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Hi everybody I need your help. My question is: what does "$t$" represent in De Casteljau's algorithm?
We have the following formula to calculate the point $Q$:
$Q=(1−t)P_1+tP_2,\;t\in[0,1]$
But what does $t$ mean here and why is it between $0$ and $1$?
Given points $P_1$ and $P_2$, the formula $$(1-t)P_1+tP_2$$ is a parametrisation of the straight line from $P_1$ to $P_2$. For every value $t\in[0,1]$, the formula gives you exactly one point on the line.
The best way to imagine this is, in my oppinion, to imagine that $t$ is time. At the beginning, you are at point $(1-0)P_1+0P_2=P_1$, then as time passes, you start to move from one point to the other. For example, at time $0.5$, you have reached the point $0.5P_1+0.5P_2$ which is the midway point. When the whole time passes, at $t=1$, you are at point $P_2$ and have drawn the whole line.
In the equation $$ \mathbf{Q} = (1-t)\mathbf{P}_1 + t\mathbf{P}_2 $$ the $t$ represents fractional distance along the line segment from $\mathbf{P}_1$ to $\mathbf{P}_2$. To see this, rewrite the equation as $$ \mathbf{Q} = \mathbf{P}_1 + t(\mathbf{P}_2 - \mathbf{P}_1) $$ This equation says ... go to $\mathbf{P}_1$, and then move by a fraction "$t$" of the vector from $\mathbf{P}_1$ to $\mathbf{P}_2$. Or, more briefly, go "$t$ of the way" from $\mathbf{P}_1$ to $\mathbf{P}_2$.
The time idea works, too. Suppose you are moving at constant speed along the line from $\mathbf{P}_1$ to $\mathbf{P}_2$, and you use a time scale such that $\text{time} = 0$ at $\mathbf{P}_1$, and $\text{time} = 1$ at $\mathbf{P}_2$. Then, at time $t$, you will be at the point $\mathbf{Q}$.
Actually, the $t$ value does not need to be between $0$ and $1$. By using a value larger than 1, you can extend a Bezier curve (rather than dividing it internally, which is what you'd get if $0 < t < 1$).
If you're studying Bezier curves and surfaces, you really need to make sure you get this idea, because it's the foundation of everything. There are some nice animations on this page that might help.
Treat this as a comment! (Since I got a new 'phone, I seem to not be connected anymore with my old cognomen (Senex Ægypti Parvi); so my reputation seems to be zeroed out, hence, comments from me are not accepted.) Anyway, although Bézier curves are usually thought of as existing where $0\le t\le 1$, the parametrization is valid outside that interval, as well. What is interesting about the curve at its endpoints ($t=0$ and $t=1$) is that the curve at those points is always tangent to the straight line connecting the points. Also, the portion of the curve defined for that interval is always completely contained within the hull comprised of all the control points.
