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From a $n$-polygon in $\mathbb{R}^3$ the set of distances between all pairs of vertices are given. (How) is it possible to reconstruct the geometric structure of the polygone?

Symbolically: For a set of coordinates $(x_i,y_i,z_i)$ with $i=1,...,n$ we have given $r_{k} = (x_i -x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2$, for $k = 1,...,n(n-1)/2$ without knowing the map $k\rightarrow (i,j)$ and want to know the set of $(x_i,y_i,z_i)$.

It seems clear, that the distances between the points determine the polygon only up to translations and rotations. So we are seeking to determine only 3$n$-6 degrees of freedom and we have $n(n-1)/2$ equations. As well the coordinates are only determined up to an arbitrary permutation.

I have absolutely now clue as to how address the question, I guess the answer will somehow involve group theory.

It is motivated by the determination of molecular structures using diffraction
methods. Its a kind of toy-version (all elements the same) of the so called inverse structure problem. I am doing research on this subject, but according to my current knowledge no one ever looked closer at the mathematical basis of the problem. It is usually solved using model structures (You just assume the tape of polygon and fit the structure parameters (coordinates) to the diffraction data.

(I am also clueless about the correct tags for the disciplines that involves, so sorry for wrong guesses in case).

For a $n$-polygon in $\mathbb{R}^3$ the set of distances between all pairs of vertices is given. (How) is it possible to reconstruct the geometric structure of the polygone?

Symbolically: For a set of coordinates $(x_i,y_i,z_i)$ with $i=1,...,n$ we have given $r_{k} = (x_i -x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2$, for $k = 1,...,n(n-1)/2$ without knowing the map $k\rightarrow (i,j)$ and want to know the set of $(x_i,y_i,z_i)$.

It seems clear, that the distances between the points determine the polygon only up to translations and rotations. So we are seeking to determine only 3$n$-6 degrees of freedom and we have $n(n-1)/2$ equations. As well the coordinates are at most determined up to an arbitrary permutation. The question of the assignment of the distances to actual vertices seems a key in the solution.

I have absolutely now clue as to how address the question, I guess the answer will somehow involve group theory.

It is motivated by the determination of molecular structures using diffraction
methods. Its a kind of toy-version (all elements the same) of the so called inverse structure problem. I am doing experimental research on this subject. It is usually solved using model structures (you just assume the type of polygon including the mapping between $k$ and $(i,j)$ and fit the structure parameters (coordinates) to the diffraction data).

(I am also clueless about the correct tags for the disciplines that involves, so sorry for wrong guesses in case).

From a $n$-polygon in $\mathbb{R}^3$ the set of all distances between all pairs of vertices are given. (How) is it possible to reconstruct the geometric structure of the polygone?

Symbolically: For a set of coordinates $(x_i,y_i,z_i)$ with $i=1,...,n$ we have given $r_{k} = (x_i -x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2$, for $k = 1,...,n(n-1)/2$ without knowing the map $k\rightarrow (i,j)$ and want to know the set of $(x_i,y_i,z_i)$.

It seems clear, that the distances between the points determine the polygon only up to translations and rotations. So we are seeking to determine only 3$n$-6 degrees of freedom and we have $n(n-1)/2$ equations. As well the coordinates are only determined up to an arbitrary permutation.

I have absolutely now clue as to how address the question, I guess the answer will somehow involve group theory.

It is motivated by the determination of molecular structures using diffraction
methods. Its a kind of toy-version (all elements the same) of the so called inverse structure problem. I am doing research on this subject, but according to my current knowledge no one ever looked closer at the mathematical basis of the problem. It is usually solved using model structures (You just assume the tape of polygon and fit the structure parameters (coordinates) to the diffraction data.

(I am also clueless about the correct tags for the disciplines that involves, so sorry for wrong guesses in case).

From a $n$-polygon in $\mathbb{R}^3$ the set of distances between all pairs of vertices are given. (How) is it possible to reconstruct the geometric structure of the polygone?

Symbolically: For a set of coordinates $(x_i,y_i,z_i)$ with $i=1,...,n$ we have given $r_{k} = (x_i -x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2$, for $k = 1,...,n(n-1)/2$ without knowing the map $k\rightarrow (i,j)$ and want to know the set of $(x_i,y_i,z_i)$.

It seems clear, that the distances between the points determine the polygon only up to translations and rotations. So we are seeking to determine only 3$n$-6 degrees of freedom and we have $n(n-1)/2$ equations. As well the coordinates are only determined up to an arbitrary permutation.

I have absolutely now clue as to how address the question, I guess the answer will somehow involve group theory.

It is motivated by the determination of molecular structures using diffraction
methods. Its a kind of toy-version (all elements the same) of the so called inverse structure problem. I am doing research on this subject, but according to my current knowledge no one ever looked closer at the mathematical basis of the problem. It is usually solved using model structures (You just assume the tape of polygon and fit the structure parameters (coordinates) to the diffraction data.

(I am also clueless about the correct tags for the disciplines that involves, so sorry for wrong guesses in case).

From a $n$-polygon in $\mathbb{R}^3$ the distances between all pairs of vertices are given. (How) is it possible to reconstruct its structure? Symbolically: For a set of coordinates $(x_i,y_i,z_i)$ with $i=1,...,N$ we have given $r_{ij} = (x_i -x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2$ and want to know $(x_i,y_i,z_i)$ for all $i$.

It seems clear, that the distances between the points determine the polygon only up to translations and rotations. So we are seeking to determine only 3$N$-6 degrees of freedom and we have $N(N-1)/2$ equations.

I have absolutely now clue as to how address the question, I guess the answer will somehow involve group theory.

It is motivated by the determination of molecular structures using diffraction
methods. Its a kind of toy-version (all elements the same) of the so called inverse structure problem. I am doing research on this subject, but according to my current knowledge no one ever looked closer at the mathematical basis of the problem. It is usually solved using model structures (You just assume the tape of polygon and fit the structure parameters (coordinates) to the diffraction data.

(I am also clueless about the correct tags for the disciplines that involves, so sorry for wrong guesses in case).

From a $n$-polygon in $\mathbb{R}^3$ the set of all distances between all pairs of vertices are given. (How) is it possible to reconstruct the geometric structure of the polygone? 

Symbolically: For a set of coordinates $(x_i,y_i,z_i)$ with $i=1,...,n$ we have given $r_{k} = (x_i -x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2$, for $k = 1,...,n(n-1)/2$ without knowing the map $k\rightarrow (i,j)$ and want to know the set of $(x_i,y_i,z_i)$.

It seems clear, that the distances between the points determine the polygon only up to translations and rotations. So we are seeking to determine only 3$n$-6 degrees of freedom and we have $n(n-1)/2$ equations. As well the coordinates are only determined up to an arbitrary permutation.

I have absolutely now clue as to how address the question, I guess the answer will somehow involve group theory.

It is motivated by the determination of molecular structures using diffraction
methods. Its a kind of toy-version (all elements the same) of the so called inverse structure problem. I am doing research on this subject, but according to my current knowledge no one ever looked closer at the mathematical basis of the problem. It is usually solved using model structures (You just assume the tape of polygon and fit the structure parameters (coordinates) to the diffraction data.

(I am also clueless about the correct tags for the disciplines that involves, so sorry for wrong guesses in case).