On the existence of two affine-equivalent frameworks with prescribed edge lengths in Euclidean $d$-space

TL;DR

This paper investigates the theoretical existence of two affine-equivalent frameworks with given edge lengths and structure in Euclidean space, establishing that such frameworks always exist in theory but without providing a practical solution.

Contribution

The paper proves the universal theoretical existence of affine-equivalent frameworks with prescribed edge lengths, highlighting the gap in practical algorithms.

Findings

Existence of affine-equivalent frameworks is always guaranteed theoretically.

No practical algorithm is currently available for constructing such frameworks.

The problem is solvable in theory but remains computationally challenging.

Abstract

We study the problem of existence of two affine-equivalent bar-and-joint frameworks in Euclidean $d$-space which have prescribed combinatorial structure and edge lengths. We prove that theoretically this problem is always solvable, but we cannot propose any practical algorithm for its solution.