From ec9c2eb3a26a603a67429ba70bee4eb74baad6c6 Mon Sep 17 00:00:00 2001 From: Levi Pearson Date: Sat, 25 Jan 2014 00:25:39 -0700 Subject: [PATCH] More notes --- math/computational_geometry/notes.page | 27 ++++++++++++++++++++++++++ 1 file changed, 27 insertions(+) diff --git a/math/computational_geometry/notes.page b/math/computational_geometry/notes.page index 88ec5dc..11212b3 100644 --- a/math/computational_geometry/notes.page +++ b/math/computational_geometry/notes.page @@ -45,3 +45,30 @@ $P$, not touching $\partial P$ except at endpoints. Two diagonals are : A decomposition of $P$ into triangles by a maximal set of noncrossing diagonals. Maximal means that no more diagonals may be added without crossing. + +**Lemma 1.3** +: Every polygon with more than three vertices has a diagonal. + +**Theorem 1.4** +: Every polygon has a triangulation. + +*polyhedron* +: A 3-d generalization of a polygon, a solid bounded by finitely many +polygons. + +*tetrahedron* +: A pyramid with a triangular base. The simplest polyhedron. + +Polygon triangularization generalizes to polyhedron +*tetrahedralization*, which is partitioning of the interior into +tetrahedrons whose edges are diagonals of the polyhedron. Not all +polyhedrons can be tetrahedralized! + +**Theorem 1.8** +: Every triangularization of a polygon $P$ with $n$ vertices has $n - +2$ triangles and $n - 3$ diagonals. + +*ear* +: 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*.