Handouts:
1/30/08: Polygons, visibility, and art galleries
I define basic terminology: simple polygon, Jordan curve theorem, visibility, clear visibiblity.
Definition of guards, visibility polygon, VP(p).
Definition of guard number, g(P), and vertex guard number, g_V(P). We also define G(n) = max g(P), where the max is over all simple n-gons P.
Computing G(3), G(4), G(5), G(6), G(7), etc. Examples.
Notion of independent "witness points" used to argue that a polygon needs at least a certain number of guards: If we can find k independent witness points (meaning that the visibility polygons are pairwise disjoint), then we know that g(P) is AT LEAST k.
Statement of Art Gallery Theorem; example of Chvatal comb, which shows that sometimes floor(n/3) guards are necessary to guard an n-gon.