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-/*    This file is part of the Gudhi Library. The Gudhi library
- *    (Geometric Understanding in Higher Dimensions) is a generic C++
- *    library for computational topology.
- *
- *    Author(s):       Vincent Rouvreau
- *
- *    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
- *    the Free Software Foundation, either version 3 of the License, or
- *    (at your option) any later version.
- *
- *    This program is distributed in the hope that it will be useful,
- *    but WITHOUT ANY WARRANTY; without even the implied warranty of
- *    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
- *    GNU General Public License for more details.
- *
- *    You should have received a copy of the GNU General Public License
- *    along with this program.  If not, see <http://www.gnu.org/licenses/>.
- */
-
-#ifndef DOC_ALPHA_COMPLEX_INTRO_ALPHA_COMPLEX_H_
-#define DOC_ALPHA_COMPLEX_INTRO_ALPHA_COMPLEX_H_
-
-// needs namespace for Doxygen to link on classes
-namespace Gudhi {
-// needs namespace for Doxygen to link on classes
-namespace alpha_complex {
-
-/**  \defgroup alpha_complex Alpha complex
- * 
- * \author    Vincent Rouvreau
- *
- * @{
- * 
- * \section definition Definition
- * 
- * Alpha_complex is a <a target="_blank" href="https://en.wikipedia.org/wiki/Simplicial_complex">simplicial complex</a>
- * constructed from the finite cells of a Delaunay Triangulation.
- * 
- * The filtration value of each simplex is computed as the square of the circumradius of the simplex if the
- * circumsphere is empty (the simplex is then said to be Gabriel), and as the minimum of the filtration
- * values of the codimension 1 cofaces that make it not Gabriel otherwise.
- * 
- * All simplices that have a filtration value strictly greater than a given alpha squared value are not inserted into
- * the complex.
- * 
- * \image html "alpha_complex_representation.png" "Alpha-complex representation"
- * 
- * Alpha_complex is constructing a <a target="_blank"
- * href="http://doc.cgal.org/latest/Triangulation/index.html#Chapter_Triangulations">Delaunay Triangulation</a>
- * \cite cgal:hdj-t-15b from <a target="_blank" href="http://www.cgal.org/">CGAL</a> (the Computational Geometry
- * Algorithms Library \cite cgal:eb-15b) and is able to create a `SimplicialComplexForAlpha`.
- * 
- * The complex is a template class requiring an Epick_d <a target="_blank"
- * 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
- * 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
- * `Simplex_tree` with it.
- * 
- * Then, it is asked to display information about the simplicial complex.
- * 
- * \include Alpha_complex/Alpha_complex_from_points.cpp
- * 
- * When launching:
- * 
- * \code $> ./Alpha_complex_example_from_points
- * \endcode
- *
- * the program output is:
- * 
- * \include Alpha_complex/alphaoffreader_for_doc_60.txt
- * 
- * \section createcomplexalgorithm Create complex algorithm
- * 
- * \subsection datastructure Data structure
- * 
- * In order to create the simplicial complex, first, it is built from the cells of the Delaunay Triangulation.
- * The filtration values are set to NaN, which stands for unknown value.
- * 
- * In example, :
- * \image html "alpha_complex_doc.png" "Simplicial complex structure construction example"
- *
- * \subsection filtrationcomputation Filtration value computation algorithm
- * <br>
- * \f$
- * \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
- * 
- * From the example above, it means the algorithm looks into each triangle ([0,1,2], [0,2,4], [1,2,3], ...),
- * computes the filtration value of the triangle, and then propagates the filtration value as described
- * here :
- * \image html "alpha_complex_doc_420.png" "Filtration value propagation example"
- * 
- * \subsubsection dimension1 Dimension 1
- * 
- * Then, the algorithm looks into each edge ([0,1], [0,2], [1,2], ...),
- * computes the filtration value of the edge (in this case, propagation will have no effect).
- * 
- * \subsubsection dimension0 Dimension 0
- * 
- * Finally, the algorithm looks into each vertex ([0], [1], [2], [3], [4], [5] and [6]) and
- * sets the filtration value (0 in case of a vertex - propagation will have no effect).
- * 
- * \subsubsection nondecreasing Non decreasing filtration values
- * 
- * As the squared radii computed by CGAL are an approximation, it might happen that these alpha squared values do not
- * quite define a proper filtration (i.e. non-decreasing with respect to inclusion).
- * We fix that up by calling `SimplicialComplexForAlpha::make_filtration_non_decreasing()`.
- * 
- * \subsubsection pruneabove Prune above given filtration value
- * 
- * The simplex tree is pruned from the given maximum alpha squared value (cf.
- * `SimplicialComplexForAlpha::prune_above_filtration()`).
- * In the following example, the value is given by the user as argument of the program.
- * 
- * 
- * \section offexample Example from OFF file
- * 
- * This example builds the Delaunay triangulation in a dynamic kernel, and initializes the alpha complex with it.
- * 
- * 
- * Then, it is asked to display information about the alpha complex.
- * 
- * \include Alpha_complex/Alpha_complex_from_off.cpp
- * 
- * When launching:
- * 
- * \code $> ./Alpha_complex_example_from_off ../../data/points/alphacomplexdoc.off 32.0
- * \endcode
- *
- * the program output is:
- * 
- * \include Alpha_complex/alphaoffreader_for_doc_32.txt
- * 
- */
-/** @} */  // end defgroup alpha_complex
-
-}  // namespace alpha_complex
-
-namespace alphacomplex = alpha_complex;
-
-}  // namespace Gudhi
-
-#endif  // DOC_ALPHA_COMPLEX_INTRO_ALPHA_COMPLEX_H_