2. Repeated linear interpolation

now that we've looked at linear interpolation let's see how we can get smoother motion using Bezier curves the shape of each segment of this curve is controlled by four points so how can we write an equation that gives us a smooth curve out of these four points you may remember we faced a similar problem in the environment modeling lesson there we were trying to make curved blades of grass we saw how to use three points to define a parabolic arc using the string art method so let's review how that string art method actually works let's label our points a B and C we've also got a parameter will call T which is how far along each line segment we are first we can calculate a point on a B using a weighted average of these two endpoints this is another kind of linear interpolation but instead of using the slope-intercept form we're using what's called a parametric form the parameter is T which tells us how far along the line we are as T goes from 0 to 1 our new point let's call it Q goes from A to B let's do the same thing for the other line segment calculating a point R between B and C finally we'll use the same method between Q and R to calculate P which is a point on our curve as T goes from 0 to 1 P traces out the smooth curve you can think of this construction method as repeated linear interpolation since Q R and P are all computed using linear functions of T this method of repeated linear interpolation is called decassle Joe's algorithm it's named after Paul - Kastle Joe who actually discovered the math for this a few years before pierre bézier but wasn't able to publish it until after Bessie had scooped him we've seen how de caso Joe's algorithm can be used to make a smooth curve out of three points but for animation we want to use four points to control the curve take a few minutes with pencil and paper and see if you can work out how to get a smooth curve starting with four points instead of three