3. De Casteljau's algorithm

- So, how'd it go? Did you figure out how to extend a Casteljau's algorithm to 4 points? It's not so easy, so don't worry if you had some trouble. Here's what De Casteljau came up with. First, we use linear interpolation along with our parameter t, to find a point on each of the 3 line segments. Now we have a 3-point polygon, just like the grass blade. As before, we find a point on each of the new segments using linear interpolation and the same t value. Now we have a 2-point polygon, or a line. We find a point on our line using linear interpolation, one more time. As we vary the parameter t, this final point traces out our smooth curve. These are the kind of curves we typically use to control the motion of our characters as we animate. We use something called a graph editor, which lets us manipulate the control points of these curves to get smooth motion between poses. This is the graph editor that we use at Pixar. In between each one of my key poses is a Bézier curve. We also tend to group the adjacent segments so they maintain the slope of the curve across the key frame. This prevents sudden jerks in the motion. You can use the next interactive excercise to get some experience with Bézier curves that have 3, 4, or even more points.