Handouts: HW5, course survey
3/25/09: More on convex hulls in 3 or more dimensions

We return to discussing convex hulls in 3 or more dimensions, stating that the time it takes to compute a hull in d dimensions for d>=4 is O(n^{floor(d/2)}), which is the worst-case complexity of the output size (ie., the hull may have that many faces).
Combinatorics: planar graphs, Euler's formula, linearity of the number of edges and faces (as a function of n, the number of vertices).
There is a nice simple proof of Euler's formula based on the "dual trees" method: Consider any spanning tree, T, of the n vertices of the graph. Let E' be the edges of the graph that are NOT in T. Then, the duals to E' form a spanning tree, T', in the dual graph (a cycle would disconnect T, and two connected components would imply a cycle in T). Now, T has n-1 edges and T' has f-1 edges, so the total number of edges is e=n-1+f-1. This is Euler's formula.
Platonic solids: The book describes the platonic solids in 2, 3, 4, and more dimensions. In 3D, it is easy to show (as we do in class) that there are exactly 5 different Platonic solids. (I brought Zometool models of them to the next class.)
Data structures to represent a polyhderon: I briefly discussed the winged edge data structure (O'Rourke) and the doubly-connected edge list (DCEL) from the graduate text (BCKO).
I started to introduce convex hull algorithms in 3 or more dimensions. In 3D, the convex hull can be computed in time O(n log n), which is worst-case optimal in terms of n. (There is also an "utlimate" convex hull bound of O(n log h), due to T. Chan.)