mirror of https://github.com/CGAL/cgal
Improved SMP's documentation
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Mael Rouxel-Labbé
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@ -65,8 +65,7 @@ This package implements all common border parameterization methods:
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- For fixed border methods:
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- the user can select a border parameterization among two common methods:
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uniform or arc-length parameterizations.
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- one convex shape specified by:
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- one shape among a set of standard ones (circle, square).
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- the user can select a convex shape among a set of standard ones (circle, square).
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- For free border methods: at least two constraints (the pinned vertices).
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- `CGAL::Circular_border_uniform_parameterizer_3<TriangleMesh>`
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@ -36,11 +36,10 @@ planar domain (convex polygon vs. free border) and by the guarantees
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provided in terms of bijective mapping.
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The package proposes an interface for any model of the concept `FaceGraph`,
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that is classes as `Polyhedron_3`, `Surface_mesh`, as well as mesh
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such as the classes `Surface_mesh`, `Polyhedron_3`, or the mesh
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classes of OpenMesh.
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Since parameterizing meshes require efficient representation of sparse
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Since parameterizing meshes requires an efficient representation of sparse
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matrices and efficient iterative or direct linear solvers, we provide
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a unified interface to linear solvers as described in Chapter
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\ref PkgSolverSummary.
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@ -48,9 +47,8 @@ a unified interface to linear solvers as described in Chapter
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Note that linear solvers commonly use double precision floating point
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numbers. Therefore, this package is intended to be used with a \cgal %Cartesian kernel with doubles.
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\cgalFigureBegin{Surface_mesh_parameterizationfigintroduction,introduction.jpg}
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Texture mapping via Least Squares Conformal Maps parameterization. Top: original mesh and texture. Bottom: parameterizedmesh (left: parameter space, right: textured mesh).
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Texture mapping via Least Squares Conformal Maps parameterization. Top: original mesh and texture. Bottom: parameterized mesh (left: parameter space, right: textured mesh).
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\cgalFigureEnd
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\section Surface_mesh_parameterizationBasics Basics
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@ -65,34 +63,32 @@ template <typename TriangleMesh>
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Parameterizer_traits<TriangleMesh>::Error_code
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parameterize(TriangleMesh & mesh,
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boost::graph_traits<TriangleMesh>::halfedge_descriptor bhd,
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VertexUvMap uvm);
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VertexUVMap uvm);
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\endcode
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The function `parameterize()` applies a default surface parameterization
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method, namely Floater Mean Value Coordinates \cgalCite{cgal:f-mvc-03},
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to the connected component of the mesh of type `TriangleMesh` with the border given by the halfedge `bhd`.
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This border is parameterized with an arc-length circular border parameterization.
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The sparse linear solver used is from the \ref thirdpartyEigen library.
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The sparse linear solver of the \ref thirdpartyEigen library is used.
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The mesh of type `TriangleMesh` which must be a model of the concept
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`FaceGraph`, additionally must be triangulated, 2-manifold, oriented,
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and homeomorphic to a disc (possibly with holes). We will later show
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how to parameterize a mesh that is not a topological disk.
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The mesh of type `TriangleMesh` must be a model of the concept
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`FaceGraph` and must additionally be triangulated, 2-manifold, oriented,
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and homeomorphic to a disc (possibly with holes). The last requirement
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is not hard and we will later show how to parameterize a mesh that is
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not a topological disk (Section \ref Surface_mesh_parameterizationComputingaCut).
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The result is stored in a property map (whose type is here VertexUVMap)
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for the mesh vertices. See also Chapter \ref PkgProperty_mapSummary.
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The result is stored in a property map for the mesh vertices. See also
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Chapter \ref PkgProperty_mapSummary.
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`Parameterizer_traits_3<TriangleMesh>` is a base class of all surface
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The class `Parameterizer_traits_3<TriangleMesh>` is a base class of all surface
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parameterizations and defines the error codes.
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\subsection Surface_mesh_parameterizationDefaultExample Default Parameterization Example
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In the following example, we apply the default parameterization to a
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`Polyhedron_3` mesh, and we use a `Unique_hash_map` to store the uv-coordinates
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of each vertex.
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\cgalExample{Surface_mesh_parameterization/Simple_parameterization.cpp}
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`Surface_mesh` mesh. We store the UV-coordinates as a vertex property
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using the `Surface_mesh` built-in property mechanism.
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\cgalExample{Surface_mesh_parameterization/simple_parameterization.cpp}
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\subsection Surface_mesh_parameterizationEnhancedparameterize Choosing a Parameterization Algorithm
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@ -105,22 +101,22 @@ Parameterizer_traits<TriangleMesh>::Error_code
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parameterize(TriangleMesh& mesh,
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ParameterizerTraits_3 parameterizer,
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boost::graph_traits<TriangleMesh>::halfedge_descriptor bhd,
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VertexUvMap uvm);
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VertexUVMap uvm);
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\endcode
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It computes a one-to-one mapping from a 3D triangle surface mesh to a simple 2D domain.
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The mapping is piecewise linear on the triangle mesh. The result is a pair (u,v)
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of parameter coordinates for each vertex of the input mesh.
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A one-to-one mapping may be guaranteed or not, depending on the chosen
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ParameterizerTraits_3 algorithm.
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In the following example, we apply the circular border parameterizations
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and then the discrete authalic parameterization to a
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`Surface_mesh`. We store the uv-coordinates as a vertex property
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using the built-in property mechanism.
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In the following example, we use a circular uniform border parameterization
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and then use the discrete authalic parameterizer with a
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`Surface_mesh`. We store the UV-coordinates as a vertex property
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using the `Surface_mesh` built-in property mechanism.
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\cgalExample{Surface_mesh_parameterization/discrete_authalic.cpp}
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\section secSurfaceParameterizationMethods Surface Parameterization Methods
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This \cgal package implements surface parameterization methods, such
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@ -224,40 +220,35 @@ Border parameterizations for fixed border surface parameterizations
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are a family of methods to define a set of constraints, namely two
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\f$ u,v\f$ coordinates for each vertex along the border.
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Different choices are offered to the user when choosing border parameterizers
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for fixed border methods:
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<UL>
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<LI>The user can select a border parameterization among
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two commonly used methods: uniform or arc-length parameterization.
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<I>Usage:</I>
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Uniform border parameterization is more stable, although it gives
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poor visual results. The
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arc-length border parameterization is used by default.
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<I>Usage:</I> Uniform border parameterizations are more stable,
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although they give poor visual results. The arc-length border
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parameterization is used by default.
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<LI>One convex shape specified by one shape among two standard ones:
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a circle or a square.
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<I>Usage:</I>
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The circular border parameterization is used by default as it
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<I>Usage:</I> The circular border parameterization is used by default as it
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corresponds to the simplest convex shape. The square border
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parameterization is commonly used for texture mapping.
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</UL>
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`Circular_border_arc_length_parameterizer_3<TriangleMesh>`
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`Circular_border_uniform_parameterizer_3<TriangleMesh>`
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`Square_border_arc_length_parameterizer_3<TriangleMesh>`
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`Square_border_uniform_parameterizer_3<TriangleMesh>`
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\cgalFigureBegin{Surface_mesh_parameterizationfigcircular_border,border.png}
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Left: Julius Cesar mask parameterization with Authalic/circular border. Right: Julius Cesar mask's image with Floater/square border.
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\cgalFigureEnd
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All combinations of uniform/arc-length and circle/square are provided by
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the following classes:
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`Circular_border_arc_length_parameterizer_3<TriangleMesh>`
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`Circular_border_uniform_parameterizer_3<TriangleMesh>`
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`Square_border_arc_length_parameterizer_3<TriangleMesh>`
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`Square_border_uniform_parameterizer_3<TriangleMesh>`
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\subsection Surface_mesh_parameterizationFreeBorderSurface Free Border Surface Parameterizations
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\subsubsection Surface_mesh_parameterizationLeastSquares Least Squares Conformal Maps
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@ -289,35 +280,37 @@ parameterization methods define only two constraints: the pinned vertices.
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<LI>`Two_vertices_parameterizer_3<TriangleMesh>`
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<I>Usage:</I>
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`Two_vertices_parameterizer_3<TriangleMesh>` is the default
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<I>Usage:</I> `Two_vertices_parameterizer_3<TriangleMesh>` is the default
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free border parameterization, and is the only one available
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in the current version of this package.
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</UL>
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\section secCuttingaMesh Cutting a Mesh
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\subsection Surface_mesh_parameterizationComputingaCut Computing a Cut Graph
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All surface parameterization methods proposed in this package only
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The surface parameterization methods proposed in this package only
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deal with meshes which are homeomorphic (topologically equivalent) to
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discs. Nevertheless meshes with arbitrary topology and number of
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connected components car be parameterized, provided that the user
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discs. Nevertheless, meshes with arbitrary topology and number of
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connected components can be parameterized, provided that the user
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specifies a cut graph (a set of edges), which defines the
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border of a topological disc. These edges can be passed together with
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the mesh to a `Seam_mesh`.
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a mesh to a `Seam_mesh`.
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\cgalFigureBegin{Surface_mesh_parameterizationfigcut,cut.png}
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Cut Graph
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\cgalFigureEnd
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In the following example, we apply the default parameterization to a
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`Polyhedron_3`-based `Seam_mesh`. We store the UV-coordinates as a
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halfedge property using a `Unique_hash_map`.
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\cgalExample{Surface_mesh_parameterization/seam.cpp}
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Note that vertices on a seam are duplicated in a `Seam_mesh` structure
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and thus the UV-coordinates are here associated to halfedges of the
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seam mesh.
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\cgalExample{Surface_mesh_parameterization/seam_Polyhedron_3.cpp}
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\section Surface_mesh_parameterizationComplexity Complexity and Guarantees
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@ -1,6 +1,6 @@
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/*!
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\example Surface_mesh_parameterization/Simple_parameterization.cpp
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\example Surface_mesh_parameterization/simple_parameterization.cpp
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\example Surface_mesh_parameterization/discrete_authalic.cpp
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\example Surface_mesh_parameterization/seam.cpp
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\example Surface_mesh_parameterization/seam_Polyhedron_3.cpp
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\example Surface_mesh_parameterization/lscm.cpp
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*/
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@ -216,7 +216,7 @@ protected:
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}
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}
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/// Compute w_ij = (i, j) coefficient of matrix A for j neighbor vertex of i.
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/// Compute w_ij, coefficient of matrix A for j neighbor vertex of i.
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/// Implementation note: Subclasses must at least implement compute_w_ij().
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///
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/// \param mesh a triangulated surface.
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