Bilinear filtering

Bilinear filtering is a texture filtering method used to smooth textures when displayed larger or smaller than they actually are.

Most of the time, when drawing a textured shape on the screen, the texture is not displayed exactly as it is stored, without any distortion. Because of this, most pixels will end up needing to use a point on the texture that is "between" texels, assuming the texels are points (as opposed to, say, squares) in the middle (or on the upper left corner, or anywhere else; it does not matter, as long as it is consistent) of their respective "cells". Bilinear filtering uses these points to perform bilinear interpolation between the four texels nearest to the point that the pixel represents (in the middle or upper left of the pixel, usually).

In a mathematical context, bilinear interpolation is the problem of finding a function f(x,y) of the form

satisfying

The usual, and usually computationally least expensive way to compute ${\displaystyle f}$ is through linear interpolation used twice, for example to compute two functions ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ satisfying

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