The example is like this: Given 3 points $p(0)=2,p(1)=1,p(3)=1$.The question asks us to apply the Casteljau algorithm to evaluate the Bezier curve b(u) for the given Bezier polygon at $u=2$. I did the following:
- Change of variables: $t=\frac{u-0}{3-0}=\frac{u}{3}$
- By applying Casteljau algorithm, I got a vector like this $\left( \frac{2}{3}u+\frac{1}{9}u^2,2-\frac{2}{3}u+\frac{u^2}{9}\right)$. In Matlab, if I set $u=0:0.01:3$ and plot all these points then I will get the Bezier curve. But now it comes to my the confusing part for me: why the $u$ coordinate of this vector is $\frac{2}{3}u+\frac{1}{9}u^2$ rather than u? What does "evaluate the Bezier curve b(u) for the given Bezier polygon at $u=2$" mean exactly? Does it mean that I just need to plug in $u=2$ and get the point or I should set $ \frac{2}{3}u+\frac{1}{9}u^2=2$ first and then find the root of $u$ and then evaluate $p$ coordinate $2-\frac{2}{3}u+\frac{u^2}{9}$.
- By this example can we say that every point on the control polygon is just bijective to the point on the Bezier curve, but it doesn't necessarily mean that the point on the Bezier curve and the control polygon has to be evaluated at the same $u$ coordinate? Can anyone also explain to me the theory behind it?
Really thanks for the help!