From a6c9fde4254a9e3961666f1c69a9ac3dbdf72c67 Mon Sep 17 00:00:00 2001 From: Levi Pearson Date: Sat, 25 Jan 2014 00:36:19 -0700 Subject: [PATCH] Changes --- math/computational_geometry/notes.page | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) diff --git a/math/computational_geometry/notes.page b/math/computational_geometry/notes.page index 11212b3..7efa6b7 100644 --- a/math/computational_geometry/notes.page +++ b/math/computational_geometry/notes.page @@ -72,3 +72,23 @@ polyhedrons can be tetrahedralized! : Three consecutive vertices $a, b, c$ form an *ear* of a polygon if $a c$ is a diagonal of the polygon. The vertex $b$ is called the ear *tip*. + +**Corollary 1.9** +: Every polygon with more than three vertices has at least two ears. + +A vertex of a polygon is *reflex* if its angle is greater than $\pi$ +and *convex* if its angle is less than or equal to $\pi$. It is *flat* +if its angle is exactly $\pi$ and *strictly convex* if its angle is +strictly less than $\pi$. + +A polygon $P$ is a *convex polygon* if all of its vertices are +strictly convex. + +**Lemma 1.18** +: A diagonal exists between any two nonadjacent vertices of a polygon +$P$ if and only if $P$ is a convex polygon. + +**Theorem 1.19** +: The number of triangulations of a convex polygon with $n + 2$ +vertices is the Catalan number $$ C_n = \frac{1, n + 1}{n + 1 \choose +n} $$