I always seem to see the patterns in the numbers, and while playing around with the result of this challenge, I managed to extend the solution beyond parabolic curves:
[math]P_x(t)=(x_0-2\,x_1+x_2)\,t^2-2(x_0-x_1)\,t+x_0[/math] [math]P_y(t)=(y_0-2\,y_1+y_2)\,t^2-2(y_0-y_1)\,t+y_0[/math]
where [math]\{t \in \mathbb{R} : 0\le t\le 1\}[/math]
If you look at the above result, you should notice some patterns that seem to be a recurring theme. As a bonus, I'll give anyone rep who can tell me what type of curve this new pattern entails. It's interesting, because I should've noticed this when I first worked the problem