UV Mapping (UV Transformation)

UV mapping, or unwrapping, in three-dimensional graphics (UV map) is the correspondence between coordinates on the surface of a three-dimensional object (X, Y, Z) and coordinates on a texture (U, V). The values of U and V typically range from 0 to 1. Unwrapping can be done either manually or automatically — for example, 3Ds Studio MAX offers several algorithms for automatic model unwrapping.

Modern three-dimensional hardware assumes that the UV transformation within a single triangle is affine — consequently, it is sufficient to specify U and V for each vertex of each triangle. However, it is the 3D modeler who chooses exactly how to join the triangles together, and the ability to build a successful unwrap is one of the indicators of the modeler's skill. There are several criteria for unwrap quality that conflict with one another:

• Maximal use of the texture area. However, depending on the gap between the «minimum» and «maximum» system requirements, a certain «margin» is needed at the edges of the unwrap to allow the generation of smaller-sized textures
• Absence of areas with insufficient or excessive texture detail.
• Absence of areas with excessive geometric distortion.
• Similarity to the standard viewpoints from which the object is usually drawn or photographed simplifies the work of the texture artist.
• Well-placed «seams» — lines corresponding to a single edge but located in different places on the texture. Seams are desirable where there is a natural «break» in the surface (clothing seams, edges, joints, etc.), and undesirable where there is none. Amateurs from all over the world took part in modeling the Dota 2 characters, and the modeling guidelines required that the eyes be a separate «island» of the unwrap.
• For partially symmetric objects: a successful combination of symmetric and asymmetric sections of the unwrap. Symmetry increases texture detail and simplifies the work of the texture artist; asymmetric details «bring the object to life».

Equirectangular projection — an example of mapping a two-dimensional texture (coordinates U, V) onto a three-dimensional globe (coordinates X, Y, Z)

A checkerboard texture on a sphere: on the left — a single matrix applied to the whole sphere, on the right — an unwrap with an equirectangular projection

An example of mapping a texture onto a cube. The texture is in the form of a flat unwrap.

In cartography, a cylindrical projection is a family of projections that transform meridians into verticals (U=const) and parallels into horizontals (V=const). One of the cylindrical projections, the so-called equirectangular projection (φ — latitude, [−π; π]; λ — longitude, [−½π; ½π]):

A similar transformation can also be applied in computer graphics to map a texture of the Earth's surface onto a three-dimensional sphere. According to these metrics:

• Use of texture area: excellent. The entire texture area is used.
• Absence of areas with insufficient/excessive detail: medium. There is excess detail at the poles.
• Similarity to standard viewpoints: good. This projection resembles ordinary cartographic ones, although it does not coincide with them.
• Absence of seams: excellent. There is a single seam along the meridian (between U=0 and U=1). The seam can be made as inconspicuous as possible by placing an ocean at its location; also, by cyclically shifting the texture, one can carefully draw the seam and then shift it back.
• Geometric distortion: poor. At the poles, the Earth's surface is strongly distorted.
• Symmetry: not applicable. The globe is asymmetric.

Drawn and photographic textures use only two coordinates: U and V. In the case of auto-generated textures or multitexturing, a third coordinate is possible — W, depth in the «texture continuum» . For example, W=0 might be sand, W=1 — rocks, with intermediate values being sand with rocks poking through.

Theoretically, a fourth, fifth, and so on coordinates are possible — however, graphics hardware, when applying textures, performs the transformation and discards the third coordinate. UVW mapping puts the discarded coordinate to use for transitions from one texture to another: this brings the «world behind the screen» to life without significantly changing the design of the graphics card.

See also

• Map projection
• Geodesic
• Least squares conformal mapping
• Mesh parameterization
• NURBS (Non-Uniform Rational B-Spline)
• Polygon mesh
• Radon transform
• Lightmap
• Digital sculpting
• Barycentric coordinate system