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Given a finite set $P$ of points in $\mathbb{E}^3$ , one can calculate an approximating plane either as the solution of a Total Least Squares problem or by interpreting the problem physically, assigning a unit mass to each point and taking the best-approximating plane that contains the center of gravity and, that is orthogonal to one of the prinicipal axes of the Inertia Tensor.

as both of the above mentioned methods can be formulated on basis of squared coordinate differences, I don't see any criterion for preferring one over the other or even any argument in favor or against why both methods must yield identical planes.

Questions:

  • do both methods yield identical planes?
  • if they need not be identical, what are bounds on the angle between their normals?
  • what are criteria for preferring one method over the other?
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  • $\begingroup$ I expect I'm making an embarrassing mistake by saying this, but... don't both approaches just boil down to minimising $(X\beta - r)^T(X\beta - r)$ over $\beta\in S^2$? (writing $X$ for the $n\times 3$ matrix whose $i$th row is the transpose of your $i$th data point, and $r$ is an $n$-vector whose components are all the same real number). $\endgroup$
    – DCM
    Commented Jun 22, 2019 at 16:50
  • $\begingroup$ @DCM the principal axes of the inertia tensor correspond to eigenvectors of a matrix and one question is whether that is also true for total least squares $\endgroup$
    – Manfred Weis
    Commented Jun 22, 2019 at 17:12

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