From 2ca2d4c0281c2c6e03138d15d7d509ec2feb5adf Mon Sep 17 00:00:00 2001 From: Guillaume Damiand Date: Mon, 31 Oct 2016 14:42:57 -0400 Subject: [PATCH] Remove trailing white spaces --- .../doc/Generalized_map/Generalized_map.txt | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/Generalized_map/doc/Generalized_map/Generalized_map.txt b/Generalized_map/doc/Generalized_map/Generalized_map.txt index c3558d28bb0..40eff526881 100644 --- a/Generalized_map/doc/Generalized_map/Generalized_map.txt +++ b/Generalized_map/doc/Generalized_map/Generalized_map.txt @@ -10,7 +10,7 @@ namespace CGAL { \section Generalized_mapIntroduction Introduction -A d-dimensional generalized map is a data structure representing an orientable or non-orientable subdivided d-dimensional object obtained by taking dD cells, and allowing to glue dD cells along (d-1)D cells. It provides a description of all the cells of the subdivision (for example vertices and edges), together with incidence and adjacency relationships. +A d-dimensional generalized map is a data structure representing an orientable or non-orientable subdivided d-dimensional object obtained by taking dD cells, and allowing to glue dD cells along (d-1)D cells. It provides a description of all the cells of the subdivision (for example vertices and edges), together with incidence and adjacency relationships. This package is a generalization of the \ref ChapterCombinatorialMap "combinatorial maps" data structure (which allows to describe only orientable objects) in order to be able to describe also non-orientable objects such as a Möbius strip (\cgalFigureRef{fig_gmap_non_orientable_objects} Left) or a Klein bottle (\cgalFigureRef{fig_gmap_non_orientable_objects} Right). @@ -61,7 +61,7 @@ In this section, we describe dD generalized maps in terms of data structu \subsection ssecgenmapanddarts Generalized Map and Darts A dD generalized map is a set of darts D. A dart d0 is an element that can be linked with d+1 darts by pointers called \f$ \alpha_i\f$, with 0 \f$ \leq \f$ i \f$ \leq \f$ d. %Dart d0 is said i-free when \f$ \alpha_i\f$(d0)=d0. Each \f$ \alpha_i\f$ is its own inverse, i.e. \f$ \alpha_i\f$(\f$ -\alpha_i\f$(d0))=d0. +\alpha_i\f$(d0))=d0. A generalized map is without i-boundary if there is no i-free dart, and it is without boundary if it is without i-boundary for all dimensions 1 \f$ \leq \f$ i \f$ \leq \f$ d. @@ -79,7 +79,7 @@ The first important property of a generalized map is that each dart belongs to a The second important property is that cells of a generalized map correspond to specific orbits. Given a set S\f$ \subseteq\f${\f$ \alpha_1\f$,...,\f$ \alpha_d\f$} and a dart d0, the orbit \f$ \langle{}\f$ S\f$ \rangle{}\f$(d0) is the set of darts that can be reached from d0 by following any combination of any \f$ \alpha_i\f$'s in S (to simplify notations, we can use for example \f$ \langle{}\f$\f$ \alpha_1\f$,\f$ \alpha_4\f$\f$\rangle{}\f$(d0) to denote \f$ \langle{}\f$ S\f$ \rangle{}\f$(d0) with S={\f$ \alpha_1\f$,\f$ \alpha_4\f$}). -Given a dart d0, in general, \f$ \alpha_i\f$(d0) (with 0 \f$ \leq \f$ i \f$ \leq \f$ d) belongs to the same cells as d0, only the i-cell is different. There are two exceptions: +Given a dart d0, in general, \f$ \alpha_i\f$(d0) (with 0 \f$ \leq \f$ i \f$ \leq \f$ d) belongs to the same cells as d0, only the i-cell is different. There are two exceptions:
  1. if d0 is i-free, then \f$ \alpha_i\f$(d0)=d0, the i-cell is not different;
  2. if \f$ \alpha_i\f$(d0) belongs to the same i-cell as d0 (case of multi-incidence). For example if an edge is an isolated loop, it is incident twice to the same vertex, then given a dart d0 belonging to this edge, \f$ \alpha_1\f$(d0) goes to the next edge, which is in fact the same edge.
    3. @@ -309,7 +309,7 @@ The \link GeneralizedMap::link_alpha `link_alpha`\endlink and \link GeneralizedM \cgalAdvancedEnd -Linking two darts d1 and d2 by \f$ \alpha_i\f$, with 0 \f$ \leq \f$ i \f$ \leq \f$ d and d1 \f$ \neq \f$ d2, consists in modifying two \f$ \alpha_i\f$ pointers such that \f$ \alpha_i\f$(d1)=d2 and \f$ \alpha_i\f$(d2)=d1. +Linking two darts d1 and d2 by \f$ \alpha_i\f$, with 0 \f$ \leq \f$ i \f$ \leq \f$ d and d1 \f$ \neq \f$ d2, consists in modifying two \f$ \alpha_i\f$ pointers such that \f$ \alpha_i\f$(d1)=d2 and \f$ \alpha_i\f$(d2)=d1. Reciprocally, unlinking a given dart d0 by \f$ \alpha_i\f$, with 0 \f$ \leq \f$ i \f$ \leq \f$ d, consists in modifying two \f$ \alpha_i\f$ pointers such that \f$ \alpha_i\f$(\f$ \alpha_i\f$(d0))=\f$ \alpha_i\f$(d0) and \f$ \alpha_i\f$(d0)=d0. Note that is it possible to unlink a given dart for \f$ \alpha_i\f$ only if it is not i-free. @@ -480,8 +480,8 @@ Functors `Sum_functor` and `Divide_by_two_functor` define a custom behavior: whe The output is: \verbatim -20; 7; 7; 7; 7; 7; 13; 13; 13; 13; 13; -2; 7; 7; 7; 7; 7; 10; 13; 13; 13; 13; 13; 5; 2; +20; 7; 7; 7; 7; 7; 13; 13; 13; 13; 13; +2; 7; 7; 7; 7; 7; 10; 13; 13; 13; 13; 13; 5; 2; #Darts=128, #0-cells=13, #1-cells=24, #2-cells=14, #3-cells=2, #ccs=1, orientable=true, valid=1 \endverbatim