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Suppose that I have a set of points E which belong to a manifold. Typically a surface mesh like that. $$E=\{x_{0},x_1,...,x_n\}$$ where the $x_i$ are the points coordinates in 3D such that : $x_i = (x_{i}^x,x_{i}^y,x_{i}^z)$

I would like to compute the centroid $c$ such that it's coordinates are the mean of the coordinates of all the points belonging to this manifold. But I don't want the metric to be the Euclidian distance (because the centroid will not be necessarily on the manifold).

Any help will be appreciated.

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    $\begingroup$ If your manifold is $S^1$ and $E = \{(1,0), (-1,0)\}$, where should the centroid be? $\endgroup$
    – Jason DeVito - on hiatus
    Commented Nov 30, 2018 at 15:31
  • $\begingroup$ "the centroid will not be necessarily on the manifold" - that sounds to me as if you simply want the $\mathbb{R}^3$ coordinate wise arithmetic mean of the points? $\endgroup$
    – Coolwater
    Commented Dec 1, 2018 at 16:28
  • $\begingroup$ Maybe you can use the formula here: math.stackexchange.com/questions/1145964/centroid-of-manifold ? $\endgroup$
    – Aneesh Durg
    Commented Mar 28, 2019 at 21:56

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