Linear Interpolation

From GPWiki

Tutorial by Nicholas Gorski GMan

This tutorial will discuss linear interpolation.

Linear Interpolation

What is it?

Linear interpolation is the process of determining an unknown point that lies between two known points. If the unknown point lies outside of the two known points, then it is call exterpolation. The linear interpolation is commonly expressed in the form:

Where p₀ and p₁ are the points that you are interpolating between, and u is a number between 0 and 1 representing the part of the distance between p₀ and p₁ at which p_u lies.

How do you use it?

Sample 1

Let's say you have two numbers and you want to find the number 1/4 of the way inbetween (u = 1/4). For example, let p₀ be 5.3 and p₁ be 24.9. Plugging this into our equation we get:

Sample 2

Linear interpolation can also be used for predicting. As with the previous example, let p₀ be 5.3 and p₁ be 24.9. To find values between these two points u has to be between 0 and 1. If u is bigger than 1 p_u lies beyond p₁.

Sample 3

To interpolate in more than one dimension you can use the same method. Let us look at a two dimensional square, where we know the values at the corners.

10 ----- 15
|    x    |
|         |
00 ----- 20

Say x lies halfway the horizontal width of this square, and 1/3 down the vertical height. To get the value at point x you first determine the values halfway the top and bottom edges of the square, let's call those points p_top and p_bottom:

To get the value at x you interpolate along a vertical line between p_top and p_bottom:

Uses

Try this: Make a point. Then make another point. Interpolate the points position, so the point moves from its starting position to the other point. More interesting, use a line. Linear interpolation can also be used to animate things.

Done

I hope that helps explain Linear Interpolation. If you find an error, feel free to fix it.