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Commit b1553795 authored by Vincent Nivoliers's avatar Vincent Nivoliers
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presentation up to perturbation

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Code/hull.cpp
+ 10
− 6
View file @ b1553795
......@@ -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) ;
}
......
Presentation/Figures/hull.svg
+ 244
− 45
View file @ b1553795
......@@ -27,6 +27,92 @@
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Presentation/Figures/orient.pdf 0 → 100644
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Presentation/presentation.tex
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......@@ -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}
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