the correctness and time complexity of
  divide-and-conquer algorithm

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More reference on the web.