Two Convex Hull Algorithms

• Graham's Scan
  • Observe, that if we travel along the convex hull, we will only make left (or right) hand turns.
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  • Calculating if it takes a left or right turn to traverse along a line can be done as cross product.
    • a x b = |a||b|sin(θ) n.
    • Where θ is the angle between the two vectors.
    • and n is the normal vector a and b.
  • This algorithm works by keeping a stack of potential vertexes on the hull.
  • First find the lowest, then left most point (P1 in this example)
  • Then sort all of the other points by polar angle with P1
    • This is the angle in polar coordinate system.
    • x axis is 0
    • y axis is π/2
    • If the angles are the same, drop the closest point.
  • In this case, p2 will be dropped since it is co linear with p1 and p3
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  • Push the first three points onto the stack.
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    • We will assume a modified stack with the following operations
    • TOP() returns the top element on the stack, but does not change the stack.
    • NEXT_TO_TOP() returns the second element on the stack.
  • for all other points p in order
    • while p, S.TOP() and S.NEXT_TO_TOP() make a non left hand turn
    • S.POP()
  • 5-4-3 make a right hand turn so pop 4
  • 5-3-1 do not make a right hand turn, so push 5
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  • Nothing interesting happens at point 6.
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  • Nor at 7,8 and 9.
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  • But at point 10, we end up popping off 9,8 and 7
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  • Which leads to
  • 
  • And it ends with
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  • GRAHAM_SCAN(Q)
    
     Initialize a stack S
     p1 ← lowest-leftmost point in Q
     POLAR_SORT(Q,p1)
     Q.push(p1)
     Q.push(Q[2])
     Q.push(Q[3])
     for i ← 4 to n do
       p ← Q[i]
       while not IS_LEFT_TURN(p, S.TOP(),S.NEXT_TO_TOP()) do
           S.POP()
       s.PUSH(p)
     return S
    
    	

  • Terminate?
  • Work?
  • Performance?
• Jarvis March
  • Also known as the gift wrapping algorithm.
  • Remove co linear points again.
  • p₀ ← find the lowest point in the set.
  • While we have not returned to the starting point
    • Select the point which makes the minimal polar angle with respect to the last line (or x axis for first point.
  • 3 is selected first.
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  • 5 is next
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  • Then 6
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  • And ends here:
  • 

• JARVIS_MARCH(Q)
  
   p ← FIND_MINIMUM(Q)
   plast ← p-(1,0)
   L.insert(p)
   do
       tmp ← FIND_MINIMAL_POLAR_ANGLE(plast,p, Q)
       if tmp != L[0]
           L.insert(tmp)
       plast ← p
       p ← tmp;
    while p != L[0]