# Planar Parameterization of Triangulated Surface Meshes

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## Planar Parameterization of Triangulated Surface Meshes

 Hi, I’d like to perform a surface parametrization of a triangle mesh. I tried using some of CGAL’s algorithms, e.g. ARAP, Discrete Conformal Map etc. The problem is that the surface parameterization methods proposed by CGAL only deal with meshes which are homeomorphic to discs. Meshes with arbitrary topology can be parameterized, provided that the user specifies a cut graph, which defines the border of a topological disc. So the problem now becomes – how to cut the graph properly (using CGAL’s interface). I would really appreciate a simple example.   Another option, is to use the Orbifold Tutte Embeddings algorithm, that way the mesh only has to be homeomorphic to a sphere, but I still need to specify the “cones”. So again, I’d really appreciate a simple example.   I’m attaching a small mesh which is a typical example of the sorts of meshes I need to parametrize.   Thanks a lot.   mesh.ply (423K) Download Attachment
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## Re: Planar Parameterization of Triangulated Surface Meshes

 Hello, You can use the class Seam_mesh of CGAL to create virtual borders on meshes. See this example for a simple use case of parameterization of a seam mesh. A simple example of usage of the orbifold parameterization method is offered here. The mesh you have attached has self-intersections and is not a topological disk, though. You will have to repair it first in any case. Best, Mael On 01/11/2018 10:42, Shalev Mintz wrote: Hi, I’d like to perform a surface parametrization of a triangle mesh. I tried using some of CGAL’s algorithms, e.g. ARAP, Discrete Conformal Map etc. The problem is that the surface parameterization methods proposed by CGAL only deal with meshes which are homeomorphic to discs. Meshes with arbitrary topology can be parameterized, provided that the user specifies a cut graph, which defines the border of a topological disc. So the problem now becomes – how to cut the graph properly (using CGAL’s interface). I would really appreciate a simple example.   Another option, is to use the Orbifold Tutte Embeddings algorithm, that way the mesh only has to be homeomorphic to a sphere, but I still need to specify the “cones”. So again, I’d really appreciate a simple example.   I’m attaching a small mesh which is a typical example of the sorts of meshes I need to parametrize.   Thanks a lot.
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## RE: Planar Parameterization of Triangulated Surface Meshes

 Thanks Mael.   I saw the examples you linked to, but what they don’t show is how to choose the actual vertices/edges of the cut graph. They just read a list of vertex indices from a file, but it is unclear how those indices were chosen in the first place. Same goes for choosing the cones in the Orbifold parameterization.   Is there a way I can recognize self-intersections in a mesh? How can I “repair” it in such a case and make it a topological disk/sphere?   Thanks a lot, Shalev.       From: [hidden email] [mailto:[hidden email]] On Behalf Of Mael Sent: Friday, November 2, 2018 10:15 AM To: [hidden email] Subject: Re: [cgal-discuss] Planar Parameterization of Triangulated Surface Meshes   Hello, You can use the class Seam_mesh of CGAL to create virtual borders on meshes. See this example for a simple use case of parameterization of a seam mesh. A simple example of usage of the orbifold parameterization method is offered here. The mesh you have attached has self-intersections and is not a topological disk, though. You will have to repair it first in any case. Best, Mael On 01/11/2018 10:42, Shalev Mintz wrote: Hi, I’d like to perform a surface parametrization of a triangle mesh. I tried using some of CGAL’s algorithms, e.g. ARAP, Discrete Conformal Map etc. The problem is that the surface parameterization methods proposed by CGAL only deal with meshes which are homeomorphic to discs. Meshes with arbitrary topology can be parameterized, provided that the user specifies a cut graph, which defines the border of a topological disc. So the problem now becomes – how to cut the graph properly (using CGAL’s interface). I would really appreciate a simple example.   Another option, is to use the Orbifold Tutte Embeddings algorithm, that way the mesh only has to be homeomorphic to a sphere, but I still need to specify the “cones”. So again, I’d really appreciate a simple example.   I’m attaching a small mesh which is a typical example of the sorts of meshes I need to parametrize.   Thanks a lot.