The Convex Hull

• Given a set of points $P=\{p_0, p_1, ... p_n\}$
  • $|P| \ge 3$
  • At least three points in P are not co-linear
  • Otherwise you can not form a polygon.
• Find the smallest convex polygon CH(P) such that for all $p \in P$,
  • p is in the interior of CH(P)
  • or p is on CH(P).
• Intuitively you might think about
  • The points are nails in a board.
  • The hull is a rubber band "snapped" over the nails.
• This leads to the fact at if $p\in$ CH(P) then $p \in P$
• The Graham's Scan Algorithm.
  • Define Angle(p1, p2, p3) to compute the angle between these three points.
  • Define IsNonLeftTurn(p1, p2, p3) to return true if $\overrightarrow{p_1p_2} $ to $\overrightarrow{p_2p_3}$ is a non left turn.

  • 
    GRAHAM_SCAN(P)
     Let lowest be the left most lowest point in P.
     Let right = (p.x+10, p.y)
     For each p in P
        p.angle = Angle(right, lowest, p)
     End For
     Scan P remove the closest points with duplicate angles.
     Sort P by p.angle
     S.push(P[0]) // lowest
     S.push(P[1])
     S.push(P[2])
     For i = 3 to P.size do
        While IsNonLeftTurn(S[top-1], S[top], P[i])
            S.pop()
        End While
        S.push(P[i])
     End For
     return S.
     

• Performance?
• Correct?
  • Let $Q=\{lowest, q_1, q_2, ... q_m\}$ be the points that remain after line 6.
  • lowest must be on the hull, and is placed on the stack
  • The points $q_1$ and $q_2$ are on the stack after line 10.
    • At this point the convex hull of the first three points has been computed.
    • This is because they have been sorted in counter clockwise order.
  • Consider $i = k$, or point $q_k$ in line 12-14
    • let $s_t$ be the top point on the stack
    • let $s_{t-1}$ be the next to top point on the stack.
    • When we remove a point $s_t$ in line 13
      • We know that it makes a right hand turn.
      • And this will make the polygon non-convex when considering the next available point.
      • Or the point removed $s_t$ is to the left of the line segment $\overrightarrow{s_{t-1}q_k}$
      • Thus is it inside of the hull
  • So when we add a point in line 15, the partial convex hull is preserved.