75 lines
2.3 KiB
Plaintext
75 lines
2.3 KiB
Plaintext
---
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format: markdown
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toc: yes
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title: Computational Geometry Study Notes
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...
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# Texts
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- *Discrete and Computational Geometry*, Devadoss
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# Reading Notes
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## *Devadoss*, 24 Jan 2014
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Computational Geometry is *discrete* rather than *continuous*
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Fundamental building blocks are the *point* and line *segment*.
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*polygon*
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: the closed region of the plane bounded by a finite collection of
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line segments forming a closed curve that does not intersect
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itself. The segments are called *edges* and the points where they meet
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are *vertices*. The set of vertices and edges is the *boundary*.
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**Theorem 1.1: Polygonal Jordan Curve**
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:The boundary $\partial P$ of a polygon $P$ partitions the plane into
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two parts. In particular, the two components of $\Bbb{R}^2\setminus
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\partial P$ are the bounded interior and the unbounded exterior.
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A point $x$ is *interior* if a ray through it in a fixed direction not
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parallel to an edge passes through an odd number of edges.
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A point $x$ is *exterior* if a ray through it ... etc ... passes
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through an even number of edges.
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These two facts form the basis of an algorithm for determining whether
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a point is inside a polygon.
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*diagonal*
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: A line segment of $P$ connecting two vertices and lying in the interior of
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$P$, not touching $\partial P$ except at endpoints. Two diagonals are
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*noncrossing* if they share no interior points.
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*triangulation*
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: A decomposition of $P$ into triangles by a maximal set of
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noncrossing diagonals. Maximal means that no more diagonals may be
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added without crossing.
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**Lemma 1.3**
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: Every polygon with more than three vertices has a diagonal.
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**Theorem 1.4**
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: Every polygon has a triangulation.
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*polyhedron*
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: A 3-d generalization of a polygon, a solid bounded by finitely many
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polygons.
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*tetrahedron*
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: A pyramid with a triangular base. The simplest polyhedron.
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Polygon triangularization generalizes to polyhedron
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*tetrahedralization*, which is partitioning of the interior into
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tetrahedrons whose edges are diagonals of the polyhedron. Not all
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polyhedrons can be tetrahedralized!
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**Theorem 1.8**
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: Every triangularization of a polygon $P$ with $n$ vertices has $n -
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2$ triangles and $n - 3$ diagonals.
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*ear*
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: Three consecutive vertices $a, b, c$ form an *ear* of a polygon if
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$a c$ is a diagonal of the polygon. The vertex $b$ is called the ear
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*tip*.
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