mirror of https://github.com/CGAL/cgal
Doc improvements
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Mael Rouxel-Labbé
parent
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commit
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@ -27,7 +27,6 @@ is further refined afterward.
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\attention The function template `refine_mesh_3()` may be used to refine a previously
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computed mesh, e.g.:
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\code{.cpp}
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C3T3 c3t3 = CGAL::make_mesh_3<C3T3>(domain,criteria);
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@ -61,7 +61,7 @@ The type of the
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embedding 3D triangulation.
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It is required to be
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the nested type
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`CGAL::Mesh_triangulation_3::type`, provided by the meta functor
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`CGAL::Mesh_triangulation_3::type`, provided by the class
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`CGAL::Mesh_triangulation_3<MD, GT, Concurrency_tag, Vertex_base, Cell_base>`
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where the Vertex_base and Cell_base template parameters are respectively instantiated with models
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of the concepts `MeshVertexBase_3` and
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@ -130,7 +130,8 @@ Currently, 1-dimensional features may be defined as segments and polyline segmen
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\subsection Mesh_3OutputMesh Output Mesh
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The resulting mesh is output as a subcomplex of a weighted 3D triangulation,
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The resulting mesh is output as a subcomplex of a
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\ref CGAL::Regular_triangulation_3 "weighted 3D triangulation",
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in a class providing various iterators
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on mesh elements.
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@ -149,8 +150,8 @@ and the 0-dimensional exposed features are represented by mesh vertices.
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\subsubsection Mesh_3OutputFaceGraph Output FaceGraph
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This \cgal component also provides a function to convert the reconstructed mesh to a `FaceGraph`,
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namely `facets_in_complex_3_to_triangle_mesh()`.
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This \cgal component also provides a function to extract the facets of the complex
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as a `FaceGraph`, namely `facets_in_complex_3_to_triangle_mesh()`.
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\subsection Mesh_3DelaunayRefinement Delaunay Refinement
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\anchor introsecparam
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@ -190,7 +191,7 @@ the corresponding points are inserted into the Delaunay triangulation
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from the start.
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If the domain has 1-dimensional exposed features,
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the method of protecting balls \cgalCite{cgal:cdr-drpsc-07}, \cgalCite{cgal:cdl-pdma-07}
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the method of protecting balls \cgalCite{cgal:cdl-pdma-07}, \cgalCite{cgal:cdr-drpsc-07}
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is used to achieve an accurate representation of those features in the mesh
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and to guarantee that the refinement process terminates
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whatever may be the dihedral angles formed by input surface patches incident to a
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@ -333,9 +334,9 @@ and to represent the final 3D mesh at the end
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of the procedure.
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The embedding 3D triangulation is required to be the nested type
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`CGAL::Mesh_triangulation_3::type`, provided by the meta functor
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`CGAL::Mesh_triangulation_3::type`, provided by the class
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`CGAL::Mesh_triangulation_3`. The type for this triangulation is a
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`CGAL::Regular_triangulation_3` whose vertex and cell base classes
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wrapper around the class `CGAL::Regular_triangulation_3` whose vertex and cell base classes
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are respectively models of the concepts `MeshVertexBase_3` and
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`MeshCellBase_3`.
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@ -5,7 +5,9 @@ namespace CGAL {
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The class `Implicit_periodic_3_mesh_domain_3` implements a periodic domain
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whose bounding surface is described implicitly as the zero level set
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of a function defined over the three dimensional flat torus.
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of a function defined over the three dimensional flat torus. In practice,
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the domain of definition of the function must contain the input canonical cube,
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see Section \ref Periodic_3_mesh_3InputDomain.
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The domain to be discretized is assumed to be the domain where
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the function has negative values.
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@ -22,12 +24,12 @@ This class is a model of the concept `Periodic_3MeshDomain_3`.
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`BisectionGeometricTraits_3`.
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The constructor of `Implicit_periodic_3_mesh_domain_3` takes as argument
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a cuboid (the fundamental domain) in which we construct the
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mesh (see \ref PkgPeriodic_3_mesh_3).
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a cuboid (the canonical representative of the fundamental domain) in which
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we construct the mesh (see \ref PkgPeriodic_3_mesh_3).
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This domain constructs intersection points between the surface and
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segments/rays/lines by bisection. It requires an `error_bound` such that the bisection
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process is stopped when the query segment is smaller than the error bound.
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The `error_bound`, passed as argument to the domain constructor,
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segments (duals of a facet) by bisection. It requires an `error_bound`
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such that the bisection process is stopped when the query segment is smaller
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than the error bound. The `error_bound`, passed as argument to the domain constructor,
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is a relative error bound expressed as a ratio of the longest space diagonal
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of the cuboid.
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@ -6,7 +6,9 @@ namespace CGAL {
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The class `Implicit_to_labeled_subdomains_function_wrapper` is a helper class
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designed to wrap an implicit function which describes a domain by
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[`p` is inside if `f(p)<0`] to a function that takes its values into `{1, 2}` and
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thus describes a multidomain.
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thus describes a multidomain: the subspace described by `f(p)<0` is attributed
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the subdomain index `1` and the subspace described by `f(p)>0` is attributed
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the subdomain index `2`.
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Note that for the 3D mesh generator [`f(p)=0`] means that p is outside the domain.
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Since this wrapper has values into `{1, 2}`, both the interior and the exterior of
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@ -15,8 +15,8 @@ This class includes a function that provides the index of the subdomain
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in which any query point lies. An intersection between a segment and bounding
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surfaces is detected when both segment endpoints are associated with different
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values of subdomain indices. The intersection is then constructed by bisection.
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The bisection stops when the query segment is shorter than an error bound
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`e` given by the product of the length of the diagonal of the bounding box
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The bisection stops when the query segment is shorter than an error bound `e`
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given by the product of the length of the diagonal of the cuboid
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(in world coordinates) and a relative error bound passed as argument
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to the constructor of `Labeled_periodic_3_mesh_domain_3`.
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@ -26,6 +26,8 @@ It uses a satisfactory labeling function if there is only one component to mesh.
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\tparam LabelingFunction is the type of the input function.<br />
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This parameter stands for a model of the concept `ImplicitFunction`
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described in the surface mesh generation package.<br />
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The function is defined over the canonical representation of the three dimensional flat torus,
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the input canonical cube (see Section \ref Periodic_3_mesh_3InputDomain).
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The labeling function `f` must return `0` if the point isn't located in any subdomain.
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Usually, the return types of labeling functions are integer.<br />
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Let `p` be a point.
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@ -33,7 +35,7 @@ It uses a satisfactory labeling function if there is only one component to mesh.
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<li>`f(p)=0` means that `p` is outside domain.</li>
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<li>`f(p)=a`, `a!=0` means that `p` is inside the subdomain `a`.</li>
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</ul>
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Implicit_multi_domain_to_labeling_function_wrapper is a good candidate for this template parameter
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`Implicit_multi_domain_to_labeling_function_wrapper` is a good candidate for this template parameter
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if there are several components to mesh.
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\tparam BGT is a geometric traits class that provides
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@ -24,19 +24,21 @@ is further refined afterward.
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\attention The function template `refine_periodic_3_mesh_3()` may be used
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to refine a previously computed mesh, e.g.:
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\code{.cpp}
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C3T3 c3t3 = CGAL::make_periodic_3_mesh_3<C3T3>(domain,criteria);
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CGAL::refine_periodic_3_mesh_3(c3t3, domain, new_criteria);
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\endcode
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Note that the triangulation must form at all times a simplicial complex within
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a single copy of the domain. It is the responsability of the user to provide
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\attention Note that the triangulation must form at all times a simplicial complex within
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a single copy of the domain (see Sections \ref P3Triangulation3secspace and \ref P3Triangulation3secintro
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of the manual of 3D periodic triangulations). It is the responsability of the user to provide
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a triangulation that satisfies this condition when calling the refinement
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function `refine_periodic_3_mesh_3`.
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function `refine_periodic_3_mesh_3`. The underlying triangulation of a mesh
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complex obtained through `make_periodic_3_mesh_3()` or `refine_periodic_3_mesh_3()`
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will always satisfy this condition.
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\tparam C3T3 is required to be a model of the concept `MeshComplex_3InTriangulation_3`.
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\tparam C3T3 is required to be a model of the concept `MeshComplex_3InTriangulation_3`.
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The argument `c3t3` is passed by reference as this object is modified
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by the refinement process. As the refinement process only adds points
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to the triangulation, all vertices of the triangulation of `c3t3` remain in the
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@ -15,7 +15,8 @@ curves, and corners according to their respective dimensions 3, 2, 1, and 0.
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From a syntactic point of view, `Periodic_3MeshDomainWithFeatures_3`
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refines `MeshDomainWithFeatures_3`. However, the various requirements from
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`MeshDomainWithFeatures_3` must also take into account the periodicity of the domain.
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`MeshDomainWithFeatures_3` must also take into account the periodicity of the domain
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(see Section \ref Periodic_3_mesh_3InputDomain).
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Wrapping any model of `Periodic_3MeshDomain_3` with the class
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`CGAL::Mesh_domain_with_polyline_features_3` gives a model
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@ -11,12 +11,9 @@ domain is defined over the three-dimensional flat torus. From a syntactic point
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of view, it defines exactly the same requirements as the concept `MeshDomain_3` and
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thus `Periodic_3MeshDomain_3` refines `MeshDomain_3` without any additional requirement.
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However, the oracle must take into account the periodicity of the domain.
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For instance, when evaluating the `MeshDomain_3::Do_intersect_surface` oracle for a segment,
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it may be the case that the segment does not intersect the surface in the domain,
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but its translated copy does intersect the surface in the domain.
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The oracle must thus consider translated images of primitives to correctly determine
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if an intersection exists.
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However, the oracle must take into account the periodicity of the domain and
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evaluate queries at the canonical representative a point (see Section
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\ref Periodic_3_mesh_3InputDomain).
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\cgalHasModel `CGAL::Implicit_periodic_3_mesh_domain_3<Function,BGT>`
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\cgalHasModel `CGAL::Labeled_periodic_3_mesh_domain_3<LabelingFunction,BGT>`
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@ -9,6 +9,7 @@ HTML_EXTRA_FILES = ${CGAL_PACKAGE_DOC_DIR}/fig/periodic_banner.png \
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${CGAL_PACKAGE_DOC_DIR}/fig/periodic_copies.jpg \
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${CGAL_PACKAGE_DOC_DIR}/fig/periodic_implicit_domain.png \
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${CGAL_PACKAGE_DOC_DIR}/fig/periodic_implicit_spheres.png \
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${CGAL_PACKAGE_DOC_DIR}/fig/periodic_implicit_interior_and_exterior.jpg \
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${CGAL_PACKAGE_DOC_DIR}/fig/periodic_implicit_multi_domain.jpg \
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${CGAL_PACKAGE_DOC_DIR}/fig/periodic_implicit_optimizers.png \
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${CGAL_PACKAGE_DOC_DIR}/fig/periodic_implicit_protection.jpg \
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@ -1,8 +1,8 @@
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/// \defgroup PkgPeriodic_3_mesh_3 3D Periodic Mesh Generation Reference
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/// \defgroup PkgPeriodic_3_mesh_3Concepts Main Concepts
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/// \defgroup PkgPeriodic_3_mesh_3Concepts Concepts
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/// \ingroup PkgPeriodic_3_mesh_3
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/// The main concepts of this package.
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/// The concepts of this package.
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/// \defgroup PkgPeriodic_3_mesh_3MeshClasses Mesh Classes
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/// \ingroup PkgPeriodic_3_mesh_3
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@ -31,11 +31,10 @@ flexibility that is offered in the \ref PkgMesh_3Summary package.}
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\cgalPkgManuals{Chapter_3D_Periodic_Mesh_Generation,PkgPeriodic_3_mesh_3}
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\cgalPkgSummaryEnd
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\cgalPkgShortInfoBegin
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\cgalPkgSince{4.11}
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\cgalPkgSince{4.13}
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\cgalPkgDependsOn{\ref PkgPeriodic3Triangulation3Summary, \ref PkgMesh_3Summary}
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\cgalPkgBib{cgal:btprl-p3m3}
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\cgalPkgLicense{\ref licensesGPL "GPL"}
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\cgalPkgDemo{,}
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\cgalPkgShortInfoEnd
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\cgalPkgDescriptionEnd
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@ -43,18 +42,18 @@ flexibility that is offered in the \ref PkgMesh_3Summary package.}
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A periodic mesh extends, by definition, infinitely in space. To avoid storing and
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manipulating duplicate points, well-chosen "dummy" points are inserted
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at the beginning of the meshing process, thus ensuring that the periodic triangulation
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forms at all times a simplicial complex within a single fundamental copy of the domain
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at the beginning of the meshing process, thus ensuring that the underlying periodic
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triangulation forms at all times a simplicial complex within a single copy of the periodic space \f$ \mathbb T_c^3\f$
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(see Sections \ref P3Triangulation3secspace and \ref P3Triangulation3secintro
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of the manual of 3D periodic triangulations).
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The meshing process can then be exclusively conducted in the fundamental domain.
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This single copy of the complete periodic mesh is generated using
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the three-dimensional mesh generator of %CGAL (see package: \ref PkgMesh_3Summary).
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Since this mesh generator only supports traditional (non-periodic) domains,
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it must be provided adapted oracles to handle the periodicity of the input domain
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and of the mesh.
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This package provides the necessary tools to be able to use %CGAL's \ref PkgMesh_3Summary
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with %CGAL's \ref PkgPeriodic3Triangulation3Summary.
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of the package \ref PkgPeriodic3Triangulation3Summary).
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By identifying a single copy of the flat torus \f$ \mathbb T_c^3\f$ (where `c`
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denotes the period) with a cube of side `c` in \f$ \mathbb R^3\f$, the meshing process
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can be exclusively conducted within this cube.
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The mesh within a single copy is created using %CGAL's \ref PkgMesh_3Summary package, but
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because %CGAL's \ref PkgMesh_3Summary package aims to mesh traditional (non-periodic)
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domains, an interface is necessary between %CGAL's \ref PkgMesh_3Summary package
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and %CGAL's \ref PkgPeriodic3Triangulation3Summary.
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This package offers these interfaces.
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\cgalClassifedRefPages
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@ -20,9 +20,11 @@ A cut view of a periodic mesh.
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This package is devoted to the generation of isotropic simplicial
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meshes discretizing periodic 3D domains.
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The domain to be meshed is a region of the three-dimensional flat torus
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(see Section \ref P3Triangulation3secspace).
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(see Section \ref P3Triangulation3secspace of the package \ref PkgPeriodic3Triangulation3Summary).
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The region may be connected or composed of multiple components
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and/or subdivided in several subdomains.
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The current implementation provides classes to represent
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domains bounded by isosurfaces of implicit functions defined over a cube.
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Boundary and subdivision surfaces are either
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smooth or piecewise-smooth surfaces, formed with planar or curved surface patches.
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@ -56,29 +58,31 @@ the boundary and subdivision surface patches may form \cgalCite{cgal:cdl-pdma-07
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\cgalCite{cgal:cdr-drpsc-07}. The Delaunay refinement is followed
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by a mesh optimization phase to remove slivers and provide a good quality mesh.
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\subsection Periodic_3_mesh_3Mesh_3 Relation to the 3D Mesh Generation Package
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\subsection Periodic_3_mesh_3Mesh_3 Relation to the 3D Mesh Generation and 3D Periodic Triangulation Packages
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This package is fundamentally linked to the package \ref PkgMesh_3Summary.
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This package is fundamentally linked to the package \ref PkgMesh_3Summary,
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which is devoted to the generation of isotropic simplicial
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meshes discretizing (non-periodic) 3D domains and %CGAL's \ref PkgPeriodic3Triangulation3Summary,
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which are used as underlying triangulation structures of the mesh.
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A periodic mesh extends, by definition, infinitely in space. To avoid storing and
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manipulating duplicate points, well-chosen "dummy" points are inserted
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at the beginning of the meshing process, thus ensuring that the triangulation forms
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at all times a simplicial complex within a single fundamental copy of the domain
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at the beginning of the meshing process, thus ensuring that the underlying periodic
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triangulation forms at all times a simplicial complex within a single copy of the periodic space \f$ \mathbb T_c^3\f$
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(see Sections \ref P3Triangulation3secspace and \ref P3Triangulation3secintro
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of the manual of 3D periodic triangulations).
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The meshing process can then be exclusively conducted in the fundamental domain.
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This single copy of the complete periodic mesh is generated using
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the three-dimensional mesh generator of %CGAL (see package: \ref PkgMesh_3Summary).
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Since the package \ref PkgMesh_3Summary only supports traditional (non-periodic) domains,
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we provide adapted oracles to handle the periodicity of the input domain
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and of the mesh.
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This package provides the necessary tools to be able to use %CGAL's \ref PkgMesh_3Summary
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with %CGAL's \ref PkgPeriodic3Triangulation3Summary.
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of the package \ref PkgPeriodic3Triangulation3Summary).
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By identifying a single copy of the flat torus \f$ \mathbb T_c^3\f$ (where `c`
|
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denotes the period) with a cube of side `c` in \f$ \mathbb R^3\f$, the meshing process
|
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can be exclusively conducted within this cube.
|
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The mesh within a single copy is created using %CGAL's \ref PkgMesh_3Summary package, but
|
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because %CGAL's \ref PkgMesh_3Summary package aims to mesh traditional (non-periodic)
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domains, an interface is necessary between %CGAL's \ref PkgMesh_3Summary package
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and %CGAL's \ref PkgPeriodic3Triangulation3Summary.
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This package offers these interfaces.
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\subsection Periodic_3_mesh_3InputDomain Input Domain
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The domain to be meshed is assumed to live within a (non-necessarily unit)
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periodic cube. Furthermore, it is assumed to be representable as a pure 3D complex.
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The domain to be meshed is assumed to be representable as a pure 3D complex.
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A 3D complex is a set of faces with dimension 0, 1, 2, and 3
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such that all faces are pairwise interior disjoint, and the boundary
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of each face of the complex is the union of lower-dimensional faces of the complex.
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@ -115,20 +119,80 @@ as a domain class, often called the oracle,
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that provides predicates and constructors related to the domain,
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the subdomains, and the boundary surface patches.
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Mainly, the oracle provides a predicate to test
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if a given query point belongs to the periodic domain or not
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if a given query point belongs to the domain or not
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and to find in which subdomain it lies in the affirmative case.
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The domain class also provides predicates and constructors to test the intersection of a query line segment
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with the boundary surface patches and to construct intersection points, if any.
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The current implementation provides classes to represent
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domains bounded by isosurfaces of implicit functions.
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\subsubsection Periodic_3_mesh_3InputDomainPeriodicity Periodicity of the Input Domain
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As described in Section \ref Periodic_3_mesh_3Mesh_3, the periodic mesh is in fact
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constructed over a single copy of the periodic space. This instance of the flat torus
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\f$ \mathbb T_c^3\f$ (`c` being the period) is identified with a cube of side `c`
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in \f$ \mathbb R^3\f$. The origin (given by three coordinates \f$ \alpha\f$, \f$ \beta\f$,
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and \f$ \gamma\f$) of this cube and the period `c` are input parameters chosen
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by the user. The cube \f$ [\alpha,\alpha+c)\times[\beta,\beta+c)\times[\gamma,\gamma+c)\f$
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contains exactly one representative of each element in \f$ \mathbb T_c^3\f$
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and is called the <I>canonical</I> or <I>fundamental instance</I> of the periodic space.
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Note that this is defined as <I>original domain</I> in the package \ref PkgPeriodic3Triangulation3Summary,
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but this name will not be used in this package to avoid creating confusion between
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the different types of "domains".
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The mesh is only constructed over the canonical instance and periodicity
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is handled internally: oracles described above evaluate queries at a point
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by instead evaluating at the representative that belongs in the canonical instance.
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||||
For example, if considering the unit cube as canonical instance, an oracle answering
|
||||
a query such as <I>"what is the value of the implicit function at this point?"</I>
|
||||
at the point `(2.5, 2.5, 2.5)` will be in fact evaluated at the canonical position,
|
||||
`(0.5, 0.5, 0.5)`.
|
||||
|
||||
Consequently It is thus not required to provide an input domain that is periodic over the
|
||||
whole space or oracles that can answer queries for all points of \f$ \mathbb R^3\f$, but only
|
||||
over the chosen canonical instance (the input cube).
|
||||
|
||||
This approach allows to construct periodic meshes of domains that are not intrinsically
|
||||
periodic, for example a sphere (see
|
||||
\cgalFigureRef{Periodic_3_mesh_3FromCanonicalToWholeDiscard}
|
||||
and \cgalFigureRef{Periodic_3_mesh_3Periodic_implicit_sphere}) or a cone
|
||||
(see Section \ref Periodic_3_mesh_3ConeWithSharpFeatures).
|
||||
|
||||
\cgalFigureBegin{Periodic_3_mesh_3FromCanonicalToWhole, periodicity_base.svg}
|
||||
Illustration in 2D (cut view) of a domain defined by an implicit function.
|
||||
Only the values of the input domain that are in the canonical instance matter:
|
||||
the values of the implicit functions at `P` and `P'` are obtained by evaluating
|
||||
instead at `Q` and `Q'`, yielding the periodic domain on the right.
|
||||
\cgalFigureEnd
|
||||
|
||||
In practice, the domain of definition of an implicit function provided by the user
|
||||
is likely larger than the canonical instance (in general, it is even \f$ \mathbb R^3\f$).
|
||||
Note that since queries are always answered by looking at the canonical
|
||||
position, all information provided by the input domain that is outside of this
|
||||
canonical instance is unused. \cgalFigureRef{Periodic_3_mesh_3FromCanonicalToWhole}
|
||||
gives an example of such domain where some information is discarded.
|
||||
|
||||
\cgalFigureBegin{Periodic_3_mesh_3FromCanonicalToWholeDiscard, periodicity.svg}
|
||||
Illustration in 2D (cut view) of a domain defined by an implicit function.
|
||||
Any value outside of the canonical cube is unused.
|
||||
\cgalFigureEnd
|
||||
|
||||
For the resulting mesh to make sense, the input domain should behave
|
||||
nicely at the border of the canonical instance. For example, the isovalues of an
|
||||
implicit function should be continuous and without intersections.
|
||||
\cgalFigureRef{Periodic_3_mesh_3ContinuityIssue}
|
||||
|
||||
\cgalFigureBegin{Periodic_3_mesh_3ContinuityIssue, periodicity_issue.svg}
|
||||
Illustration in 2D (cut view) of a domain defined by an implicit function.
|
||||
Good continuity at the border of the domain is marked with green segments,
|
||||
and bad continuity is marked in red.
|
||||
\cgalFigureEnd
|
||||
|
||||
\subsection Periodic_3_mesh_3OutputMesh Output Mesh
|
||||
|
||||
The resulting mesh is output as a subcomplex of a weighted periodic 3D triangulation,
|
||||
The resulting mesh is output as a subcomplex of a
|
||||
\ref CGAL::Periodic_3_regular_triangulation_3 "weighted periodic 3D triangulation",
|
||||
in a class that provides various iterators on mesh elements.
|
||||
|
||||
The periodic 3D triangulation provides approximations of the subdomains,
|
||||
This periodic 3D triangulation provides approximations of the subdomains,
|
||||
surface patches, curves, and corners according to the restricted
|
||||
Delaunay triangulation paradigm. This means that each subdomain is approximated
|
||||
by the union of the tetrahedral cells whose circumcenters are located inside
|
||||
|
|
@ -141,8 +205,8 @@ are represented by mesh vertices.
|
|||
|
||||
\subsubsection Periodic_3_mesh_3OutputFaceGraph Output FaceGraph
|
||||
|
||||
This \cgal component also provides a function to convert the reconstructed mesh to a `FaceGraph`,
|
||||
namely `facets_in_complex_3_to_triangle_mesh()`.
|
||||
This \cgal component also provides a function to extract the facets of the complex
|
||||
as a `FaceGraph`, namely `facets_in_complex_3_to_triangle_mesh()`.
|
||||
|
||||
\subsection Periodic_3_mesh_3DelaunayRefinement Delaunay Refinement
|
||||
|
||||
|
|
@ -175,7 +239,7 @@ the corresponding points are inserted into the Delaunay triangulation
|
|||
from the start.
|
||||
|
||||
If the domain has 1-dimensional exposed features,
|
||||
the method of protecting balls \cgalCite{cgal:cdr-drpsc-07}, \cgalCite{cgal:cdl-pdma-07}
|
||||
the method of protecting balls \cgalCite{cgal:cdl-pdma-07}, \cgalCite{cgal:cdr-drpsc-07}
|
||||
is used to achieve an accurate representation of those features in the mesh
|
||||
and to guarantee that the refinement process terminates
|
||||
whatever may be the dihedral angles formed by input surface patches incident to a
|
||||
|
|
@ -183,8 +247,8 @@ given 1-feature or the angles formed by two 1-features incident to a 0-feature.
|
|||
See Section \ref Mesh_3Protectionof0and1dimensionalExposed in the documentation
|
||||
of the package \ref PkgMesh_3Summary for further information.
|
||||
|
||||
How to prescribe sharp features and examples of periodic meshes with features are presented
|
||||
in Section \ref Periodic_3_mesh_3MeshingDomainswithSharpFeatures.
|
||||
Section \ref Periodic_3_mesh_3MeshingDomainswithSharpFeatures details how
|
||||
to prescribe sharp features and examples of periodic meshes with features
|
||||
|
||||
\subsection Periodic_3_mesh_3OptimizationPhase Optimization Phase
|
||||
|
||||
|
|
@ -234,9 +298,12 @@ of the input domain, while the function `refine_periodic_3_mesh_3()` refines
|
|||
an existing periodic mesh of the input domain.
|
||||
|
||||
\warning The triangulation must form at all times a simplicial complex within
|
||||
a single copy of the domain. It is the responsibility of the user to provide
|
||||
a single copy of the domain (see Sections \ref P3Triangulation3secspace and \ref P3Triangulation3secintro
|
||||
of the manual of 3D periodic triangulations). It is the responsibility of the user to provide
|
||||
a triangulation that satisfies this condition when calling the refinement
|
||||
function `refine_periodic_3_mesh_3()`.
|
||||
function `refine_periodic_3_mesh_3()`. The underlying triangulation of a mesh
|
||||
complex obtained through `make_periodic_3_mesh_3()` or `refine_periodic_3_mesh_3()`
|
||||
will always satisfy this condition.
|
||||
|
||||
\subsection Periodic_3_mesh_3TheDataStructure The Data Structure
|
||||
|
||||
|
|
@ -244,13 +311,13 @@ The template parameter `C3T3` is required to be a model of
|
|||
the concept `MeshComplex_3InTriangulation_3`. This data structure is devised to
|
||||
represent a three-dimensional complex embedded in a periodic 3D triangulation.
|
||||
In both functions, an instance of type `C3T3` is used to maintain the current
|
||||
approximating simplicial mesh and to represent a single instance of
|
||||
approximating simplicial mesh and to represent a single copy of
|
||||
the final periodic 3D mesh at the end of the procedure.
|
||||
|
||||
The embedding periodic 3D triangulation is required to be the nested type
|
||||
`CGAL::Periodic_3_mesh_triangulation_3::type`, provided by the meta functor
|
||||
`CGAL::Periodic_3_mesh_triangulation_3::type`, provided by the class
|
||||
`CGAL::Periodic_3_mesh_triangulation_3`. The type for this triangulation is a
|
||||
wrapper around `CGAL::Periodic_3_regular_triangulation_3` whose vertex and cell
|
||||
wrapper around the class `CGAL::Periodic_3_regular_triangulation_3` whose vertex and cell
|
||||
base classes are respectively models of the concepts `MeshVertexBase_3` and
|
||||
`MeshCellBase_3`.
|
||||
|
||||
|
|
@ -353,32 +420,37 @@ It provides an upper bound on the circumradii of the
|
|||
mesh tetrahedra.
|
||||
</UL>
|
||||
|
||||
Note that if a sizing field is used in the edge criterion, it must naturally
|
||||
have the same periodicity as the fundamental periodic domain.
|
||||
|
||||
\section Periodic_3_mesh_3_section_examples Examples
|
||||
|
||||
This section presents various use cases of the periodic mesh generator.
|
||||
|
||||
\subsection Periodic_3_mesh_3SubMultipleCopies Outputting Multiple Instances of a Periodic Mesh
|
||||
\subsection Periodic_3_mesh_3SubMultipleCopies Outputting Multiple Copies of a Periodic Mesh
|
||||
|
||||
Generated meshes can be output to the `.mesh` file format, which can be read
|
||||
with `medit`. The function `output_to_medit()` takes a stream, a mesh
|
||||
complex, and - optionally - the number of periodic instances that should be drawn,
|
||||
complex, and - optionally - the number of periodic copies that should be drawn,
|
||||
making it easier to observe the periodicity of the result.
|
||||
\cgalFigureRef{Periodic_3_mesh_3Periodic_copies} illustrates the different output
|
||||
for the three possible number of instances: `1`, `2`, `4`, and `8`.
|
||||
for the three possible number of copies: `1`, `2`, `4`, and `8`.
|
||||
|
||||
\cgalFigureAnchor{Periodic_3_mesh_3Periodic_copies}
|
||||
<center>
|
||||
<img src="periodic_copies.jpg" style="max-width:90%;"/>
|
||||
</center>
|
||||
\cgalFigureCaptionBegin{Periodic_3_mesh_3Periodic_copies}
|
||||
Example of a periodic mesh output in `1`, `4`, or `8` instances.
|
||||
Example of a periodic mesh output in `1`, `4`, or `8` copies.
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
In the following examples (except on \cgalFigureRef{Periodic_3_mesh_3Periodic_multi_domain}),
|
||||
each instance of a periodic mesh will be attributed a unique color.
|
||||
each copy of a periodic mesh will be attributed a unique color.
|
||||
|
||||
\attention A single copy of the periodic mesh does not necessarily form a visually-pleasing mesh:
|
||||
there can be uninteded holes, disconnected elements, non-manifold locations, etc.
|
||||
This is because each copy is only made of unique elements and the periodic mesh
|
||||
only has meaning when considered with other neighboring instances:
|
||||
a strange hole on the "left" of a copy is filled by a tetrahedron on its "right".
|
||||
\cgalFigureRef{Periodic_3_mesh_3Periodic_protection} is an example of such phenomenon:
|
||||
These (purely visual) issues disappear when considering multiple copiess together.
|
||||
|
||||
\subsection Periodic_3_mesh_33DDomainsImplicitIsosurfaces 3D Periodic Domains Bounded by Implicit Isosurfaces
|
||||
The following code produces a 3D periodic mesh for a domain whose boundary surface
|
||||
|
|
@ -394,7 +466,7 @@ constructor of the `Mesh_criteria` instance.
|
|||
<img src="periodic_implicit_domain.png" style="max-width:90%;"/>
|
||||
</center>
|
||||
\cgalFigureCaptionBegin{Periodic_3_mesh_3Periodic_implicit_shape}
|
||||
A single instance of a periodic mesh produced from an implicit domain (left).
|
||||
A single copy of a periodic mesh produced from an implicit domain (left).
|
||||
Cut view (middle). Another cut is shown, with 7 copies (right).
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
|
|
@ -424,6 +496,42 @@ Periodic mesh of an implicit sphere that is entirely contained in the fundamenta
|
|||
shown with 7 copies (left). A cut view along one of the axes (right).
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
\subsubsection Periodic_3_mesh_33DDomainsImplicitIsosurfacesBothSides Meshing the Interior and the Exterior of a Domain Bounded by an Isosurface
|
||||
In the case of periodic mesh generation, the exterior of an isosurface defined by
|
||||
an implicit function can also be meshed because it is a finite space.
|
||||
The class `Implicit_to_labeled_subdomains_function_wrapper` is a convenient wrapper
|
||||
to achieve this by internally transforming the interior and the exterior as simply
|
||||
two subdomains to be meshed. It can be simply plugged in the definition of the domain,
|
||||
replacing the line
|
||||
\code{.cpp}
|
||||
...
|
||||
typedef CGAL::Implicit_periodic_3_mesh_domain_3<Function, K> Periodic_mesh_domain;
|
||||
...
|
||||
\endcode
|
||||
in the previous example, with:
|
||||
\code{.cpp}
|
||||
...
|
||||
typedef CGAL::Implicit_to_labeled_subdomains_function_wrapper<Function, K> Function_wrapper;
|
||||
typedef CGAL::Implicit_periodic_3_mesh_domain_3<Function, K, Function_wrapper> Periodic_mesh_domain;
|
||||
...
|
||||
\endcode
|
||||
|
||||
\cgalFigureRef{Periodic_3_mesh_3Periodic_implicit_interior_and_exterior} shows
|
||||
the periodic mesh obtained after this change. Note that both subdomains are meshed
|
||||
with different cell sizing fields, which can be obtained using the class
|
||||
`Mesh_constant_domain_field_3`. See Example \ref Periodic_3_mesh_3/mesh_implicit_shape_with_subdomains.cpp
|
||||
"mesh_implicit_shape_with_subdomains.cpp" for the complete code.
|
||||
|
||||
\cgalFigureAnchor{Periodic_3_mesh_3Periodic_implicit_interior_and_exterior}
|
||||
<center>
|
||||
<img src="periodic_implicit_interior_and_exterior.jpg" style="max-width:70%;"/>
|
||||
</center>
|
||||
\cgalFigureCaptionBegin{Periodic_3_mesh_3Periodic_implicit_interior_and_exterior}
|
||||
A periodic mesh (4 copies, left) of the same implicit function
|
||||
as in \cgalFigureRef{Periodic_3_mesh_3Periodic_implicit_shape}), where both sides
|
||||
of the isosurface are now meshed. On the right, a cut view.
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
\subsection Periodic_3_mesh_3MeshingMultipleDomains Meshing Multiple Domains
|
||||
|
||||
The following code produces a 3D periodic mesh for a domain consisting of a combination
|
||||
|
|
@ -442,7 +550,7 @@ where `si` is the sign of the function `fi(p)` at a point `p` of the subdomain.
|
|||
<center>
|
||||
<img src="periodic_implicit_multi_domain.jpg" style="max-width:90%;"/>
|
||||
</center>
|
||||
\cgalFigureCaptionBegin{Periodic_3_mesh_3Periodic_implicit_sphere}
|
||||
\cgalFigureCaptionBegin{Periodic_3_mesh_3Periodic_multi_domain}
|
||||
A periodic mesh produced by a combination of implicit functions (with 7 copies, left).
|
||||
Cut view without (middle) and with (right) the internal tetrahedra.
|
||||
\cgalFigureCaptionEnd
|
||||
|
|
@ -487,10 +595,10 @@ and corners that are specified by the user:
|
|||
- A curve cannot self-intersect, except at its endpoint (it then forms a so-called `cycle`).
|
||||
- Two curves cannot intersect, except at a common endpoint.
|
||||
|
||||
\warning Curves and corners do not need to be restricted to the fundamental
|
||||
periodic domain, but users should be mindful that curves and corners will exist
|
||||
in all copies of the domain and the requirements described above must still be verified.
|
||||
For example, if considering a unit cube, it is not a valid input
|
||||
\warning For conveniency, curves and corners do not need to be restricted
|
||||
to the fundamental domain, but users should be mindful that curves and corners will exist
|
||||
in all periodic instances and the requirements described above must be satisfied.
|
||||
For example, if considering the unit cube, it is not a valid input
|
||||
to prescribe a corner at the point `(1.5, 1.5, 1.5)` and the curve `(2,2,2)--(3,3,3)`
|
||||
because the corner intersects the curve in the middle.
|
||||
|
||||
|
|
@ -502,6 +610,8 @@ that the apex will appear in the mesh.
|
|||
\cgalFigureRef{Periodic_3_mesh_3Periodic_protection} shows the comparison between
|
||||
the same domain meshed with the same criteria when protection is disabled and enabled.
|
||||
|
||||
Note the use of a more complex intrinsically non-periodic implicit function as input domain.
|
||||
|
||||
\cgalExample{Periodic_3_mesh_3/mesh_implicit_shape_with_protection.cpp}
|
||||
|
||||
\cgalFigureAnchor{Periodic_3_mesh_3Periodic_protection}
|
||||
|
|
@ -509,10 +619,13 @@ the same domain meshed with the same criteria when protection is disabled and en
|
|||
<img src="periodic_implicit_protection.jpg" style="max-width:90%;"/>
|
||||
</center>
|
||||
\cgalFigureCaptionBegin{Periodic_3_mesh_3Periodic_protection}
|
||||
A periodic mesh obtained with a cone-like periodic implicit function (8 instances
|
||||
A periodic mesh obtained with a cone-like periodic implicit function (8 copies
|
||||
are drawn). The base and the apex of the cone
|
||||
are badly approximated when protection is unused (left).
|
||||
Using protection, we obtain a faithful approximation.
|
||||
Facets seemingly disconnected from other facets of the same color are due to the fact that
|
||||
a single copy of the mesh does not necessarily form a visually pleasing mesh by itself
|
||||
(see Section \ref Periodic_3_mesh_3SubMultipleCopies for more details).
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
\subsubsection Periodic_3_mesh_3InfiniteFeatures Example of Infinite Protecting Curves
|
||||
|
|
@ -520,8 +633,36 @@ Using protection, we obtain a faithful approximation.
|
|||
It is possible to prescribe features that will, due to periodicity, extend indefinitely
|
||||
in space. A simple example of such occurrence is the segment `(0,0,0) -- (1,1,1)` when
|
||||
considering the unit cube as fundamental domain.
|
||||
\cgalFigureRef{Periodic_3_mesh_3Periodic_protection} shows a domain where sharp edges
|
||||
can be protected to obtain a nicer result.
|
||||
\cgalFigureRef{Periodic_3_mesh_3Periodic_protection} shows an implicit function
|
||||
describing a square-based prism such that the axis of the prism is the (Oz) axis.
|
||||
(The prism is thus 'cut' into 4 pieces when considered within a single instance of the periodic space.)
|
||||
The fundamental domain is the unit cube and the following polylines have been specified to protect edges:
|
||||
\code
|
||||
// These are four vertical edges (orthogonal to xOy), strictly in the domain.
|
||||
// They correspond to the four sharp edges of the prism.
|
||||
LV0: ( 0.3, 0.3, -1) -- ( 0.3, 0.3, 0) // canonically, (0.3, 0.3, 0) -- (0.3, 0.3, 1)
|
||||
LV1: ( 1.7, 1.7, 2) -- ( 1.7, 1.7, 3) // canonically, (0.7, 0.7, 0) -- (0.7, 0.7, 1)
|
||||
LV2: ( 0.7, 0.3, 4) -- ( 0.7, 0.3, 5) // canonically, (0.7, 0.3, 0) -- (0.7, 0.3, 1)
|
||||
LV3: (-0.7, -0.3, 6) -- (-0.7, -0.3, 7) // canonically, (0.3, 0.7, 0) -- (0.3, 0.7, 1)
|
||||
|
||||
// These are four vertical edges (orthogonal to xOy) on the border of the domain.
|
||||
// The prism does not intrinsically possess a sharp feature at these locations;
|
||||
// rather, these edges are protected to artifically create features on the border of the domain.
|
||||
LV4: (0.3, 0, 0) -- (0.3, 0, 1)
|
||||
LV5: (0, 1.3, 0) -- (0, 1.3, 1) // canonically, (0., 0.3, 0) -- (0., 0.3, 1)
|
||||
LV6: (0, 0.7, 0) -- (0, 0.7, 1)
|
||||
LV7: (2.7, 2, 0) -- (2.7, 2, 1) // canonically, (0.7, 0., 0) -- (0.7, 0., 1)
|
||||
|
||||
// These are four horizontal edges (in the plane xOy) on the border of the domain.
|
||||
// As the last four polylines, these polylines do not correspond to a sharp feature
|
||||
// on the prism, but are rather just for aesthetics.
|
||||
//
|
||||
// Note that these polylines cross the border of the fundamental domain.
|
||||
LH0: ( 0.3, 0.3, 0) -- ( 0.3, -0.3, 0)
|
||||
LH1: ( 0.3, -0.3, 1) -- (-0.3, -0.3, 1)
|
||||
LH2: (-0.3, -0.3, 2) -- (-0.3, 0.3, 2)
|
||||
LH3: (-0.3, 0.3, 3) -- ( 0.3, 0.3, 3)
|
||||
\endcode
|
||||
|
||||
\cgalFigureAnchor{Periodic_3_mesh_3Periodic_protection_infinite}
|
||||
<center>
|
||||
|
|
|
|||
|
|
@ -59,7 +59,7 @@ public:
|
|||
/**
|
||||
* Constructor
|
||||
* @param f the function which negative values defines the domain
|
||||
* @param cuboid a bounding box of the domain, periodic domain
|
||||
* @param cuboid the fundamental domain
|
||||
* @param error_bound the error bound relative to the sphere radius
|
||||
*/
|
||||
Implicit_periodic_3_mesh_domain_3(const Function& f,
|
||||
|
|
|
|||
|
|
@ -30,7 +30,7 @@
|
|||
|
||||
// Main differences with Mesh_3's version:
|
||||
// - A map [Vertex_handle] --> [position in full space] because:
|
||||
// * This position might not be in the canonical domain
|
||||
// * This position might not be (in full space coordinates) in the canonical instance
|
||||
// * Different curves might represent the same corner at different positions
|
||||
// - change_ball_size()'s signature is different: it can return false if we failed
|
||||
// to change the ball size (if the triangulation cover changes during the process).
|
||||
|
|
@ -534,7 +534,7 @@ private:
|
|||
|
||||
// The correspondence map is a map that keeps track of the position of the points
|
||||
// in the domain class, which might differ due to periodicity. For example,
|
||||
// someone might be using the unit cube as periodic domain and add the vertex(1,0,0)
|
||||
// someone might be using the unit cube as fundamental domain and add the vertex(1,0,0)
|
||||
// as corner. If 'v' is the vertex handle of that corner, we have v->point() = (0,0,0).
|
||||
//
|
||||
// This map must be kept perfectly up to date: removing a vertex from the triangulation
|
||||
|
|
|
|||
|
|
@ -446,7 +446,7 @@ public:
|
|||
// Warning : This function finds which offset 'Oq' should be applied to 'q' so
|
||||
// that the distance between 'p' and '(q, Oq)' is minimal.
|
||||
//
|
||||
// \pre 'p' lives in the canonical domain.
|
||||
// \pre 'p' lives in the canonical instance.
|
||||
Bare_point get_closest_point(const Bare_point& p, const Bare_point& q) const
|
||||
{
|
||||
Bare_point rq;
|
||||
|
|
|
|||
|
|
@ -21,7 +21,7 @@ typedef P3DT3::Locate_type Locate_type;
|
|||
|
||||
int main(int, char**)
|
||||
{
|
||||
Iso_cuboid domain(-1,-1,-1,2,2,2); // the cube for the periodic domain
|
||||
Iso_cuboid domain(-1,-1,-1,2,2,2); // the fundamental domain
|
||||
|
||||
// construction from a list of points :
|
||||
std::list<Point> L;
|
||||
|
|
|
|||
|
|
@ -22,7 +22,7 @@ typedef P3RT3::Locate_type Locate_type;
|
|||
|
||||
int main(int, char**)
|
||||
{
|
||||
Iso_cuboid domain(-1,-1,-1, 2,2,2); // the cube for the periodic domain
|
||||
Iso_cuboid domain(-1,-1,-1, 2,2,2); // the fundamental domain
|
||||
|
||||
// construction from a list of weighted points :
|
||||
std::list<Weighted_point> L;
|
||||
|
|
|
|||
|
|
@ -687,7 +687,7 @@ public:
|
|||
}
|
||||
|
||||
// The following functions return the "real" position in space (unrestrained
|
||||
// to the original periodic domain) of the vertices v and c->vertex(idx),
|
||||
// to the fundamental domain) of the vertices v and c->vertex(idx),
|
||||
// respectively
|
||||
|
||||
Point point(Vertex_handle v) const
|
||||
|
|
|
|||
|
|
@ -999,7 +999,7 @@ public:
|
|||
}
|
||||
|
||||
// The following functions return the "real" position in space (unrestrained
|
||||
// to the original periodic domain) of the vertices v and c->vertex(idx),
|
||||
// to the fundamental domain) of the vertices v and c->vertex(idx),
|
||||
// respectively
|
||||
|
||||
Weighted_point point(Vertex_handle v) const
|
||||
|
|
|
|||
|
|
@ -643,7 +643,7 @@ public:
|
|||
}
|
||||
|
||||
// The following functions return the "real" position in space (unconstrained
|
||||
// to the original periodic domain) of the vertices v and c->vertex(idx),
|
||||
// to the fundamental domain) of the vertices v and c->vertex(idx),
|
||||
// respectively
|
||||
template <class ConstructPoint>
|
||||
Point point(Vertex_handle v, ConstructPoint cp) const {
|
||||
|
|
@ -2583,9 +2583,9 @@ template < class GT, class TDS >
|
|||
inline typename Periodic_3_triangulation_3<GT,TDS>::Vertex_handle
|
||||
Periodic_3_triangulation_3<GT,TDS>::create_initial_triangulation(const Point& p)
|
||||
{
|
||||
/// Virtual vertices, one per periodic domain
|
||||
/// Virtual vertices, one per periodic instance
|
||||
Vertex_handle vir_vertices[3][3][3];
|
||||
/// Virtual cells, 6 per periodic domain
|
||||
/// Virtual cells, 6 per periodic instance
|
||||
Cell_handle cells[3][3][3][6];
|
||||
|
||||
// Initialise vertices:
|
||||
|
|
|
|||
|
|
@ -39,7 +39,7 @@ class Periodic_3_triangulation_tetrahedron_iterator_3 {
|
|||
// - UNIQUE: output exactly one periodic copy of each primitive, no matter
|
||||
// whether the current tds stores a n-sheeted covering for n!=1.
|
||||
// - STORED_COVER_DOMAIN: output each primitive whose intersection with the
|
||||
// actually used periodic domain is non-zero.
|
||||
// fundamental domain is non-zero.
|
||||
// - UNIQUE_COVER_DOMAIN: output each primitive whose intersection
|
||||
// with the original domain that the user has given is non-zero
|
||||
//
|
||||
|
|
@ -265,7 +265,7 @@ private:
|
|||
int get_drawing_offsets() {
|
||||
Offset off0, off1, off2, off3;
|
||||
// Choose edges that are to be duplicated. These are edges that
|
||||
// intersect the boundary of the periodic domain. In UNIQUE mode
|
||||
// intersect the boundary of the fundamental domain. In UNIQUE mode
|
||||
// this means that the offset with respect to drawing should
|
||||
// differ in some entries. Otherwise we consider the offsets
|
||||
// internally stored inside the cell telling us that this cell
|
||||
|
|
@ -344,7 +344,7 @@ class Periodic_3_triangulation_triangle_iterator_3 {
|
|||
// - UNIQUE: output exactly one periodic copy of each primitive, no matter
|
||||
// whether the current tds stores a n-sheeted covering for n!=1.
|
||||
// - STORED_COVER_DOMAIN: output each primitive whose intersection with the
|
||||
// actually used periodic domain is non-zero.
|
||||
// fundamental domain is non-zero.
|
||||
// - UNIQUE_COVER_DOMAIN: output each primitive whose intersection
|
||||
// with the original domain that the user has given is non-zero
|
||||
//
|
||||
|
|
@ -568,7 +568,7 @@ private:
|
|||
int get_drawing_offsets() {
|
||||
Offset off0, off1, off2;
|
||||
// Choose edges that are to be duplicated. These are edges that
|
||||
// intersect the boundary of the periodic domain. In UNIQUE mode
|
||||
// intersect the boundary of the fundamental domain. In UNIQUE mode
|
||||
// this means that the offset with respect to drawing should
|
||||
// differ in some entries. Otherwise we consider the offsets
|
||||
// internally stored inside the cell telling us that this cell
|
||||
|
|
@ -634,7 +634,7 @@ class Periodic_3_triangulation_segment_iterator_3 {
|
|||
// - UNIQUE: output exactly one periodic copy of each primitive, no matter
|
||||
// whether the current tds stores a n-sheeted covering for n!=1.
|
||||
// - STORED_COVER_DOMAIN: output each primitive whose intersection with the
|
||||
// actually used periodic domain is non-zero.
|
||||
// fundamental domain is non-zero.
|
||||
// - UNIQUE_COVER_DOMAIN: output each primitive whose intersection
|
||||
// with the original domain that the user has given is non-zero
|
||||
//
|
||||
|
|
@ -842,7 +842,7 @@ private:
|
|||
int get_drawing_offsets() {
|
||||
Offset off0, off1;
|
||||
// Choose edges that are to be duplicated. These are edges that
|
||||
// intersect the boundary of the periodic domain. In UNIQUE mode
|
||||
// intersect the boundary of the fundamental domain. In UNIQUE mode
|
||||
// this means that the offset with respect to drawing should
|
||||
// differ in some entries. Otherwise we consider the offsets
|
||||
// internally stored inside the cell telling us that this cell
|
||||
|
|
@ -894,7 +894,7 @@ class Periodic_3_triangulation_point_iterator_3 {
|
|||
// - UNIQUE: output exactly one periodic copy of each primitive, no matter
|
||||
// whether the current tds stores a n-sheeted covering for n!=1.
|
||||
// - STORED_COVER_DOMAIN: output each primitive whose intersection with the
|
||||
// actually used periodic domain is non-zero.
|
||||
// fundamental domain is non-zero.
|
||||
// - UNIQUE_COVER_DOMAIN: output each primitive whose intersection
|
||||
// with the original domain that the user has given is non-zero
|
||||
//
|
||||
|
|
|
|||
|
|
@ -98,7 +98,7 @@ construct_periodic_point_exact(const typename Gt_::Point_3& p,
|
|||
}
|
||||
|
||||
// Given a point `p` in space, compute its offset `o` with respect
|
||||
// to the canonical domain and returns (p, o)
|
||||
// to the canonical instance and returns (p, o)
|
||||
template <typename Gt_>
|
||||
std::pair<typename Gt_::Point_3, typename Gt_::Periodic_3_offset_3>
|
||||
construct_periodic_point(const typename Gt_::Point_3& p, bool& encountered_issue, const Gt_& gt)
|
||||
|
|
|
|||
|
|
@ -19,9 +19,9 @@
|
|||
#include <vector>
|
||||
|
||||
// A basic test to check that P3RT3 and RT3 produces the same regular triangulations
|
||||
// In the case of RT3, the fake periodic domain is obtained by addding 26 copies
|
||||
// In the case of RT3, a fake periodicity is obtained by addding 26 copies
|
||||
// of the same input point set (similarly to how we copy in P3RT3 when it cannot
|
||||
// yet be converted to 1 sheet)
|
||||
// yet be converted to 1 sheet) around the cube.
|
||||
|
||||
typedef CGAL::Exact_predicates_inexact_constructions_kernel Epick;
|
||||
typedef CGAL::Periodic_3_regular_triangulation_traits_3<Epick> PRTT_Inexact;
|
||||
|
|
@ -33,7 +33,7 @@ typedef P3RT3::Offset Offset;
|
|||
typedef P3RT3::Weighted_point Weighted_point;
|
||||
|
||||
// The info() is a unique color per point + a boolean to mark those in the "fake"
|
||||
// canonical domain (the cuboid surrounded by 26 copies)
|
||||
// instances (the cuboid surrounded by 26 copies)
|
||||
typedef CGAL::Regular_triangulation_vertex_base_3<Epick> Vb0;
|
||||
typedef CGAL::Triangulation_vertex_base_with_info_3<
|
||||
std::pair<int, bool>, Epick, Vb0> Vb;
|
||||
|
|
@ -82,7 +82,7 @@ int main (int, char**)
|
|||
|
||||
if(v != Vertex_handle())
|
||||
v->info() = std::make_pair(id /*unique id*/,
|
||||
(i==0 && j==0 && k==0) /*canonical domain?*/);
|
||||
(i==0 && j==0 && k==0) /*canonical instance?*/);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
|
@ -103,7 +103,7 @@ int main (int, char**)
|
|||
Finite_vertices_iterator vit = rt3.finite_vertices_begin();
|
||||
for(; vit!=rt3.finite_vertices_end(); ++vit)
|
||||
{
|
||||
if(vit->info().second) // is in the canonical domain
|
||||
if(vit->info().second) // is in the canonical instance
|
||||
unique_vertices_from_rt3.insert(vit->info().first);
|
||||
}
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue