3. De Casteljau's algorithm

so how'd it go did you figure out how to extend to castle Joe's algorithm to four points it's not so easy so don't worry if you had some trouble here's what to Castle Joe came up with first we use linear interpolation along with our parameter T to find a point on each of the three line segments now we have a three 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 two 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 Bezier curve we also tend to group the adjacent segments so they maintain the slope of the curve across the keyframe this prevents sudden jerks in the motion you can use the next interactive exercise to get some experience with Bezier curves that have three four or even more points