How to calculate the projection of point onto a line

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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!
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回答(2 个)

Alan Stevens
Alan Stevens 2021-3-3

darova
darova 2021-3-4
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
  2 个评论
Elisa Iovene
Elisa Iovene 2021-3-4
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
darova 2021-3-5
  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

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