# Linear Interpolation

**Linear interpolation** is a method for constructing new data points between two existing data points in a linear fashion (analytically speaking using linear polynomials, geometrically speaking on a stright line between two points).

## Formula

The formula for such an interpolation is

where **a** and **b** are the two points to interpolate between and *u* indicates the ratio along the line from **a** to **b** of the desired point. If *u* lies outside the range [0, 1] this is known as *extrapolation*.

It may be helpful to note the following:

This formula applies to points of any dimensionality.

For example, in **R**^{3}:

A new point **p** at distance *u* would then be

Sometimes it is necessary to find both coordinates of a point on a line when only one coordinate is known (in addition to two points on the line). This can be accomplished by expanding the two-dimensional case. Given the desired *x* coordinate, we can find the corresponding *y* value:

Solving the first equation for *u* in terms of *x*:

and substituting into the second equation:

*TODO: Insert image to demonstrate usage.*

## Applications

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Linear interpolation has a few common applications. It is frequently used as a way to interpolate between values in a table. It can also be used as a way to animate an object moving between two points by using time as the ratio *u*.

*TODO: Discuss bilinear interpolation.*