This question emanates directly from This previous question I made, but from a more general perspective.
Given some function $f(t)\colon \mathbb{R^1}\to\mathbb{R^n}$ parametrized in some arbitrary way that draws some curve, can I eyeball the acceleration from the curve? can I at least eyeball where the acceleration would NOT point to?
For example, the following curve in RED is the image of some function $f(t)\colon \mathbb{R^1}\to\mathbb{R^2}$. The velocity is marked in ORANGE at some point and is tangent to the curve (that we already know). The purple lines indicate possible acceleration vectors for that curve in that point.
The question:
Given only the image of the function (the curve). Are all the purple acceleration vectors equally possible? Can I discard some general direction? Why?
Here I'm interested only in direction, since I understand that the magnitude is impossible to infer.
The relevant formula you want is this: If $\upsilon$ is the speed of the particle, then
$$\vec a = \upsilon’ \vec T + \kappa\upsilon^2 \vec N,$$
where $\vec T$ and $\vec N$ are the unit tangent and principal normal, respectively. It follows that the acceleration vector must be in the half-plane determined by the tangent line and the principal normal. (To derive this, differentiate $\upsilon\vec T$ with respect to time.)