Given two surfaces with the same topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a bijective mapping between them exists. On triangular mesh

Polygon mesh

A polygon mesh or unstructured grid is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling...

 surfaces, the problem of computing this mapping is called mesh parameterization. The parameter domain is the surface that the mesh is mapped onto.

Parameterization was mainly used for mapping textures

Texture mapping

Texture mapping is a method for adding detail, surface texture , or color to a computer-generated graphic or 3D model. Its application to 3D graphics was pioneered by Dr Edwin Catmull in his Ph.D. thesis of 1974.-Texture mapping:...

 to surfaces. Recently, it has become a powerful tool for many applications in mesh processing. Various techniques are developed for different types of parameter domains with different parameterization properties.

Applications

• Texture mapping
  Texture mapping
  Texture mapping is a method for adding detail, surface texture , or color to a computer-generated graphic or 3D model. Its application to 3D graphics was pioneered by Dr Edwin Catmull in his Ph.D. thesis of 1974.-Texture mapping:...
  
  
• Normal mapping
• Detail transfer
• Morphing
  Morphing
  Morphing is a special effect in motion pictures and animations that changes one image into another through a seamless transition. Most often it is used to depict one person turning into another through technological means or as part of a fantasy or surreal sequence. Traditionally such a depiction...
  
  
• Mesh completion
• Mesh Editing
• Mesh Databases
• Remeshing
• Surface fitting

Implementations

• A fast and simple stretch-minimizing mesh parameterization
• Graphite: ABF, ABF++, DPBF, LSCM
  Least Squares Conformal Map
  A Least squares conformal map is a 2-D representation of a 3-D shape created using the Least Squares Conformal Mapping Method. By using the map as a guide when creating a new 2-D image, the colors of the 2-D image can be applied to the original 3-D model.LSCM is used in computer graphics as a...
  
  , HLSCM, Barycentric, mean-value coordinates, L2 stretch, spectral conformal, Periodic Global Parameterization, Constrained texture mapping, texture atlas generation
• Linear discrete conformal parameterization
• Discrete Exponential Map

External Links

"Mesh Parameterization: theory and practice"