# Force of Gravity of a stationary object acting on another object moving through space

Story: I have x and y positions of a spaceship as it moves through space to some goal location at [1000, 700].

A record of its movement is given by:

x = [800, 850, 900, 950, 1000]
y = [600, 625, 650, 675, 700]


with each observation of its position recorded 4 seconds apart.

Now let's say there is also a stationary object (let's say it's a planet) at [894, 444]. I'd like to quantify the 'pull' of this object on the spaceship as it moves through space. The captain of the spaceship is assumed to fight this force to make sure she arrives at her goal location.

If the force is weak, then there should be no record of her making small corrections. If the force is strong, then there should be a clear record of her correcting course.

Real problem: I have a log of eye movements. I am trying to quantify the 'pull' of an irrelevant object in space as eyes move towards a goal location.

As I only have a math background, my initial instinct was to use regression to compute a fit for the above data (to get a polynomial model of the points), compute the curvature (ϰ) of this fitted function and look to see if the maximal curvature occured at roughly the same place as the object as the eyes moved towards a goal.

...but this seems to miss the point in my mind. How else can I accomplish this?

• How many different trajectories do you have to work with? Is it just the one? – lemon Mar 28 '16 at 11:45
• Yes. Just the one. Also, the data I give above is just to illustrate the point. The real data is not always so linear. – lnNoam Mar 28 '16 at 11:49
• I'm not sure the rocket analogy is especially helpful; you may get more helpful responses to your real problem on cross validated. – lemon Mar 28 '16 at 12:19
• First, thank you for the answer. Second, I am inclined to agree, however I am not trying to make any predictions or generalizations about future behaviour at this point. If you think I should remove the analogy I am happy to, but it seems that you have provided a very fitting answer. Thanks again. – lnNoam Mar 28 '16 at 12:25

Suppose the target is $t$, the 'distractor' is $d$, and the current position is $x_n$. You can compute the following three angles (w.r.t. any axis):
• $\theta_n$ angle of $x_{n+1}-x_n$
• $\theta_{n}^t$ angle of $t-x_n$
• $\theta_n^d$ angle of $d-x_n$
If the user is distracted by $d$ then $\theta_n$ should be closer to $\theta_n^d$ than to $\theta_n^t$. So you could define, say, a Gaussian-like function $f$ such that $f(\theta_n^d; \theta_n^t,\theta_n^d)=1$ and $f(\theta_n^t; \theta_n^t,\theta_n^d)\approx 0$, and then integrate this along the path to get a total 'distraction quantity':
$$D=\sum_n f(\theta_n; \theta_n^t,\theta_n^d)$$
I have another idea which I haven't fully fledged because I think it would only be suitable if you had multiple trajectories: you could imagine an energy surface expressed as a radial basis function centred on $d$ and $t$ and then treat the trajectories as the brownian-ish motion of a particle along the energy surface and optimise the energy surface. You then quantify the 'distraction' based on the height of the $d$ contribution to the energy surface.