From 4997461b9c772efe1fc1bd176f25be5b43256259 Mon Sep 17 00:00:00 2001
From: Vincent Nivoliers <vincent.nivoliers@univ-lyon1.fr>
Date: Mon, 18 Nov 2019 14:21:57 +0100
Subject: [PATCH] added sos
---
Presentation/Figures/generalized_orient.pdf | Bin 0 -> 5800 bytes
Presentation/Figures/generalized_orient.svg | 311 ++++++++++++++++++++
Presentation/presentation.tex | 83 ++++++
3 files changed, 394 insertions(+)
create mode 100644 Presentation/Figures/generalized_orient.pdf
create mode 100644 Presentation/Figures/generalized_orient.svg
diff --git a/Presentation/Figures/generalized_orient.pdf b/Presentation/Figures/generalized_orient.pdf
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diff --git a/Presentation/presentation.tex b/Presentation/presentation.tex
index a943089..8374480 100644
--- a/Presentation/presentation.tex
+++ b/Presentation/presentation.tex
@@ -225,4 +225,87 @@
}
}
+\frame{
+ \frametitle{Calcul exact}
+ \begin{center}
+ \begin{myblock}{0.8\li}
+ \centering
+ Il peut arriver que le déterminant ne soit pas nul\\
+ mais que l'arrondi en virgule flottante le soit.
+ \end{myblock}
+ \end{center}
+
+ \visible<2>{
+ \vspace{8mm}
+ \textbf{Calculer en précision exacte :}
+ \vspace{3mm}
+ \begin{itemize}
+ \item entiers sans limite de taille ;
+ \item nombre rationels ;
+ \item opérations dont le résultat est rationnel (pas de racine carrée !) ;
+ \item le prédicat ne doit prendre en entrée que des nombres initiaux.
+ \end{itemize}
+ }
+}
+
+\frame{
+ \frametitle{Généralisation de l'orientation}
+ \centering
+ \includegraphics[width=0.3\li]{Figures/generalized_orient}
+
+ \vspace{8mm}
+ \begin{myblock}{0.8\li}
+ \centering
+ examiner le signe de :
+ $
+ \det{
+ \begin{pmatrix}
+ x_0 & y_0 & 1 \\
+ x_1 & y_1 & 1 \\
+ x_2 & y_2 & 1
+ \end{pmatrix}
+ }
+ $
+ \end{myblock}
+}
+
+\frame{
+ \frametitle{Simulation de simplicité (Edelsbrnner \& Mücke 90)}
+
+ \begin{center}
+ \begin{myblock}{0.8\li}
+ Utiliser des \eps{} avec une puissance \hl{exponentielle} en $i$ et $j$ :
+ \begin{equation*}
+ \eps_{i,j} = \eps^{2^{i*d - j}}
+ \end{equation*}
+ \end{myblock}
+ \end{center}
+
+ \vspace{5mm}
+ \textbf{Principe :}
+ \begin{itemize}
+ \item $\sum_{k=0}^n 2^k = 2^{k+1} - 1$
+ \item $\prod_{k=0}^n\eps^{2^k} = \eps^{\sum_{k=0}^n2^k} = \eps^{2^{k+1} - 1} >> \eps^{2^{k+1}}$
+ \end{itemize}
+}
+
+\frame{
+ \frametitle{Extraire les termes}
+
+ \begin{equation*}
+ \begin{pmatrix}
+ p_{0,0} + \eps{0,0} & p_{0,1} + \eps{0,1} & \cdots & p_{0,d} + \eps{0,d-1} & 1 \\
+ p_{1,0} + \eps{1,0} & p_{1,1} + \eps{1,1} & \cdots & p_{1,d} + \eps{1,d-1} & 1 \\
+ \vdots & \vdots & \ddots & \vdots \\
+ p_{d,0} + \eps{d,0} & p_{d,1} + \eps{d,1} & \cdots & p_{d,d-1} + \eps{d,d-1} & 1
+ \end{pmatrix}
+ \end{equation*}
+
+ \vspace{5mm}
+ \begin{itemize}
+ \item pour chaque point aligner la perturbation voulue sur la diagonale ;
+ \item développement par les cofacteurs.
+ \end{itemize}
+}
+
\end{document}
--
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