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edited Jun 7 2011 at 16:43

WhitAngl
249●6

Hi,

I have a set of points $p_i \in \mathbb S^2$.

Is there a way to find a function $F : \mathbb S^2 \rightarrow \mathbb R^3$ of class $\mathcal C^1$, minimizing $$\sum_{i,j} $\int_{\mathbb S^2\times\mathbb S^2} (d(F(p_i),F(p_j)) d(F(x),F(y)) - \delta(p_i,p_j))^2$$ delta(x,y))^2 ~dx ~dy$$ , where $d$ stands for the euclidean distance in $\mathbb R^3$ and $\delta$ the geodesic distance on the sphere $\mathbb S^2$?

I tried to perform a Multi-Dimensional Scaling to get this least square solution for a finite set of point, but it seems that the solution is was just the identity... is that right?

Thanks!

deleted 158 characters in body

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edited Jun 7 2011 at 15:58

WhitAngl
249●6

Hi,

I have a set of points $p_i$ on a sphere, identified by their 3d coordinates $(x_i,y_i,z_i)$.p_i \in \mathbb S^2$.
Is there a way to find a function $F : \mathbb S^2 \rightarrow \mathbb R^3$ , and points of class $q_i = F(p_i)$\mathcal C^1$, so that minimizing $\forall $\sum_{i,j} (i,j),~ d(q_i,q_j) = d(F(p_i),F(p_j)) - \delta(p_i,p_j)$, delta(p_i,p_j))^2$$ , where $d$ stands for the euclidean distance in $\mathbb R^3$ and $\delta$ the geodesic distance on the sphere $\mathbb S^2$?

I don't need high regularity on $F$ ($\mathcal C^1$ is fine), and a least square solution of the problem is fine either.

I tried to perform a Multi-Dimensional Scaling to get the this least square solution, but it seems that the solution is just the identity... is that right?

Thanks!

full rewrite; added 25 characters in body; added 35 characters in body

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edited Jun 7 2011 at 15:40

WhitAngl
249●6

Hi,

I have some a set of points $p_i$ on a sphere, identified by their 3d coordinates (x,y,z). $(x_i,y_i,z_i)$.
Is there a way to transform these find a function $F : \mathbb S^2 \rightarrow \mathbb R^3$, and points $q_i = F(p_i)$, so that their $\forall (i,j),~ d(q_i,q_j) = \delta(p_i,p_j)$, where $d$ stands for the euclidean distance in 3D matches as much as possible their $\mathbb R^3$ and $\delta$ the geodesic distance on the sphere $\mathbb S^2$?

The new points would have coordinates
I don't need high regularity on $F$ (f(x,y,z), g(x,y,z), h(x,y,z))$\mathcal C^1$ is fine), and the euclidean distance between any two such points would correspond to the geodesic distance between the original points on the sphere. An approximate a least square solution would be of the problem is fine either.

I tried to perform a Multi-Dimensional Scaling to get the least square solution, but it seems that the solution is just the identity... is that right?

Thanks!