# 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

$\mathbf{p}(u) = \mathbf{a} + u \times (\mathbf{b} - \mathbf{a}) \;\;\;\; 0 \leq u \leq 1$

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:

$\mathbf{p}(0) = \mathbf{a} \;\;\;\; \mathbf{p}(1) = \mathbf{b}$

This formula applies to points of any dimensionality.

For example, in R3:

$\mathbf{a} = (x_0, y_0, z_0) \;\;\;\; \mathbf{b} = (x_1, y_1, z_1)$
$x(u) = x_0 + u \times (x_1 - x_0)$
$y(u) = y_0 + u \times (y_1 - y_0)$
$z(u) = z_0 + u \times (z_1 - z_0)$

A new point p at distance u would then be

$\mathbf{p}(u) = (x(u), y(u), z(u))$

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:

$\mathbf{a} = (x_0, y_0) \;\;\;\; \mathbf{b} = (x_1, y_1)$
$x(u) = x_0 + u \times (x_1 - x_0)$
$y(u) = y_0 + u \times (y_1 - y_0)$

Solving the first equation for u in terms of x:

$u(x) = \frac{x - x_0}{x_1 - x_0}$

and substituting into the second equation:

$y(x) = y_0 + (x - x_0) \frac{y_1 - y_0}{x_1 - x_0}$
TODO: Insert image to demonstrate usage.