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added example of non-simplicial decompisition of the Bolza surface

This commit is contained in:
Iordan Iordanov 2018-09-04 11:52:55 +02:00
parent 958f1190ac
commit 10f7ad2c54
2 changed files with 15 additions and 1 deletions
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3
Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Doxyfile.in
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@ -12,7 +12,8 @@ HTML_EXTRA_FILES = ${CGAL_PACKAGE_DOC_DIR}/fig/octagon_identification.svg \
${CGAL_PACKAGE_DOC_DIR}/fig/dt-construction.svg \
${CGAL_PACKAGE_DOC_DIR}/fig/periodic_face.svg \
${CGAL_PACKAGE_DOC_DIR}/fig/ds_cgal.svg \
${CGAL_PACKAGE_DOC_DIR}/fig/dummy-points.png
${CGAL_PACKAGE_DOC_DIR}/fig/dummy-points.png \
${CGAL_PACKAGE_DOC_DIR}/fig/non-triangulation.svg \
HTML_EXTRA_STYLESHEET = ${CGAL_PACKAGE_DOC_DIR}/css/customstyle.css

13
Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Periodic_4_hyperbolic_triangulation_2.txt
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@ -237,6 +237,19 @@ Some point sets do not admit a triangulation of \f$\mathcal M\f$.
For instance, a single point does not define a triangulation of \f$\mathcal M\f$, as the result
would not be a simplicial complex.
\cgalModifEnd
\cgalFigureAnchor{P4HNonSimplicialExample}
<center>
<img src="non-triangulation.svg" style="max-width:45%; width=45%; display: inline-block;"/>
</center>
\cgalFigureCaptionBegin{P4HNonSimplicialExample}
Example of a non-simplicial decomposition of the Bolza surface. Note that with the three
points in the central octagon, we obtain a decomposition that is <b>not</b> a triangulation,
since it is non-simplicial. On the figure a few cycles of length 2 are shown in color. Such
cycles are double edges in the triangulation. Note also the pink edge between the two blue
vertices: it corresponds to a loop on the surface.
\cgalFigureCaptionEnd
For this reason, we initialize a triangulation of \f$\mathcal M\f$ with a predetermined set of
14 points, called <i>dummy points</i>, whose triangulation in \f$\mathcal M\f$ is a simplicial
complex and has the empty circle property. The set of dummy points has been proposed in