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Diffstat (limited to 'doc/Alpha_complex/Intro_alpha_complex.h')
| -rw-r--r-- | doc/Alpha_complex/Intro_alpha_complex.h | 90 |
1 files changed, 30 insertions, 60 deletions
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$\textbf{for all } \tau face of \sigma \textbf{ do} \f$ - * // 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 * |