How to calculate the projection of point onto a line

Hello,

I would like to calculate the trajectory volatility (TV) defined as the standard deviation between the real trajectory and the ideal trajectory.

The formula is the following:

where xi (i= 1,2,3...N, the order of point on the trajectory) is the actual measurement point sampled from the trajectory, and pi is the corresponding projection onto the ideal trajectory as shown in the pic

Both actual and ideal trajectories are 100x3 matrices of x,y and z coordinate.

Here a pic for a better understanding:

(purple one actual, green line ideal)

Do you have any suggestion on how to do calculate pi, so the projection on the ideal trajectory given xi?

Thanks!

[darova]

Are you asking of how to calcualte ?

[Elisa Iovene]

Sorry, my fault. I'm asking how to calculate pi so the projection on my ideal trajectory, given xi which is a point on my real one.

Thanks

[darova]

Do you have line parameters?

[Elisa Iovene]

Yes, I have!

Answer 1

[Alan Stevens]

Try

doc std

Answer 2

[darova]

Here are some exaplanations

Let be some point on the line

cross product:

distance you want: →

Let me know if you need more explanations or some example

[Elisa Iovene]

Thank you for the answer!

I still have some doubts:

1. Ai is a random point on my real trajectory?
2. Should Ai change if I choose another xi?
3. Since I have 3D points, should I apply the formula you mentioned, considering each axis separately?

[darova]

1. Yes, is any point on a real trajectory
2. No, you don't need to choose another
3. What do you mean by "each axis separately"?

Sorry, i forgot you to explain: component is needed to create unit vector. So point here is another point on the line