TL;DR

Where I re-discover a useful source for information, and that contributions might get lost.

For reasons that will hopefully be clear in a few days, I’m taking (again) a look at Bézier curves. There’s an excellent resource about them online, namely A Primer on Bézier Curves.

But… wait a minute! I already knew this place from before, because… I actually contributed to it about five years ago (I still have an oline version of the old site). Alas, in the meantime it underwent some re-writing, and it seems that my old contribution got lost 🤔

The contribution was actually a minor one, but it was enough to tickle the not-so-little nit-picker in me at the time. In section Splitting curves using matrices there is:

[…] the new end point is a mixture that looks oddly similar to a Bernstein polynomial of degree two

and my point is that the new end point is a Bernstein polynomial.

The key in this insight is that $z$ is actually the free variable in the parametric equations, which ranges in $[0,1]$. For this reason, $(z-1)$ is better expressed as $-(1-z)$, because it gives you an immediate view of what’s the real sign of the expression.

For this reason, then, the following expression:

$$z^2 \cdot P_3 - 2 \cdot z \cdot (z-1) \cdot P_2 + (z - 1)^2 \cdot P_1$$

is best put as:

$$(1-z)^2 \cdot P_1 + 2 \cdot (1-z) \cdot z \cdot P_2 + z^2 \cdot P_3$$

which also reveals its… Bernstein nature.

So there you have it… I know (where to find info on) Bézier curves! (And now you do too).