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- Now that we've looked at linear interpolation, let's see how we can get smother motion using Bézier 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 modelling lesson. There we were trying to make curved blades of grass. We saw how to use three point 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 we'll call t, which is how far along each line segment we are. First, we can calculate a point on AB, 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 deCastlejau's algorithm. It's named after Paul deCastlejau, who actually discovered the math for this a few years before Pierre Bézier, but wasn't able to publish it until after Bézier had scooped him. We've seen how deCastlejau'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.