If I only know $x$- and $y$- coordinates of every point on a trajectory without knowledge of time information, is there any way to approximate Cartesian acceleration angle at each point? Time interval between every two points is very small, ~0.03 second.
Answer
Yes. Provided you are only interested in the direction of the acceleration, and not it's magnitude. And further assuming your time samples are equally spaced, you can take the second derivative of the path and this will be proportional to the acceleration.
A decent method in practice would be to use a second order central finite difference scheme wherein you say that:
$$ a_x(t) = x(t-1) - 2x(t) + x(t+1) $$ and $$ a_y(t) = y(t-1) - 2y(t) + y(t+1) $$ this will give you decent estimates for the cartesian components of acceleration at every time, caveat to an overall scaling in magnitude that you won't know without knowing the actual timing, but the direction should be alright.
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