mirror of https://github.com/CGAL/cgal
Updated documentation
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Mael Rouxel-Labbé
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namespace CGAL {
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/*!
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\ingroup PkgPeriodic_3_mesh_3Domains
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The class `Periodic_3_function_wrapper` is a helper class designed to wrap
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an (a priori non-periodic) implicit function describing a domain through the relationship
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[`p` is inside if `f(p)<0`] and defined over the canonical cube to a function
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defined over the whole Euclidean space and periodic, with the same period as the canonical cube.
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More precisely, if `f` is the real function defined either over \f$ \mathbb R^3\f$
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or over the canonical cube, we construct the periodic real function \f$ f^{\ast} \f$ defined over
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\f$ \mathbb R^3\f$ as follows:
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- For any point \f$(x,y,z)\f$ in the canonical cube, \f$ f^{\ast}(x,y,z) = f(x,y,z)\f$
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- For any point \f$(x,y,z)\f$ outside the canonical cube, there is a unique canonical representative
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\f$(x_0,y_0,z_0)\f$ of \f$(x,y,z)\f$ in the canonical cube, i.e.,
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\f$(x,y,z)=(x_0 + k_x c, y_0 + k_y c, z_0 + k_z c)\f$ with \f$(k_x,k_y,k_z)\f$ in \f$ \mathbb Z^3\f$,
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and \f$ f^{\ast}(x,y,z) = f(x_0,y_0,z_0) \f$.
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For example, if considering the unit cube as canonical cube, an oracle answering a
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query such as <I>"what is the value of the implicit function at this point?"</I>
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at the point `(2.5, 2.5, 2.5)` will be in fact evaluated at the canonical representative, that is
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`(0.5, 0.5, 0.5)`.
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Consequently, it is then not required to provide an input domain that is defined over the whole
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space or periodic, but only defined over the canonical cube.
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\cgalFigureBegin{Periodic_3_mesh_3FromCanonicalToWhole, periodicity_base.svg}
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Illustration in 2D (cut view) of a domain defined by an implicit function that is transformed
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into a periodic implicit function.
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Only the values of the implicit function that are in the canonical cube are used:
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the values of the implicit function at \f$ P \f$ and \f$ Q \f$ are obtained by evaluating
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instead at \f$ P' \f$ and \f$ Q' \f$, as shown on the right.
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\cgalFigureEnd
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In practice, the implicit function provided by the user is likely defined
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over a larger domain than the canonical cube (in general, it is \f$ \mathbb R^3\f$).
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Note that -- when constructing artifically periodic functions -- all the values of the implicit function
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for points outside this canonical cube are unused since queries are always answered by looking at the canonical representative.
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\cgalFigureRef{Periodic_3_mesh_3FromCanonicalToWholeDiscard} gives an example of such domain where some information is discarded.
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\cgalFigureBegin{Periodic_3_mesh_3FromCanonicalToWholeDiscard, periodicity.svg}
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Illustration in 2D (cut view) of a domain defined by an implicit function made artifically periodic.
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Any value of the function outside of the canonical cube is ignored.
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\cgalFigureEnd
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Note also that when construction artificially periodic functions, it is the responsability of the user
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to provide an input function that is compatible with the canonical cube (that is, whose isovalues
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are <em>periodicially</em> continuous and without intersections).
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\cgalFigureRef{Periodic_3_mesh_3ContinuityIssue} is an example of a bad choice
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of input function and canonical cube: there is no continuity of the isovalues
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at the border of the canonical cube. In such configuration, the mesher might
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or might not finish and the result is likely to be non-manifold and to contain self-intersections.
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\cgalFigureBegin{Periodic_3_mesh_3ContinuityIssue, periodicity_issue.svg}
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Illustration in 2D (cut view) of a domain defined by an implicit function made artifically periodic.
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The zero isovalue of the implicit function does not form a continuous curve.
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\cgalFigureEnd
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\tparam Function provides the definition of the function.
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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.
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The number types `Function::FT` and `BGT::FT` are required to match.
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\tparam BGT is a geometric traits class that provides
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the basic operations to implement
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intersection tests and intersection computations
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through a bisection method. This parameter must be instantiated
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with a model of the concept `BisectionGeometricTraits_3`.
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*/
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template<class Function, class BGT>
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class Periodic_3_function_wrapper
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{
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public:
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/// \name Types
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/// @{
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//!
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typedef typename BGT::FT FT;
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typedef typename BGT::Point_3 Point_3;
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typedef typename BGT::Iso_cuboid_3 Iso_cuboid_3;
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/// @}
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/// \name Creation
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/// @{
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/*!
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* \brief Construction from an implicit function and the canonical cube.
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*/
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Periodic_3_function_wrapper(const Function& f, const Iso_cuboid_3& domain);
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/// @}
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/// \name Operations
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/// @{
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/*!
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* Evaluates the function \f$ f \f$ passed in input at the canonical representative of \f$ p \f$.
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*/
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FT operator()(const Point_3& p) const;
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/// @}
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}; /* end Implicit_to_labeled_subdomains_function_wrapper */
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} /* end namespace CGAL */
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@ -23,9 +23,9 @@ Wrapping any model of `Periodic_3MeshDomain_3` with the class
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of `Periodic_3MeshDomainWithFeatures_3`.
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\cgalHasModel `CGAL::Mesh_domain_with_polyline_features_3<
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CGAL::Implicit_periodic_3_mesh_domain_3<Function,BGT> >`
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\cgalHasModel `CGAL::Mesh_domain_with_polyline_features_3<
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CGAL::Labeled_periodic_3_mesh_domain_3<LabelingFunction,BGT> >`
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CGAL::Labeled_mesh_domain_3<BGT> >`
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\sa `CGAL::Periodic_3_function_wrapper<Function,BGT>`
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\sa `CGAL::make_periodic_3_mesh_3()`
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\sa `CGAL::refine_periodic_3_mesh_3()`
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@ -9,14 +9,20 @@ object to be discretized.
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The concept `Periodic_3MeshDomain_3` is the concept to be used when the input
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domain is defined over the three-dimensional flat torus.
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From a syntactic point of view, it defines almost exactly the same requirements
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as the concept `MeshDomain_3` and thus `Periodic_3MeshDomain_3` refines `MeshDomain_3`.
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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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From a syntactic point of view, it defines almost the same requirements
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as the concept `MeshDomain_3` and thus `Periodic_3MeshDomain_3` refines `MeshDomain_3`:
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the concept `Periodic_3MeshDomain_3` additionally requires an access to the user-defined
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canonical cube via the function `bounding_box`.
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However, the oracle must take into account the periodicity of the domain (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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The class `CGAL::Labeled_mesh_domain_3<BGT>` paired with a periodic labeling function
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is a model of this concept. It is possible to create artificially periodic functions
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through the class `CGAL::Periodic_3_function_wrapper<Function,BGT>`.
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\cgalHasModel `CGAL::Labeled_mesh_domain_3<BGT>`
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\sa `CGAL::Labeled_mesh_domain_3<BG>`
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\sa `CGAL::make_periodic_3_mesh_3()`
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\sa `CGAL::refine_periodic_3_mesh_3()`
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@ -26,14 +32,14 @@ class Periodic_3MeshDomain_3
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{
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public:
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/*!
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Cuboid type.
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The canonical cube type.
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*/
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typedef unspecified_type Iso_cuboid_3;
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/*!
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Return the user-chosen cube that is the canonical instance of the flat torus.
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returns the user-chosen cube that is the canonical instance of the flat torus.
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*/
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const Iso_cuboid_3& canonical_periodic_domain();
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const Iso_cuboid_3& bounding_box();
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/// @}
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@ -66,8 +66,7 @@ This package offers these interfaces.
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This package relies entirely on the \ref PkgMesh_3Concepts
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and the \ref PkgMesh_3SecondaryConcepts described in the \ref PkgMesh_3Summary package.
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The following concepts are essentially identical to the concepts `MeshDomain_3`
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and `MeshDomainWithFeatures_3` in the package \ref PkgMesh_3Summary,
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but are nevertheless defined for clarity:
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and `MeshDomainWithFeatures_3` in the package \ref PkgMesh_3Summary :
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- `Periodic_3MeshDomain_3`
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- `Periodic_3MeshDomainWithFeatures_3`
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@ -78,9 +77,8 @@ The following class provides the interface between %CGAL's periodic triangulatio
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and %CGAL's three-dimensional mesh generator:
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- `CGAL::Periodic_3_mesh_triangulation_3<MD, K, Vertex_base, Cell_base>`
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The following classes are models of the periodic domain concept:
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- `CGAL::Implicit_periodic_3_mesh_domain_3<Function,BGT>`
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- `CGAL::Labeled_periodic_3_mesh_domain_3<LabelingFunction,BGT>`
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The following class allows to construct a periodic implicit function from an implicit function that is not periodic:
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- `CGAL::Periodic_3_function_wrapper<Function,BGT>`
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The following class allows to split the canonical cube in two subdomains,
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separated by the zero-level of an implicit function:
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@ -105,6 +103,4 @@ and \ref PkgMesh_3Parameters.
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## Input/Output Functions ##
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- \link PkgPeriodic_3_mesh_3IOFunctions `CGAL::output_periodic_mesh_to_medit()` \endlink
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*/
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@ -62,7 +62,7 @@ by a mesh optimization phase to remove slivers and provide a good quality mesh.
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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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meshes discretizing (non-periodic) 3D domains and to the \ref PkgPeriodic3Triangulation3Summary of %CGAL,
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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. We consider the flat torus \f$ \mathbb T_c^3\f$,
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@ -73,9 +73,10 @@ are inserted at the beginning of the meshing process, ensuring that the projecti
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of the periodic triangulation into the flat torus \f$ \mathbb T_c^3\f$ forms at all times a simplicial complex
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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 be exclusively conducted within the canonical cube.
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The mesh is created using the \ref PkgMesh_3Summary package of %CGAL,
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which originally aims to mesh non-periodic domains of \f$ \mathbb R^3\f$, and so, an interface
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Thanks to this "scaffholding" structure, the meshing process can be exclusively conducted
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within the canonical cube.
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The mesh can then be created using the \ref PkgMesh_3Summary package of %CGAL.
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As this package originally aims to mesh non-periodic domains of \f$ \mathbb R^3\f$, an interface
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is necessary between the packages \ref PkgMesh_3Summary and \ref PkgPeriodic3Triangulation3Summary.
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This package provides this interface.
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@ -102,7 +103,7 @@ that are called vertices, edges, facets and cells.
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Note that the input complex faces are not required to be linear nor smooth.
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Surface patches, for instance, may be smooth surface patches,
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or portions of surface meshes with boundaries.
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Curves may be for instance straight segments, parameterized curves
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Curves may be for instance straight segments, parameterized curves,
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or polylines. Each of those features will be accurately represented in the final mesh.
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The 0 and 1-dimensional features of the input domain are usually singular points
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@ -153,64 +154,21 @@ of the flat torus \f$ \mathbb T_c^3\f$. The origin (given by three coordinates \
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and \f$ \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$.
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Although the mesh is only constructed over the canonical cube, some of the oracles
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used during the generation of the mesh must sometimes be evaluated outside of the canonical cube.
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The implicit function describing the domain to be meshed must thus be defined
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over the whole Euclidean space and be periodic, with a period compatible with the canonical cube.
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The mesh is only constructed over the canonical cube and periodicity
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is handled internally: oracles described above evaluate queries at a point
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by instead evaluating at the representative that belongs in the canonical cube.
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More precisely, if `f` is a real function defined either over \f$ \mathbb R^3\f$
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or over the canonical cube, we define the periodic real function `f^{\ast}` defined over
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\f$ \mathbb R^3\f$ as follows:
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- For any point \f$(x,y,z)\f$ in the canonical cube, \f$ f^{\ast}(x,y,z) = f(x,y,z)\f$
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- For any point \f$(x,y,z)\f$ outside the canonical cube, there is a unique canonical representative
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\f$(x_0,y_0,z_0)\f$ of \f$(x,y,z)\f$ in the canonical cube, i.e.,
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\f$(x,y,z)=(x_0 + k_x c, y_0 + k_y c, z_0 + k_z c)\f$ with \f$(k_x,k_y,k_z)\f$ in \f$ \mathbb Z^3\f$,
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and \f$ f^{\ast}(x,y,z) = f(x_0,y_0,z_0) \f$.
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\subsubsection Periodic_3_mesh_3ArtificialPeriodicity Artifical Domain Periodicity
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For example, if considering the unit cube as canonical cube, an oracle answering a
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query such as <I>"what is the value of the implicit function at this point?"</I>
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at the point `(2.5, 2.5, 2.5)` will be in fact evaluated at the canonical position,
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`(0.5, 0.5, 0.5)`.
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Consequently It is thus not required to provide an input domain that is periodic over the
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whole space or oracles that can answer queries for all points of \f$ \mathbb R^3\f$, but only
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for points in the canonical cube.
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This approach allows to construct periodic meshes of domains that are not intrinsically
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periodic, for example a sphere (see
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\cgalFigureRef{Periodic_3_mesh_3FromCanonicalToWholeDiscard}
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and \cgalFigureRef{Periodic_3_mesh_3Periodic_implicit_sphere}) or a cone
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(see Section \ref Periodic_3_mesh_3ConeWithSharpFeatures).
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\cgalFigureBegin{Periodic_3_mesh_3FromCanonicalToWhole, periodicity_base.svg}
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Illustration in 2D (cut view) of a domain defined by an implicit function.
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Only the values of the implicit function that are in the canonical cube matter:
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the values of the implicit function at `P` and `Q` are obtained by evaluating
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instead at `P'` and `Q'`, as shown on the right.
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\cgalFigureEnd
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In practice, the implicit function provided by the user is likely defined
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over a larger domain than the canonical cube (in general, it is even \f$ \mathbb R^3\f$).
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Note that since queries are always answered by looking at the canonical
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position, all information provided by the input domain that is outside this
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canonical cube is unused. \cgalFigureRef{Periodic_3_mesh_3FromCanonicalToWholeDiscard}
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gives an example of such domain where some information is discarded.
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\cgalFigureBegin{Periodic_3_mesh_3FromCanonicalToWholeDiscard, periodicity.svg}
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Illustration in 2D (cut view) of a domain defined by an implicit function.
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Any value outside of the canonical cube is unused.
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\cgalFigureEnd
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Note also that it is the responsability of the user to provide an input function
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whose isovalues are <em>periodicially</em> continuous and without intersections.
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\cgalFigureRef{Periodic_3_mesh_3ContinuityIssue} is an example of a bad choice
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of input function and canonical cube: there is no continuity of the isovalues
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at the border of the canonical cube. In such configuration, the mesher might
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or might not finish and the result is likely to not be non-manifold and to
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contain self-intersections.
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\cgalFigureBegin{Periodic_3_mesh_3ContinuityIssue, periodicity_issue.svg}
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Illustration in 2D (cut view) of a domain defined by an implicit function.
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The zero isovalue of the implicit function does not form a continuous curve.
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\cgalFigureEnd
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The specifications of the input implicit function described in the previous section are quite restrictive.
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To relax these requirements, this package also offers a wrapper class,
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`CGAL::Periodic_3_function_wrapper`, to artifically construct
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periodic functions compatible with the user-defined canonical cube,
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from the values of an implicit function over the canonical cube.
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It is thus possible to construct periodic domains described by implicit functions
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that are not intrinsically periodic, for example a sphere (see \cgalFigureRef{Periodic_3_mesh_3Periodic_implicit_sphere})
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or a cone (see Section \ref Periodic_3_mesh_3ConeWithSharpFeatures).
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\subsection Periodic_3_mesh_3OutputMesh Output Mesh
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@ -365,10 +323,9 @@ the point lies, if inside.
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If the domain description includes 0 and 1-dimensional features
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that have to be accurately represented in the final mesh,
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the template parameter `PeriodicMeshDomain` is required to be
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of a model of the refined concept `Periodic_3MeshDomainWithFeatures_3`.
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The concept `Periodic_3MeshDomainWithFeatures_3` mainly provides the incidence
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graph of 0, 1 and 2-dimensional features, and a member function to construct
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sample points on curves.
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of a model of the refined concept `Periodic_3MeshDomainWithFeatures_3`,
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which mainly provides the incidence graph of 0, 1 and 2-dimensional features,
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and a member function to construct sample points on curves.
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Users whose domain is a model of `Periodic_3MeshDomainWithFeatures_3`
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can choose to have the corners and curves of the domain
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@ -456,10 +413,10 @@ protecting ball centers that are consecutive on a 1-feature. This parameter has
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This section presents various use cases of the periodic mesh generator.
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\subsection Periodic_3_mesh_3SubMultipleCopies Outputting Multiple Copies of a Periodic Mesh
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\subsection Periodic_3_mesh_3SubMultipleCopies Visualizing Multiple Copies of a Periodic Mesh
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Generated meshes can be output to the `.mesh` file format, which can be read
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with `medit`. The function \link PkgPeriodic_3_mesh_3IOFunctions `CGAL::output_periodic_mesh_to_medit()` \endlink
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Generated meshes can be output to the `.mesh` file format, which can be visualized with the demo
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of the package \ref PkgPolyhedronSummary. The function \link PkgPeriodic_3_mesh_3IOFunctions `CGAL::output_periodic_mesh_to_medit()` \endlink
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takes a stream, a mesh complex, and - optionally - the number of periodic copies that should be drawn,
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making it easier to observe the periodicity of the result.
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\cgalFigureRef{Periodic_3_mesh_3Periodic_copies} illustrates the different output
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@ -503,17 +460,23 @@ Cut view (middle). Another cut is shown, using 8 copies (right).
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\cgalFigureCaptionEnd
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While the implicit function used in the previous example is defined and periodic
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over the complete space, it is also possible to consider an implicit function
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defined entirely within the canonical cube. It will then be periodically
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duplicated. For example, replacing the previous domain and with the following
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implicit function (a sphere):
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over the complete space, it is also possible to consider non-periodic implicit functions
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defined entirely within the canonical cube (or over the whole space) by using the wrapper class
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`CGAL::Periodic_3_function_wrapper`. Values will then be periodically duplicated, creating
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a periodic function. For example, replacing the previous domain with the following
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non-periodic implicit function (a sphere):
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\code{.cpp}
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...
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typedef CGAL::Periodic_3_function_wrapper<Function, K> Periodic_function;
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...
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// A sphere centered on (0.5, 0.5, 0.5) with radius sqrt(0.2)
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FT sphere_function (const Point& p) { return CGAL::squared_distance(p, Point(0.5, 0.5, 0.5)) - 0.2; }
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...
|
||||
Periodic_mesh_domain domain(sphere_function, CGAL::Iso_cuboid_3<K>(0, 0, 0, 1, 1, 1));
|
||||
Iso_cuboid_3 canonical_cube(0, 0, 0, 1, 1, 1);
|
||||
Periodic_mesh_domain domain =
|
||||
Periodic_mesh_domain::create_implicit_mesh_domain(
|
||||
Periodic_3_function(sphere_function, canonical_cube), canonical_cube );
|
||||
...
|
||||
\endcode
|
||||
|
||||
|
|
@ -533,18 +496,18 @@ In the case of periodic mesh generation, the exterior of an isosurface defined b
|
|||
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:
|
||||
two subdomains to be meshed. Compared to the previous example, it can be simply achieved
|
||||
by adding the line
|
||||
\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
|
||||
and wrapping the function as follows:
|
||||
\code{.cpp}
|
||||
...
|
||||
Function_wrapper wrapper(schwarz_p);
|
||||
Periodic_mesh_domain domain(wrapper, canonical_cube);
|
||||
...
|
||||
\endcode
|
||||
|
||||
|
|
@ -570,7 +533,7 @@ The following code produces a 3D periodic mesh for a domain consisting of a comb
|
|||
of the two implicit functions used in the previous example: a sphere is encompassed
|
||||
within the implicit domain of \cgalFigureRef{Periodic_3_mesh_3Periodic_implicit_shape}.
|
||||
The class `Implicit_multi_domain_to_labeling_function_wrapper` is used as model
|
||||
of `ImplicitFunction`, required by the class `Labeled_periodic_3_mesh_domain_3`.
|
||||
of `ImplicitFunction`, required by the class `Labeled_mesh_domain_3`.
|
||||
The set of subdomains is given by a vector of vector of signs.
|
||||
Each subdomain corresponds to a sign vector `[s1, s2, ..., sn]`,
|
||||
where `si` is the sign of the function `fi(p)` at a point `p` of the subdomain.
|
||||
|
|
|
|||
Loading…
Reference in New Issue