Incremental Algorithm

The main motivation to study an incremental algorithm for convex hulls is to eventually develop an algorithm for 3D. It is hard to extend Graham's algorithm to 3D.

The main ideas behind the incremental algorithms are:

1. Add the points one at a time. Initially we start with an empty set.
2. At each step construct the hull of the first k points. Use that hull to incorporate the next point.

Given and we assume that no three points are collinear.

The first hull is the triangle corresponding to convex hull of . After that, the algorithm proceeds inductively. Let correspond to the convex hull of and let . The computation of the convex hull of naturally falls into two cases, depending on whether or does not lie in .

1. : discard .
2. does not lie in : To compute convex hull of , find 2 lines of tangency from to . These lines of tangency and the corresponding intersection points on are part of convex hulls of .

Complexity: .