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Tutte Embeddings of Tetrahedral Meshes.

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Title: Tutte Embeddings of Tetrahedral Meshes.
Authors: Alexa, Marc1 (AUTHOR) marc.alexa@tu-berlin.de
Source: Discrete & Computational Geometry. Jan2025, Vol. 73 Issue 1, p197-207. 11p.
Subjects: Harmonic maps, Embedding theorems, Polyhedra, Triangles, Planar graphs
Abstract: Tutte's embedding theorem states that every 3-connected graph without a K 5 - or K 3 , 3 -minor (i.e., a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of their neighbors. We show that this result extends to simply connected tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if the outer polyhedron is in convex position and the interior vertices are convex combination of their neighbors it is sufficient (but not necessary) that the graph of the tetrahedral mesh contains no K 6 and no K 3 , 3 , 1 , and all triangles incident on three boundary vertices are boundary triangles. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Tutte's embedding theorem states that every 3-connected graph without a K 5 - or K 3 , 3 -minor (i.e., a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of their neighbors. We show that this result extends to simply connected tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if the outer polyhedron is in convex position and the interior vertices are convex combination of their neighbors it is sufficient (but not necessary) that the graph of the tetrahedral mesh contains no K 6 and no K 3 , 3 , 1 , and all triangles incident on three boundary vertices are boundary triangles. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00454-023-00494-0
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      – Code: eng
        Text: English
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        PageCount: 11
        StartPage: 197
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      – SubjectFull: Harmonic maps
        Type: general
      – SubjectFull: Embedding theorems
        Type: general
      – SubjectFull: Polyhedra
        Type: general
      – SubjectFull: Triangles
        Type: general
      – SubjectFull: Planar graphs
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      – TitleFull: Tutte Embeddings of Tetrahedral Meshes.
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            – D: 01
              M: 01
              Text: Jan2025
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              Y: 2025
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