UV conversion

Equidistant projection - an example of applying a two-dimensional texture ( U , V coordinates) to a three-dimensional globe ( X , Y , Z coordinates)

Chess texture on a sphere: on the left is a common matrix for the entire sphere, on the right is a scan with an equal-intermediate projection

An example of texture mapping on a cube . The texture has the form of a flat scan .

UV conversion or development in a three-dimensional graph ( English UV map ) - the correspondence between the coordinates on the surface of a three-dimensional object ( X , Y , Z ) and the coordinates on the texture ( U , V ). The values of U and V usually vary from 0 to 1 ^[1] . The scan can be built either manually or automatically - for example, in 3D Studio MAX there are several algorithms for the automatic deployment of the model.

Modern three-dimensional hardware considers that UV-conversion within the limits of one triangle is affine - therefore it is enough to set U and V for each vertex of each of the triangles. However, the 3D modeler chooses exactly how to join the triangles with each other, and the ability to build a successful scan is one of the indicators of his class. There are several conflicting scan quality indicators:

Maximum use of the texture area. However, depending on the gap between the "minimum" and "maximum" system requirements, the edges of the scan texture need a certain "allowance" for generating smaller textures . ^[2]
Lack of areas with insufficient or excessive texture detail.
Lack of areas with excessive geometric distortion.
The similarity with the standard angles from which an object is usually drawn or photographed simplifies the artist’s work on textures .
Successfully located “seams” are lines corresponding to one edge, but located in different places of the texture. Seams are desirable if there is a natural “gap” in the surface (seams of clothing, edges, joints, etc.), and undesirable if there are none. Amateurs from all over the world participated in the modeling of Dota 2 characters, and the modeling guide required that the eyes be a separate “island” of development. ^[2]
For partially symmetrical objects: a successful combination of symmetric and asymmetric scan areas. Symmetry increases the detail of the texture and simplifies the work of the artist on the textures; asymmetric details “enliven” the object.

Example: Equidistant Projection

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

u={\frac {\varphi }{2\pi }}+0{,}5

{\ displaystyle u = {\ frac {\ varphi} {2 \ pi}} + 0 {,} 5}

v={\frac {\lambda }{\pi }}+0{,}5

{\ displaystyle v = {\ frac {\ lambda} {\ pi}} + 0 {,} 5}

A similar transformation can be applied in computer graphics to superimpose the texture of the earth's surface on a three-dimensional ball. By these metrics:

Use of texture area : excellent. Used the entire area of the texture.
Lack of areas with insufficient / excessive detail : medium. The poles have an excess of detail.
Similarities to standard angles : good. Such a projection is similar to ordinary cartographic, although it does not coincide with them.
Lack of seams : excellent. There is one seam along the meridian (between U = 0 and U = 1). The seam can be made as invisible as possible by placing the ocean in its place; also, cyclically shifting the texture, you can accurately draw a seam, and then shift it back.
Geometric distortion : bad. At the poles, the Earth's surface is severely distorted.
Symmetry : not applicable. The globe is asymmetrical.

UVW Conversion

Painted 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 “continuum of textures” ^[3] . For example, W = 0 can be sand, W = 1 can be stones, intermediate values can be sand from which stones peep.

The fourth, fifth, etc. coordinates are theoretically possible - however, the graphic hardware, imposing textures, carries out the transformation $\mathbb {R} ^{3}\to \mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3} \ to \ mathbb {R} ^ {3}}$ ^[4] and discards the third coordinate. The UVW transform uses the discarded coordinate to transition from one texture to another: it animates the “screen world” without seriously changing the design of the video card.

Notes

↑ There is also a mode where U and V are arbitrary, and the video card takes a fractional part , this allows you to cover the surface with a repeating texture. However, in this mode, the scan is prepared algorithmically, and not manually.
↑ ¹ ² DOTA 2 item UV mapping
↑ 3ds Max UVW Mapping - CGWiki ( unopened ) (inaccessible link) . Date of treatment July 29, 2013. Archived September 21, 2013.
↑ To be precise, the graphics accelerator operates with 4-dimensional vectors, which are the uniform coordinates of a point in R³

[1] There is also a mode where U and V are arbitrary, and the video card takes a fractional part , this allows you to cover the surface with a repeating texture. However, in this mode, the scan is prepared algorithmically, and not manually.

[valve-2] ¹ ² DOTA 2 item UV mapping

[3] 3ds Max UVW Mapping - CGWiki ( unopened ) (inaccessible link) . Date of treatment July 29, 2013. Archived September 21, 2013.

[4] To be precise, the graphics accelerator operates with 4-dimensional vectors, which are the uniform coordinates of a point in R³

UV conversion

Example: Equidistant Projection

UVW Conversion

Notes

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