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@ -26,14 +26,14 @@ transparency, reflection or light modulation maps), fitting scattered
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data, re-parameterizing spline surfaces, repairing CAD models,
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approximating surfaces and remeshing.
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This \cgal package implements
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surface parameterization methods, such as As Rigid As Possible Parameterization,
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Least Squares Conformal Maps, Discrete Conformal Map, Discrete Authalic
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Parameterization, Floater Mean Value Coordinates or Tutte Barycentric
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Mapping. These methods mainly distinguish by the distortion they
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This \cgal package implements surface parameterization methods, such as
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As Rigid As Possible Parameterization, Tutte Barycentric Mapping,
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Discrete Authalic Parameterization, Discrete Conformal Maps,
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Least Squares Conformal Maps, Floater Mean Value Coordinates, or Orbifold Tutte Embeddings.
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These methods mainly distinguish by the distortion they
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minimize (angles vs. areas), by the constrained border onto the
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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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planar domain (convex polygon vs. free border) and by the bijectivity guarantees
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of the mapping.
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The package proposes an interface for any model of the concept `FaceGraph`,
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such as the classes `Surface_mesh`, `Polyhedron_3`, or the mesh
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@ -102,12 +102,12 @@ Error_code parameterize(TriangleMesh& mesh,
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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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A one-to-one mapping may be guaranteed or not, depending on the choice of the
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Parameterizer_3 algorithm.
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In the following example, we use a circular 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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In the following example, we use the Discrete Authalic parameterizer with a
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a circular border parameterization.
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We use a `Surface_mesh` for the mesh and 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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@ -135,7 +135,7 @@ The Barycentric Mapping parameterization method has been introduced by
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Tutte \cgalCite{t-hdg-63}. In parameter space, each vertex is
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placed at the barycenter of its neighbors to achieve the so-called
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convex combination condition. This algorithm amounts to solve one
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sparse linear solver for each set of parameter coordinates, with a
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sparse linear system for each set of parameter coordinates, with a
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\#vertices x \#vertices sparse and symmetric positive definite matrix
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(if the border vertices are eliminated from the linear system).
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A coefficient \f$ (i, j)\f$ of the matrix is set to 1 for an edge linking
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@ -145,54 +145,14 @@ entry. Although a bijective mapping is guaranteed when the border is convex,
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this method does not minimize angles nor areas distortion.
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\cgalFigureBegin{Surface_mesh_parameterizationfiguniform,uniform.png}
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Left: Tutte barycentric mapping parameterization (the red line depicts the cut graph). Right: parameter space.
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\cgalFigureEnd
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\subsubsection Surface_mesh_parameterizationDiscreteConformal Discrete Conformal Map
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`Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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Discrete conformal map parameterization has been introduced to the graphics community
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by Eck et al. \cgalCite{cgal:eddhls-maam-95}. It attempts to
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lower angle deformation by minimizing a discrete version of the
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Dirichlet energy as derived by Pinkall and
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Polthier \cgalCite{cgal:pp-cdmsc-93}. A one-to-one mapping is guaranteed
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only when the two following conditions are fulfilled: the barycentric mapping
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condition (each vertex in parameter space is a convex combination if
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its neighboring vertices), and the border is convex.
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This method solves two \#vertices x \#vertices sparse linear
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systems. The matrix (the same for both systems) is sparse and symmetric definite
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positive (if the border vertices are eliminated from the linear system
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and if the mesh contains no hole),
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thus can be efficiently solved using dedicated linear solvers.
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\cgalFigureBegin{Surface_mesh_parameterizationfigconformal,conformal.png}
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Left: discrete conformal map. Right: parameter space.
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\cgalFigureEnd
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\subsubsection Surface_mesh_parameterizationFloaterMean Floater Mean Value Coordinates
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`Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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The mean value coordinates parameterization method has been introduced
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by Floater \cgalCite{cgal:f-mvc-03}. Each vertex in parameter space is
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optimized so as to be a convex combination of its neighboring
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vertices. The barycentric coordinates are this time unconditionally
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positive, by deriving an application of the mean theorem for harmonic
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functions. This method is in essence an approximation of the discrete conformal
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maps, with a guaranteed one-to-one mapping when the border is convex.
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This method solves two \#vertices x \#vertices sparse linear systems. The matrix (the
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same for both systems) is asymmetric.
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\cgalFigureBegin{Surface_mesh_parameterizationfigfloater,floater.png}
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Floater Mean Value Coordinates
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Left: Tutte Barycentric mapping parameterization (the red line depicts the cut graph). Right: parameter space.
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\cgalFigureEnd
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\subsubsection Surface_mesh_parameterizationDiscreteAuthalic Discrete Authalic Parameterization
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`Surface_mesh_parameterization::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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The discrete authalic parameterization method has been introduced by
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The Discrete Authalic parameterization method has been introduced by
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Desbrun et al. \cgalCite{cgal:dma-ipsm-02}. It corresponds to
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a weak formulation of an area-preserving method, and in essence
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locally minimizes the area distortion. A one-to-one mapping is
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@ -205,7 +165,47 @@ for both systems) is asymmetric.
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Discrete Authalic Parameterization
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\cgalFigureEnd
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\subsection secBorderParameterizationsforFixedMethods Border Parameterizations for Fixed Methods
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\subsubsection Surface_mesh_parameterizationDiscreteConformal Discrete Conformal Map
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`Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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Discrete Conformal Map parameterization has been introduced to the graphics community
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by Eck et al. \cgalCite{cgal:eddhls-maam-95}. It attempts to
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lower angle deformation by minimizing a discrete version of the
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Dirichlet energy as derived by Pinkall and
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Polthier \cgalCite{cgal:pp-cdmsc-93}. A one-to-one mapping is guaranteed
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only when the two following conditions are fulfilled: the barycentric mapping
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condition (each vertex in parameter space is a convex combination of
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its neighboring vertices), and the border is convex.
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This method solves two \#vertices x \#vertices sparse linear
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systems. The matrix (the same for both systems) is sparse and symmetric positive
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definite(if the border vertices are eliminated from the linear system
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and if the mesh contains no hole),
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thus can be efficiently solved using dedicated linear solvers.
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\cgalFigureBegin{Surface_mesh_parameterizationfigconformal,conformal.png}
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Left: Discrete Conformal Map. Right: parameter space.
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\cgalFigureEnd
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\subsubsection Surface_mesh_parameterizationFloaterMean Floater Mean Value Coordinates
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`Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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The mean value coordinates parameterization method has been introduced
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by Floater \cgalCite{cgal:f-mvc-03}. Each vertex in parameter space is
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optimized so as to be a convex combination of its neighboring
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vertices. The barycentric coordinates are this time unconditionally
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positive, by deriving an application of the mean theorem for harmonic
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functions. This method is in essence an approximation of the Discrete Conformal
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Map, with a guaranteed one-to-one mapping when the border is convex.
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This method solves two \#vertices x \#vertices sparse linear systems. The matrix (the
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same for both systems) is asymmetric.
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\cgalFigureBegin{Surface_mesh_parameterizationfigfloater,floater.png}
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Floater Mean Value Coordinates
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\cgalFigureEnd
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\subsubsection secBorderParameterizationsforFixedMethods Border Parameterizations for Fixed Methods
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Parameterization methods for borders are used as traits classes modifying the
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behavior of `Parameterizer_3` models. They are also provided as models of the
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@ -246,7 +246,8 @@ the following classes:
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</UL>
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An illustration of the use of different border parameterizers can be found in the
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example `Surface_mesh_parameterization/square_border_parameterizer.cpp`.
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example \ref Surface_mesh_parameterization/square_border_parameterizer.cpp
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"square_border_parameterizer.cpp".
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\subsection Surface_mesh_parameterizationFreeBorderSurface Free Border Surface Parameterizations
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@ -264,7 +265,7 @@ is not guaranteed by this method. It solves a (2 \f$ \times\f$
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which implies solving a symmetric matrix.
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\cgalFigureBegin{Surface_mesh_parameterizationfigLSCM,LSCM.png}
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Least squares conformal maps.
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Least Squares Conformal Maps.
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\cgalFigureEnd
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\subsubsection Surface_mesh_parameterizationARAP As Rigid As Possible Parameterization
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@ -290,7 +291,7 @@ possible) when λ goes to infinity, or a balance of both.
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As Rigid As Possible parameterization (the cut is traced in red).
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\cgalFigureCaptionEnd
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\subsection secBorderParameterizationsforFreeMethods Border Parameterizations for Free Methods
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\subsubsection secBorderParameterizationsforFreeMethods Border Parameterizations for Free Methods
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Parameterization methods for borders are used as traits classes modifying
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the behavior of `Parameterizer_3` models. They are also provided as models of
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@ -310,7 +311,7 @@ in the current version of this package.
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\subsection Surface_mesh_parameterizationBorderless Borderless Surface Parameterizations
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\subsubsection Surface_mesh_parameterizationOrbi Orbifold Tutte Embedding
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\subsubsection Surface_mesh_parameterizationOrbi Orbifold Tutte Embeddings
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`Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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@ -318,7 +319,7 @@ Orbifold-Tutte Planar Embedding was introduced by Aigerman and Lipman \cgalCite{
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and is a generalization of Tutte’s embedding to other topologies, and in particular
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spheres. The orbifold-Tutte embedding bijectively maps the original surface to
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a canonical, topologically equivalent, two-dimensional flat surface called
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a Euclidean orbifold. There are 17 Euclidean orbifolds, of which only the 4 sphere
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an Euclidean orbifold. There are 17 Euclidean orbifolds, of which only the 4 sphere
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orbifolds are implemented here.
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The orbifold-Tutte embedding yields a seamless, globally bijective parameterization that,
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@ -327,7 +328,7 @@ for its computation.
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The parameterization process internally requires the uses of seams, but the choice
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of these seams have no influence on the result. The `Seam_mesh` structure
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(see also next Section) is used for this purpose.
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(see also Section \ref secCuttingaMesh) is used for this purpose.
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\cgalFigureAnchor{Surface_mesh_parameterizationfigOrbifold}
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<center>
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@ -375,15 +376,15 @@ seam mesh.
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<LI>One-to-one mapping
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Tutte's theorem guarantees a one-to-one mapping provided that the weights are all positive
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and the border convex.
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It is the case for Tutte barycentric mapping and Floater mean value coordinates.
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It is not always the case for discrete conformal map (cotangents) and
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discrete authalic parameterization.
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and the border is convex.
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It is the case for Tutte Barycentric Mapping and Floater Mean Value Coordinates.
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It is not always the case for Discrete Conformal Map (cotangents) and
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Discrete Authalic parameterization.
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<LI>Non-singularity of the matrix
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Geshorgin's theorem guarantees the convergence of the solver if the matrix is diagonal dominant.
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This is the case with positive weights (Tutte barycentric mapping and Floater mean value coordinates).
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This is the case with positive weights (Tutte Barycentric Mapping and Floater Mean Value Coordinates).
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</UL>
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@ -403,7 +404,7 @@ and is therefore non-singular (Gram theorem).
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</UL>
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<LI> Boundaryless
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<LI> Boundary-less
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<UL>
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Mael Rouxel-Labbé