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# 2. Repeated linear interpolation

## Video transcript

- 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.