95 lines
3.0 KiB
Plaintext
95 lines
3.0 KiB
Plaintext
---
|
|
format: markdown
|
|
toc: yes
|
|
title: Computational Geometry Study Notes
|
|
...
|
|
|
|
# Texts
|
|
|
|
- *Discrete and Computational Geometry*, Devadoss
|
|
|
|
# Reading Notes
|
|
|
|
## *Devadoss*, 24 Jan 2014
|
|
|
|
Computational Geometry is *discrete* rather than *continuous*
|
|
|
|
Fundamental building blocks are the *point* and line *segment*.
|
|
|
|
*polygon*
|
|
: the closed region of the plane bounded by a finite collection of
|
|
line segments forming a closed curve that does not intersect
|
|
itself. The segments are called *edges* and the points where they meet
|
|
are *vertices*. The set of vertices and edges is the *boundary*.
|
|
|
|
**Theorem 1.1: Polygonal Jordan Curve**
|
|
:The boundary $\partial P$ of a polygon $P$ partitions the plane into
|
|
two parts. In particular, the two components of $\Bbb{R}^2\setminus
|
|
\partial P$ are the bounded interior and the unbounded exterior.
|
|
|
|
A point $x$ is *interior* if a ray through it in a fixed direction not
|
|
parallel to an edge passes through an odd number of edges.
|
|
|
|
A point $x$ is *exterior* if a ray through it ... etc ... passes
|
|
through an even number of edges.
|
|
|
|
These two facts form the basis of an algorithm for determining whether
|
|
a point is inside a polygon.
|
|
|
|
*diagonal*
|
|
: A line segment of $P$ connecting two vertices and lying in the interior of
|
|
$P$, not touching $\partial P$ except at endpoints. Two diagonals are
|
|
*noncrossing* if they share no interior points.
|
|
|
|
*triangulation*
|
|
: 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*.
|
|
|
|
**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 $$ C_n = \frac{1, n + 1}{n + 1 \choose
|
|
n} $$
|