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Given I have 3d data points (x,y,z of different elements of the face in this case), assuming I calculate the multiple linear regression equation using 4 or 5 chosen points, can I treat the given model as a plane? The goal is to then extract the normal of the plane which passes through the plane and another known point to find the face's relative angle to a different plan.

Not sure if I should address this question in the math section

HermanK
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  • What variable would you be regressing against what explanatory values? If you wish to approximate a point cloud with a plane, perhaps you would be better off using PCA. This amounts to finding the eigenvector associated with the smallest eigenvalue of the covariance matrix (this will be a normal vector to the plane) and using the fact that this plane passes through the point of averages. For details, see https://stats.stackexchange.com/a/296837/919 and https://stats.stackexchange.com/a/74328/919. – whuber Apr 22 '20 at 17:10
  • Assuming my points are the edges of the eyes, mouth, and tip of the chin, I would use construct it a follows : z (depth) = a + b1x + b2y. could it be mathematically correct? I know that there is a best fit plane, but I'm unsure of how to implement it if not via the way I suggested here

    Regardless, I'll look into your purposed solution soon after I see if there are any suggestions the use a method I'm more familiar with.

     – HermanK Apr 22 '20 at 17:45
  • The sense of "best fit" in that application is one in which the $x$ and $y$ coordinates are known precisely and all the variation relative to the plane occurs in the $z$ coordinate. If that's the case, then your approach is a good one. More likely, though, these coordinates are all measurements and perhaps all should be treated on an equal footing, in which case the PCA approach is appropriate. – whuber Apr 22 '20 at 17:56
  • all coordinates are known: x,y,z are given for my purpose. I just need the plane that passes through 5 points. However since the points are not exactly placed on the plane, I could maybe use only 3 of them and use a mathematical procedure, but maybe for better accuracy, I would like to approximate the plane via more points.

    I will study the PCA approach regardless

     – HermanK Apr 22 '20 at 18:06
  • The issue comes down to whether it makes sense to privilege the $z$ coordinates over the $x$ and $y$ coordinates when accommodating the (inevitable) discrepancies between the points and any fitted plane. In my experience with morphometric measurements it's impossible to know where keypoints on a human face lie with high precision and it's well worth using procedures that accommodate that imprecision. – whuber Apr 22 '20 at 19:04

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