How to compute the geometric center of a curved manifold defined by a set of point?
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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.
geometry differential-geometry optimization numerical-methods
asked yesterday
Florent Jousse
32
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up vote
0
down vote
favorite
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.
geometry differential-geometry optimization numerical-methods
asked yesterday
Florent Jousse
32
1
If your manifold is $S^1$ and $E = {(1,0), (-1,0)}$, where should the centroid be?
– Jason DeVito
yesterday
"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?
– Coolwater
13 hours ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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.
geometry differential-geometry optimization numerical-methods
asked yesterday
Florent Jousse
32
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.
geometry differential-geometry optimization numerical-methods
geometry differential-geometry optimization numerical-methods
asked yesterday
Florent Jousse
32
asked yesterday
Florent Jousse
32
asked yesterday
Florent Jousse
32
asked yesterday
Florent Jousse
32
asked yesterday
Florent Jousse
32
32
1
If your manifold is $S^1$ and $E = {(1,0), (-1,0)}$, where should the centroid be?
– Jason DeVito
yesterday
"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?
– Coolwater
13 hours ago
add a comment |
1
If your manifold is $S^1$ and $E = {(1,0), (-1,0)}$, where should the centroid be?
– Jason DeVito
yesterday
"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?
– Coolwater
13 hours ago
1
1
If your manifold is $S^1$ and $E = {(1,0), (-1,0)}$, where should the centroid be?
– Jason DeVito
yesterday
If your manifold is $S^1$ and $E = {(1,0), (-1,0)}$, where should the centroid be?
– Jason DeVito
yesterday
"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?
– Coolwater
13 hours ago
"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?
– Coolwater
13 hours ago
add a comment |
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1
If your manifold is $S^1$ and $E = {(1,0), (-1,0)}$, where should the centroid be?
– Jason DeVito
yesterday
"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?
– Coolwater
13 hours ago