From c524232f734de875d69e2f190f01a6c976024368 Mon Sep 17 00:00:00 2001
From: Gard Spreemann <gspreemann@gmail.com>
Date: Thu, 14 Jun 2018 20:39:01 +0200
Subject: GUDHI 2.2.0 as released by upstream in a tarball.

---
 doc/Alpha_complex/Intro_alpha_complex.h | 90 +++++++++++----------------------
 1 file changed, 30 insertions(+), 60 deletions(-)

(limited to 'doc/Alpha_complex/Intro_alpha_complex.h')

diff --git a/doc/Alpha_complex/Intro_alpha_complex.h b/doc/Alpha_complex/Intro_alpha_complex.h
index a08663ca..7a375c9f 100644
--- a/doc/Alpha_complex/Intro_alpha_complex.h
+++ b/doc/Alpha_complex/Intro_alpha_complex.h
@@ -4,7 +4,7 @@
  *
  *    Author(s):       Vincent Rouvreau
  *
- *    Copyright (C) 2015  INRIA
+ *    Copyright (C) 2015 Inria
  *
  *    This program is free software: you can redistribute it and/or modify
  *    it under the terms of the GNU General Public License as published by
@@ -57,9 +57,13 @@ namespace alpha_complex {
  * href="http://doc.cgal.org/latest/Kernel_d/index.html#Chapter_dD_Geometry_Kernel">dD Geometry Kernel</a>
  * \cite cgal:s-gkd-15b from CGAL as template parameter.
  * 
- * \remark When the simplicial complex is constructed with an infinite value of alpha, the complex is a Delaunay
+ * \remark
+ * - When the simplicial complex is constructed with an infinite value of alpha, the complex is a Delaunay
  * complex.
- * 
+ * - For people only interested in the topology of the \ref alpha_complex (for instance persistence),
+ * \ref alpha_complex is equivalent to the \ref cech_complex and much smaller if you do not bound the radii.
+ * \ref cech_complex can still make sense in higher dimension precisely because you can bound the radii.
+ *
  * \section pointsexample Example from points
  * 
  * This example builds the Delaunay triangulation from the given points in a 2D static kernel, and creates a
@@ -89,63 +93,29 @@ namespace alpha_complex {
  * \image html "alpha_complex_doc.png" "Simplicial complex structure construction example"
  *
  * \subsection filtrationcomputation Filtration value computation algorithm
- *
- *
- *
- * <ul>
- *   <li style="list-style-type: none;">\f$ \textbf{for } i : dimension \rightarrow 0 \textbf{ do} \f$
- *     <ul>
- *       <li style="list-style-type: none;">\f$\textbf{for all } \sigma of dimension i \f$
- *         <ul>
- *           <li style="list-style-type: none;">\f$\textbf{if } filtration( \sigma ) is NaN \textbf{ then} \f$
- *             <ul>
- *               <li style="list-style-type: none;">\f$ filtration( \sigma ) = \alpha^2( \sigma ) \f$
- *               </li>
- *             </ul>
- *           </li>
- *           <li style="list-style-type: none;">\f$\textbf{end if}\f$
- *           </li>
- *           <li style="list-style-type: none;">\f$\textbf{for all } \tau face of \sigma \textbf{ do} \f$
- *             &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;// propagate alpha filtration value
- *             <ul>
- *               <li style="list-style-type: none;">\f$\textbf{if } filtration( \tau ) is not NaN \textbf{ then} \f$
- *                 <ul>
- *                   <li style="list-style-type: none;">\f$ filtration( \tau ) = min ( filtration( \tau ), filtration( \sigma ) ) \f$
- *                   </li>
- *                 </ul>
- *               </li>
- *               <li style="list-style-type: none;">\f$\textbf{else}\f$
- *                 <ul>
- *                   <li style="list-style-type: none;">\f$\textbf{if } \tau is not Gabriel for \sigma \textbf{ then} \f$
- *                     <ul>
- *                       <li style="list-style-type: none;">\f$ filtration( \tau ) = filtration( \sigma ) \f$
- *                       </li>
- *                     </ul>
- *                   </li>
- *                   <li style="list-style-type: none;">\f$\textbf{end if}\f$
- *                   </li>
- *                 </ul>
- *               </li>
- *               <li style="list-style-type: none;">\f$\textbf{end if}\f$
- *               </li>
- *             </ul>
- *           </li>
- *           <li style="list-style-type: none;">\f$\textbf{end for}\f$
- *           </li>
- *         </ul>
- *       </li>
- *       <li style="list-style-type: none;">\f$\textbf{end for}\f$
- *       </li>
- *     </ul>
- *   </li>
- *   <li style="list-style-type: none;">\f$\textbf{end for}\f$
- *   </li>
- *   <li style="list-style-type: none;">\f$make\_filtration\_non\_decreasing()\f$
- *   </li>
- *   <li style="list-style-type: none;">\f$prune\_above\_filtration()\f$
- *   </li>
- * </ul>
- *
+ * <br>
+ * \f$
+ * \textbf{for } \text{i : dimension } \rightarrow 0 \textbf{ do}\\
+ * \quad \textbf{for all } \sigma \text{ of dimension i}\\
+ * \quad\quad \textbf{if } \text{filtration(} \sigma ) \text{ is NaN} \textbf{ then}\\
+ * \quad\quad\quad \text{filtration(} \sigma ) = \alpha^2( \sigma )\\
+ * \quad\quad \textbf{end if}\\
+ * \quad\quad \textbf{for all } \tau \text{ face of } \sigma \textbf{ do}\quad\quad
+ * \textit{// propagate alpha filtration value}\\
+ * \quad\quad\quad \textbf{if } \text{filtration(} \tau ) \text{ is not NaN} \textbf{ then}\\
+ * \quad\quad\quad\quad \text{filtration(} \tau \text{) = min( filtration(} \tau \text{), filtration(} \sigma
+ * \text{) )}\\
+ * \quad\quad\quad \textbf{else}\\
+ * \quad\quad\quad\quad \textbf{if } \textbf{if } \tau \text{ is not Gabriel for } \sigma \textbf{ then}\\
+ * \quad\quad\quad\quad\quad \text{filtration(} \tau \text{) = filtration(} \sigma \text{)}\\
+ * \quad\quad\quad\quad \textbf{end if}\\
+ * \quad\quad\quad \textbf{end if}\\
+ * \quad\quad \textbf{end for}\\
+ * \quad \textbf{end for}\\
+ * \textbf{end for}\\
+ * \text{make_filtration_non_decreasing()}\\
+ * \text{prune_above_filtration()}\\
+ * \f$
  *
  * \subsubsection dimension2 Dimension 2
  * 
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