Changes

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 $$ C_n = \frac{1, n + 1}{n + 1 \choose
+n} $$