mirror of https://github.com/CGAL/cgal
Add more explanations about the non orientable case.
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Guillaume Damiand
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@ -12,7 +12,11 @@ namespace CGAL {
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A <I>d</I>-dimensional generalized map is a data structure representing an \tred{orientable or non orientable} subdivided <I>d</I>-dimensional object obtained by taking <I>d</I>D cells, and allowing to glue <I>d</I>D cells along <I>(d-1)</I>D cells. It provides a description of all the cells of the subdivision (for example vertices and edges), together with incidence and adjacency relationships.
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\tred{This package is an extension of the \ref ChapterCombinatorialMap "combinatorial maps" data structure which allows to describe only orientable objects.}
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\tred{This package is an extension of the \ref ChapterCombinatorialMap "combinatorial maps" data structure which allows to describe only 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).}
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\cgalFigureBegin{fig_gmap_non_orientable_objects,gmap_non_orientable_objects.png}
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Example of two non-orientable objects. Left: A Möbius strip. Right: A Klein bottle.
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\cgalFigureEnd
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We denote <I>i</I>-cell for an <I>i</I>-dimensional cell (for example in 3D, 0-cells are <I>vertices</I>, 1-cells are <I>edges</I>, 2-cells are <I>facets</I>, and 3-cells are <I>volumes</I>). A <I>boundary relation</I> is defined on these cells, giving for each <I>i</I>-cell <I>c</I> the set of <I>(i-1)</I>-cells contained in the boundary of <I>c</I>. Two cells <I>c1</I> and <I>c2</I> are <I>incident</I> if there is a path of cells, starting from the cell of biggest dimension to the other cell, such that each cell of the path (except the first one) belongs to the boundary of the previous cell in the path. Two <I>i</I>-cells <I>c3</I> and <I>c4</I> are <I>adjacent</I> if there is an <I>(i-1)</I>-cell incident to both <I>c3</I> and <I>c4</I>. You can see an example of a 2D object and a 3D object in \cgalFigureRef{fig_gmap_example_subdivisions} showing some cells of the subdivision and some adjacency and incidence relations.
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@ -345,6 +349,15 @@ Let us consider the generalized map given in \cgalFigureRef{fig_gmap_example_3d_
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If one wants to modify a generalized map <I>manually</I>, it is possible to switch off the updating between darts and attributes by calling \link GeneralizedMap::set_automatic_attributes_management `set_automatic_attributes_management(false)`\endlink before to call \link GeneralizedMap::sew `sew<i>(dh1,dh2)`\endlink and \link GeneralizedMap::unsew `unsew<i>(dh0)`\endlink. In these cases, the generalized map obtained may be no longer valid due to incorrect associations between darts and attributes. In \cgalFigureRef{fig_gmap_example_3d_sew} (Left), if we call \link GeneralizedMap::sew `sew<3>(1,5)`\endlink, the resulting generalized map is similar to the generalized map of \cgalFigureRef{fig_gmap_example_3d_sew} (Right) (we have linked by \f$ \alpha_3\f$ the pairs of darts (1,5), (2,8), (3,7) and (4,6)), but associations between darts and attributes are not valid. Indeed, we have kept the four initial attributes and all the associations between darts and attributes, thus two darts belonging to the same 2-cell (for example darts 1 and 5) are associated with two different attributes. We can also use the \link GeneralizedMap::link_alpha `link_alpha<i>(dh1,dh2)`\endlink which links `d1` and `d2` by \f$ \alpha_i\f$ without modifying the other links. Association between darts and attributes are only modified for darts `d1` and `d2`, and similarly as for \link GeneralizedMap::sew `sew<i>`\endlink, this updating can be avoided by calling \link GeneralizedMap::set_automatic_attributes_management `set_automatic_attributes_management(false)`\endlink before to call \link GeneralizedMap::link_alpha `link_alpha<i>(dh1,dh2)`\endlink. Lastly, we can use \link GeneralizedMap::unlink_alpha `unlink_alpha<i>(dh0)`\endlink to unlink `d0` for \f$ \alpha_i\f$. In this last case, there is no modification of association between darts and attributes. In \cgalFigureRef{fig_gmap_example_3d_sew} (Left), if we call \link GeneralizedMap::link_alpha `link_alpha<3>(1,5)`\endlink, in the resulting generalized map we have now \f$ \alpha_3\f$(1)=5 and \f$ \alpha_3\f$(5)=1. This generalized map is no longer valid (for example dart 2 is 3-free and we should have \f$ \alpha_3\f$(2)=8). A call latter to \link GeneralizedMap::set_automatic_attributes_management `set_automatic_attributes_management(true)`\endlink will correct the invalid non void attributes.
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\cgalAdvancedEnd
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\cgalModifBegin
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Sewing operations can be used in order to build a non-orientable generalized map. Let us consider the 2D generalized map representing a square given in \cgalFigureRef{fig_gmap_sew_non_orientable} (Left). Two opposite edges of the square can be identified by using the \link GeneralizedMap::sew `sew<2>`\endlink operation. But there are two possibilities to make this identification. The first one, shown in \cgalFigureRef{fig_gmap_sew_non_orientable} (Middle), creates an annulus which is thus orientable. The second one, shown in \cgalFigureRef{fig_gmap_sew_non_orientable} (Right), creates a Möbius strip which is thus non-orientable. The choice of the two darts for the sew operation is thus crucial.
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\cgalFigureBegin{fig_gmap_sew_non_orientable,gmap_sew_non_orientable.svg}
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Illustration of the use of the 2-sew operation to construct a non-orientable generalized map. Left: A 2D generalized map <I>M</I> representing a square. Middle: the generalized map obtained from <I>M</I> after \link GeneralizedMap::sew `sew<2>(1,3)`\endlink. This map is orientable and corresponds to an annulus. Right: the generalized map obtained from <I>M</I> after \link GeneralizedMap::sew `sew<2>(1,4)`\endlink. This map is non-orientable and corresponds to a Möbius strip.
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\cgalFigureEnd
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\cgalModifEnd
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\subsection ssecoperationsgmap Removal and Insertion Operations
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The following high level operations are defined. All these methods ensure that given a valid generalized map and a possible operation, the modified generalized map is also valid.
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#FIG 3.2 Produced by xfig version 3.2.5c
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Portrait
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