Some Basic Computational Geometry

• We frequent need to determine the relationship between two points and a line.
• The Cross Product of two vectors is helpful here.
• Consider points p₁ = (x₁, y₁) and p₂ = (x₂, y₂)
• 

                    |x1   x2 |
      p1 × p2  =det |y1   y2 |
      

• p₁ × p₂ = - p₂ × p₁
• p₁ × p₂ = |a||b| sin(Θ)n
  • Where n is a normal vector.
• CLRS uses the formula p₁ × p₂ = x₁y₂ - x₂y₁
• So the sign of the cross product can identify
  • We can determine if we make a left or right hand turn traveling from p₁ to p₀ to p₂
  • Or clockwise or counter clockwise turn.
  • Or even the relation of a point to a segment.
  • Consider
    • A = (1,0), B = (0,1)
    • A x B = 1*1 - 0*0 = 1
    • B x A = 0*0 - 1*1 = -1
  • We can also use this to test for colinearity
  • And to see if lines intersect.
• So back to Levitin
• QuickHull
  • An alternative approach to the Convex Hull Problem.
  • Let S be a set of points (p₁, p₂ ..., p_n)
  • Sort the points in non-decreasing order on the x coordinate.
    • Use the y coordinate as a secondary key.
    • This can be done in O(n lg(n)).
    • The first element (p₁) will be the lowest left
    • The last (p_n) will be the upper right
    • 
    • These must each be on the convex hull.
    • Create the segment (p₁,p_n)
    • it should (theoretically divide the points remaining in half.
    • 
    • Any points on the segment (p₁,p_n) can not be on the hull.
    • So divide the points into two sets S₁ and S₂
    • 
    • If either is empty, then (p₁,p_n)is part of the convex hull.
    • If not, compute the upper hull and the lower hull as follows.
      • Find the point p_max which has the maximum distance from (p₁,p_n).
        • This can be done in constant time.
        • In a tie, find the point that maximizes the angle p_maxp₁p_n
      • p_max is on the hull.
      • Divide the plane into three sets
        • 
        • One set can be eliminated
        • Two sets will still contain points on the hull.
      • Repeat this process with p₁p_max and with p_maxp_n
  • Questions...
    • Does it work correctly?
    • Worst case?
    • Best Case?
    • What is the output of this algorithm?