From b155379587fd8573cbff37d3e2203f2d81f006c8 Mon Sep 17 00:00:00 2001
From: Vincent Nivoliers <vincent.nivoliers@univ-lyon1.fr>
Date: Mon, 18 Nov 2019 13:35:06 +0100
Subject: [PATCH] presentation up to perturbation
---
Code/hull.cpp | 16 +-
Presentation/Figures/hull.svg | 289 +++++++++++++---
Presentation/Figures/orient.pdf | Bin 0 -> 6438 bytes
Presentation/Figures/orient.svg | 396 ++++++++++++++++++++++
Presentation/Figures/sort_comparison.pdf | Bin 0 -> 8865 bytes
Presentation/Figures/sweep_comparison.pdf | Bin 0 -> 9288 bytes
Presentation/presentation.tex | 161 ++++++++-
7 files changed, 803 insertions(+), 59 deletions(-)
create mode 100644 Presentation/Figures/orient.pdf
create mode 100644 Presentation/Figures/orient.svg
create mode 100644 Presentation/Figures/sort_comparison.pdf
create mode 100644 Presentation/Figures/sweep_comparison.pdf
diff --git a/Code/hull.cpp b/Code/hull.cpp
index 25249b6..00c06ba 100644
--- a/Code/hull.cpp
+++ b/Code/hull.cpp
@@ -40,14 +40,18 @@ struct vec {
//}}}
+int sign(float x) {
+ if(x > 0) return 1 ;
+ if(x < 0) return -1 ;
+ return 0 ;
+}
+
int orient(const vec& v0, const vec& v1) {
//determinant
float d = v0.x*v1.y - v0.y*v1.x ;
//encode as sign
- if(d > 0) return 1 ;
- if(d < 0) return -1 ;
- return 0 ;
+ return sign(d) ;
}
struct vec_compare {
@@ -90,13 +94,13 @@ void compute_hull(std::vector<vec>& points, std::vector<vec>& hull) {
for(size_t i = 2; i < points.size(); ++i) {
//check whether the last segment creates a wrong turn
const vec& p = points[i] ;
- vec v1 = p - *(hull.end() - 1) ;
+ vec v1 = p - *(hull.end() - 2) ;
vec v2 = *(hull.end() - 1) - *(hull.end() - 2) ;
while(hull.size() > 1 && orient(v1, v2) < 0) {
//in case of wrong turn, rease the last point of the hull and iterate
hull.erase(hull.end() - 1) ;
- v1 = p - *(hull.end() - 1) ;
- v2 = p - *(hull.end() - 2) ;
+ v1 = p - *(hull.end() - 2) ;
+ v2 = *(hull.end() - 1) - *(hull.end() - 2) ;
}
hull.push_back(p) ;
}
diff --git a/Presentation/Figures/hull.svg b/Presentation/Figures/hull.svg
index 7a44f37..e177c81 100644
--- a/Presentation/Figures/hull.svg
+++ b/Presentation/Figures/hull.svg
@@ -27,6 +27,92 @@
</metadata>
<defs
id="defs1894">
+ <marker
+ inkscape:isstock="true"
+ style="overflow:visible;"
+ id="marker5177"
+ refX="0.0"
+ refY="0.0"
+ orient="auto"
+ inkscape:stockid="Arrow2Send">
+ <path
+ transform="scale(0.3) rotate(180) translate(-2.3,0)"
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+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ id="path5175" />
+ </marker>
+ <marker
+ inkscape:isstock="true"
+ style="overflow:visible;"
+ id="marker4465"
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+ inkscape:stockid="Arrow2Mend">
+ <path
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+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ id="path4463" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Mend"
+ orient="auto"
+ refY="0.0"
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+ style="overflow:visible;"
+ inkscape:isstock="true"
+ inkscape:collect="always">
+ <path
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+ transform="scale(0.6) rotate(180) translate(0,0)" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Mstart"
+ orient="auto"
+ refY="0.0"
+ refX="0.0"
+ id="Arrow2Mstart"
+ style="overflow:visible"
+ inkscape:isstock="true">
+ <path
+ id="path841"
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
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+ transform="scale(0.6) translate(0,0)" />
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+ <marker
+ inkscape:isstock="true"
+ style="overflow:visible;"
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+ refX="0.0"
+ refY="0.0"
+ orient="auto"
+ inkscape:stockid="Arrow2Mend">
+ <path
+ transform="scale(0.6) rotate(180) translate(0,0)"
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+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
+ id="path3853" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Mend"
+ orient="auto"
+ refY="0.0"
+ refX="0.0"
+ id="marker3809"
+ style="overflow:visible;"
+ inkscape:isstock="true"
+ inkscape:collect="always">
+ <path
+ id="path844"
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#000000;stroke-opacity:1;fill:#000000;fill-opacity:1"
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+ transform="scale(0.6) rotate(180) translate(0,0)" />
+ </marker>
<marker
inkscape:stockid="Arrow2Send"
orient="auto"
@@ -34,7 +120,8 @@
refX="0.0"
id="Arrow2Send"
style="overflow:visible;"
- inkscape:isstock="true">
+ inkscape:isstock="true"
+ inkscape:collect="always">
<path
id="path2197"
style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#f57900;stroke-opacity:1;fill:#f57900;fill-opacity:1"
@@ -55,6 +142,21 @@
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transform="scale(0.6) rotate(180) translate(0,0)" />
</marker>
+ <marker
+ inkscape:stockid="Arrow2Mstart"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow2Mstart-2"
+ style="overflow:visible"
+ inkscape:isstock="true">
+ <path
+ inkscape:connector-curvature="0"
+ id="path841-8"
+ style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
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+ transform="scale(0.6)" />
+ </marker>
</defs>
<sodipodi:namedview
pagecolor="#ffffff"
@@ -70,10 +172,11 @@
id="namedview1892"
showgrid="false"
inkscape:zoom="1.3796869"
- inkscape:cx="454.51793"
- inkscape:cy="483.23057"
- inkscape:current-layer="layer6"
- inkscape:snap-object-midpoints="true" />
+ inkscape:cx="224.13977"
+ inkscape:cy="621.88734"
+ inkscape:current-layer="layer7"
+ inkscape:snap-object-midpoints="true"
+ inkscape:snap-global="true" />
<g
inkscape:groupmode="layer"
id="layer4"
@@ -780,7 +883,7 @@
inkscape:groupmode="layer"
id="layer6"
inkscape:label="partial_hull"
- style="display:inline">
+ style="display:none">
<circle
cx="56"
cy="481"
@@ -824,67 +927,68 @@
inkscape:connector-curvature="0" />
</g>
<g
- inkscape:groupmode="layer"
- id="layer7"
- inkscape:label="adding_point_1"
- style="display:inline">
+ style="display:none"
+ inkscape:label="adding_point_1 copy"
+ id="g4215"
+ inkscape:groupmode="layer">
<path
- inkscape:connector-curvature="0"
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<circle
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style="display:inline;fill:none;stroke:#000000;stroke-width:2"
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+ id="circle4199" />
<circle
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r="12"
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r="12"
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inkscape:connector-curvature="0"
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inkscape:groupmode="layer"
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@@ -1086,6 +1190,101 @@
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diff --git a/Presentation/Figures/orient.pdf b/Presentation/Figures/orient.pdf
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diff --git a/Presentation/presentation.tex b/Presentation/presentation.tex
index d70438f..a943089 100644
--- a/Presentation/presentation.tex
+++ b/Presentation/presentation.tex
@@ -25,29 +25,29 @@
\frame {
\frametitle{Enveloppe convexe}
\centering
- \includegraphics[width=0.6\li]{Figures/hull.pdf}
+ \includegraphics[width=0.6\li]{Figures/hull}
}
\frame {
\frametitle{1 -- trier par angle}
\centering
- \includegraphics[width=0.6\li]{Figures/hull_sort.pdf}
+ \includegraphics[width=0.6\li]{Figures/hull_sort}
}
\frame {
\frametitle{2 -- balayage}
\centering
- \only<1>{\includegraphics[width=0.6\li]{Figures/partial_hull.pdf}}%
- \only<2>{\includegraphics[width=0.6\li]{Figures/increment_hull_1.pdf}}%
- \only<3>{\includegraphics[width=0.6\li]{Figures/increment_hull_2.pdf}}%
- \only<4>{\includegraphics[width=0.6\li]{Figures/increment_hull_3.pdf}}%
- \only<5>{\includegraphics[width=0.6\li]{Figures/hull.pdf}}%
+ \only<1>{\includegraphics[width=0.6\li]{Figures/partial_hull}}%
+ \only<2>{\includegraphics[width=0.6\li]{Figures/increment_hull_1}}%
+ \only<3>{\includegraphics[width=0.6\li]{Figures/increment_hull_2}}%
+ \only<4>{\includegraphics[width=0.6\li]{Figures/increment_hull_3}}%
+ \only<5>{\includegraphics[width=0.6\li]{Figures/hull}}%
}
\frame {
\frametitle{Robustesse : alignements de points}
\centering
- \includegraphics[width=0.6\li]{Figures/hull_aligned.pdf}
+ \includegraphics[width=0.6\li]{Figures/hull_aligned}
}
\frame {
@@ -80,4 +80,149 @@
\end{ucblalgo}
}
+\section{Perturbation}
+
+\frame{
+ \frametitle{Prédicat : orientation}
+ \centering
+ \includegraphics[width=0.5\li]{Figures/orient}
+
+ \vspace{8mm}
+ \begin{myblock}{0.8\li}
+ \centering
+ $
+ \det{
+ \left(
+ \begin{array}{cc}
+ x_0 & x_1 \\
+ y_0 & y_1 \\
+ \end{array}
+ \right)
+ }
+ $ : sinus de l'angle si vecteurs normalisés.
+
+ \hl{positif, négatif ou nul}
+ \end{myblock}
+}
+
+\frame{
+ \frametitle{Utilisation pour l'enveloppe convexe}
+ \centering
+ \only<1>{\includegraphics[width=0.6\li]{Figures/sort_comparison}}%
+ \only<2>{\includegraphics[width=0.6\li]{Figures/sweep_comparison}}
+}
+
+\newcommand{\eps}{\ensuremath{\varepsilon}}
+
+\frame{
+ \frametitle{Perturbation}
+
+ \begin{align*}
+ \det{
+ \left(
+ \begin{array}{cc}
+ x_0 + \eps_{0,0} & x_1 + \eps_{1,0} \\
+ y_0 + \eps_{0,1} & y_1 + \eps_{1,1} \\
+ \end{array}
+ \right)
+ }
+ &=
+ \det{
+ \left(
+ \begin{array}{cc}
+ x_0 & x_1 \\
+ y_0 & y_1 \\
+ \end{array}
+ \right)
+ } \\
+ &
+ + \eps_{0,0} y_1
+ + \eps_{1,1} x_0
+ - \eps_{0,1} x_1
+ - \eps_{1,0} y_0
+ \\
+ &
+ + \eps_{0,0}\eps_{1,1}
+ - \eps_{0,1}\eps_{1,0}
+ \end{align*}
+
+ \vspace{5mm}
+ \begin{myblock}{0.8\li}
+ \begin{enumerate}
+ \item systématiquement plus de zéro ?
+ \item facilité d'évaluation ?
+ \end{enumerate}
+ \end{myblock}
+}
+
+\frame{
+ \frametitle{Rester dans le pétrin}
+
+ Posons que $ \forall(i,j), \eps_{i,j} = \eps > 0$
+
+ \begin{equation*}
+ \eps y_1
+ + \eps x_0
+ - \eps x_1
+ - \eps y_0
+ + \eps\eps
+ - \eps\eps
+ \end{equation*}
+
+ \vspace{5mm}
+ Si $(x_0,y_0) = (0,1)$ et $(x_1,y_1) = (0,2)$
+
+ \begin{equation*}
+ 2\eps
+ - \eps
+ + \eps\eps
+ - \eps\eps
+ = \eps
+ \quad \Rightarrow \mbox{positif !}
+ \end{equation*}
+
+ \visible<2>{
+ \vspace{5mm}
+ Si $(x_0,y_0) = (1,1)$ et $(x_1,y_1) = (2,2)$
+
+ \begin{equation*}
+ 2\eps
+ + \eps
+ - 2\eps
+ - \eps
+ + \eps\eps
+ - \eps\eps
+ = 0
+ \quad \Rightarrow \mbox{arf.}
+ \end{equation*}
+ }
+}
+
+\frame{
+ \frametitle{Sortir du pétrin}
+
+ Avec la perturbation $ \begin{pmatrix}\eps^1 & \eps^2 \\ \eps^2 & \eps^2\end{pmatrix}$
+
+ \begin{align*}
+ & \eps^1 y_1
+ + \eps^2 x_0
+ - \eps^2 x_1
+ - \eps^2 y_0
+ + \eps^1\eps^2
+ - \eps^2\eps^2 \\
+ &= \eps y_1 + \eps^2(x_0 - x_1 - y_0) + \eps^3 - \eps^4
+ \end{align*}
+
+ \vspace{8mm}
+ \visible<2->{
+ \begin{myblock}{0.8\li}
+ \begin{itemize}
+ \item<1-> si $y_1$ est non nul, renvoyer son signe ;
+ \item<3-> si $x_0 - x_1 - y_0$ est non nul, renvoyer son signe ;
+ \item<4-> sinon renvoyer positif.
+ \end{itemize}
+ \end{myblock}
+ }
+}
+
\end{document}
--
GitLab