What is the correct notation for curves?

Question

What is the correct math notation to use is when referring to linear interpolation, curves, and points on curves?

For instance, let's say we are talking about a quadratic Bezier curve. The control points for the curve are $P_0$, $P_1$ and $P_2$.

The De Casteljeau algorithm tells us that to evaluate a point on that curve at time $t$, we calculate the linear interpolation between $P_0$ and $P_1$ at time $t$ to get the value $\overline{P_0P_1}$.

We then calculate the linear interpolation between $P_1$ and $P_2$ at time $t$ to get the value $\overline{P_1P_2}$.

Lastly, we linear interpolate between the values $\overline{P_0P_1}$ and $\overline{P_1P_2}$ to get our final curve point $\stackrel{\frown}{P_0P_1P_2}$.

Then, if we were evaluating a cubic Bezier curve, we would linearly interpolate between $\stackrel{\frown}{P_0P_1P_2}$ and $\stackrel{\frown}{P_1P_2P_3}$ at time $t$ to get the value $\stackrel{\frown}{P_0P_1P_2P_3}$.

I know that I could (should?) just use other letters. Like $Q$ instead of $\overline{P_0P_1}$, but it's real easy to get lost in letters, and it seems using the type of notation I'm using makes it easier to follow what the values actually are and where they come from. Plus of course, you can run out of letters!

What's the correct thing to do here? Would I be laughed at (not taken seriously, etc) if using this sort of notation, or is it acceptable?

Thanks!

Answer 1

We often use a capital letter such as $P$ or $Q$ to represent a point or a vector in the 3D space. For a curve, it is often denoted as $C(t)=(x(t), y(t), z(t))$ in parametric form or $y=f(x)$ in explicit form. It will be confusing to use $\overline{P_0P_1}$ to denote a point interpolated between $P_0$ and $P_1$ as we often use the same notation to denote the line segment defined between $P_0$ to $P_1$ and use $\overrightarrow{P_0P_1}$ to denote the vector defined from $P_0$ to $P_1$.