79 lines
1.6 KiB
Markdown
79 lines
1.6 KiB
Markdown
---
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title: Convex Hull
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tags:
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- authorship/other-for-now
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- type/encyclopedia-entry
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---
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# Convex Hull
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In [[geometry]],
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the **convex hull** of a shape
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(also **convex envelope** or **convex closure**)
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is the smallest [convex set](https://en.wikipedia.org/wiki/Convex_set "Convex set")
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that contains it.[^1]
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[^1]: [Wikipedia](https://en.wikipedia.org/wiki/Convex_hull)
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```tikz
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\usepackage{pgfplots}
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\pgfplotsset{compat=1.16}
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\begin{document}
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\begin{tikzpicture}
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\begin{axis}[
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axis equal,
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xmin=0, xmax=3.5,
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ymin=0, ymax=3.4,
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grid=both,
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xlabel={$x$}, ylabel={$y$},
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tick style={black!60},
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title={Convex Hull of a Point Set}
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]
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% Point cloud
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\addplot[
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only marks,
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mark=*,
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mark size=1.8pt
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] coordinates {
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(0.5,0.5)
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(1.0,2.0)
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(2.0,1.0)
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(2.5,2.2)
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(3.0,0.5)
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(1.8,3.0)
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(0.2,2.5)
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(3.2,1.5)
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};
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% Convex hull (listed in counterclockwise order)
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\addplot[
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thick,
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fill opacity=0.15,
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fill=blue
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] coordinates {
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(0.5,0.5)
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(3.0,0.5)
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(3.2,1.5)
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(1.8,3.0)
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(0.2,2.5)
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} -- cycle;
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% label hull vertices
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\addplot[only marks, mark=*, mark size=2.2pt] coordinates {
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(0.5,0.5)
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(3.0,0.5)
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(3.2,1.5)
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(1.8,3.0)
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(0.2,2.5)
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};
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\node[anchor=north west] at (axis cs:0.5,0.5) {$H_1$};
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\node[anchor=north east] at (axis cs:3.0,0.5) {$H_2$};
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\node[anchor=west] at (axis cs:3.2,1.5) {$H_3$};
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\node[anchor=south] at (axis cs:1.8,3.0) {$H_4$};
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\node[anchor=east] at (axis cs:0.2,2.5) {$H_5$};
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\end{axis}
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\end{tikzpicture}
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\end{document}
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```
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