In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables. The key idea is to perform linear interpolation first in one direction, and then in the other direction. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Linear interpolation is a process employed in mathematics, and numerous applications including computer graphics. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ...

The four red dots show the data points and the green dot is the point at which we want to interpolate.

Suppose that we want to find the value of the unknown function f at the point P = (x, y). It is assumed that we know the value of f at the four points Q₁₁ = (x₁, y₁), Q₁₂ = (x₁, y₂), Q₂₁ = (x₂, y₁), and Q₂₂ = (x₂, y₂). This picture accompagnies the article on bilinear interpolation. ... This picture accompagnies the article on bilinear interpolation. ...

We first do linear interpolation in the x-direction. This yields

We proceed by interpolating in the y-direction.

This gives us the desired estimate of f(x, y).

If we choose a coordinate system in which the four points where f is known are (0, 0), (0, 1), (1, 0), and (1, 1), then the interpolation formula simplifies to

Contrary to what the name suggests, the interpolant is not linear, but a product of two linear functions in x and y respectively. So it is of the form

which can also be written as

In both cases, the number of constants (four) correspond to the number of data points where f is given.

Note that the result of bilinear interpolation is independent of the order of interpolation. If we had first performed the linear interpolation in the y-direction and then in the x-direction, the resulting approximation would be the same.

The obvious extension of bilinear interpolation to three dimensions is called trilinear interpolation. Trilinear interpolation is the process of taking a three-dimensional set of numbers and interpolating the values linearly, finding a point using a weighted average of eight values. ...