Gift wrapping algorithm

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The gift wrapping algorithm is a simple algorithm for computing the convex hull of a given set of points.

[edit] Planar case

In the two-dimensional case the algorithm is also known as Jarvis march, by the name of the author, and has O(nh) time complexity, where n is the number of points and h is the number of points on the convex hull. Its real-life performance compared with other convex hull algorithms is favorable when n is small or h is expected to be very small with respect to n. In general case the algorithm is outperformed by many others.

Jarvis march computing the convex hull.

The gift wrapping algorithm begins with i=0 and a point p₀ known to be on the convex hull, e.g., the leftmost point, and selects the point p_i+1 such that all points are to the right of the line p_i p_i+1. This point may be found on O(n) time by comparing polar angles of all points with respect to point p₀ taken for the center of polar coordinates. Letting i=i+1, and repeating with until one reaches p_h=p₀ again yields the convex hull in h steps. The gift wrapping algorithm is exactly analogous to the process of winding a string (or wrapping paper) around the set of points.

def jarvis(P)
  i = 0
  p[0] = leftmost point of P
  do
    p[i+1] = point such that all other points in P are to the 
                                 right of the line p[i]p[i+1]
    i = i + 1
  while p[i] != p[0]
  return p

The approach is extendable to higher dimensions.