In 3D graphics, UV mapping (also known as UVW mapping) means simply mapping of vertex coordinates on texture - saying what point at texture corresponds to which vertex of a face.

Basically, you have a 3D mesh that consists of faces (polygons), which in turn have vertices as their corners. You also have a texture, which is a 2-dimensional bitmap.

Each vertex in your model has (x, y, z) coordinate. This tells you where that point is in your scene.

Now, each vertex also has (u, v) coordinate (or sometimes (u, v, w) coordinate). This tells you where this vertex specifically is on the texture (the (x, y) coordinate on texture's coordinate system). Typically this is a 2-dimensional coordinate because textures are 2D. Apparently some apps do 3D coordinates, still.

Say, you have a triangle in your scene with vertices A, B, C. You plop into UV editing mode of your 3D application. A triangle appears on the UV editor, may or may not be the same size and shape as the actual triangle in the scene. You move the corner points to coincide with the texture. On your scene you see the texture fill the triangle likewise.

UV coordinates are important when you are assigning a bitmap texture on mesh that is irregular in shape (that is, cannot be reduced to a simple rectangle) - in other words, in almost any situation when you must use a bitmap texture on a 3D model. They are extremely handy when making 3D models for games, because you can make one texture file that has textures for the whole creature or person or whatever, and just assign that on different faces of the mesh. Texturing with UV means you can get fairly good-looking creatures and objects with fairly low polycount and 3D rendering effort.

Most modern modeling tools support UV mapping (Blender, my favorite, since freeware version 1.8 and even earlier for commercial version...)

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