# Mesh parameterization

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> Markdown URL: https://mediated.wiki/source/Mesh_parameterization.md
> Source: https://en.wikipedia.org/wiki/Mesh_parameterization
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Given two [surface](/source/surface_(topology))s with the same [topology](/source/topology), a [bijective mapping](/source/bijective_mapping) between them exists. On [triangular mesh](/source/Polygon_mesh) surfaces, the problem of computing this mapping is called mesh [parameterization](/source/Parametrization_(atmospheric_modeling)). The parameter domain is the surface that the mesh is mapped onto.

Parameterization was mainly used for [mapping textures](/source/Texture_mapping) to surfaces. Recently, it has become a powerful tool for many applications in mesh processing.{{Citation needed|date=July 2011}} Various techniques are developed for different types of parameter domains with different parameterization properties.

== Applications ==
* [Texture mapping](/source/Texture_mapping)
* Normal mapping
* Detail transfer
* [Morphing](/source/Morphing)
* Mesh completion
* Mesh Editing
* Mesh Databases
* [Remeshing](/source/Remeshing)
* [Surface fitting](/source/Surface_fitting)

== Techniques ==
* Barycentric Mappings
* Differential Geometry Primer
* Non-Linear Methods

== Implementations ==
* [http://www.riken.jp/brict/Yoshizawa/Research/Param.html A fast and simple stretch-minimizing mesh parameterization]
* [http://alice.loria.fr/index.php/software/3-platform/22-graphite.html Graphite]: ABF++, [LSCM](/source/Least_squares_conformal_map), Spectral [LSCM](/source/Least_squares_conformal_map)
* [https://web.archive.org/web/20100624001221/http://www.cs.caltech.edu/~keenan/project_dgp.html Linear discrete conformal parameterization]
* [http://www.dgp.toronto.edu/~rms/software/expmapdemo.html Discrete Exponential Map]
* [https://geometrycollective.github.io/boundary-first-flattening Boundary First Flattening]
* [https://github.com/MichaelRabinovich/Scalable-Locally-Injective-Mappings Scalable Locally Injective Mappings]
* [https://doc.cgal.org/latest/Manual/packages.html#PkgSurfaceMeshParameterization Triangulated Surface Mesh Parameterization], a chapter of [CGAL](/source/CGAL), the Computational Geometry Algorithms Library

==See also==

* [Parametrization](/source/Parametrization_(geometry))
* [Texture atlas](/source/Texture_atlas)
* [UV Mapping](/source/UV_Mapping)

==External links==
* [http://www.inf.usi.ch/hormann/parameterization/index.html "Mesh Parameterization: theory and practice"]

Category:3D computer graphics

{{Computer-graphics-stub}}

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Adapted from the Wikipedia article [Mesh parameterization](https://en.wikipedia.org/wiki/Mesh_parameterization) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Mesh_parameterization?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.