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On the Existence of Two Affine-Equivalent Frameworks with Prescribed Edge Lengths in Euclidean \( d \)-Space

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Abstract

We study the existence of the two affine-equivalent bar-and-joint frameworks in Euclidean \( d \)-space which have some prescribed combinatorial structure and edge lengths. We show that the existence problem is always solvable theoretically and explain why to propose a practical algorithm for solving the problem is impossible.

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References

  1. Connelly R. and Guest S.D., Frameworks, Tensegrities, and Symmetry, Cambridge University, Cambridge (2022).

    Book  MATH  Google Scholar 

  2. Alexandrov V., “How to decide whether two convex octahedra are affinely equivalent using their natural developments only,” J. Geom. Graph., vol. 26, no. 1, 29–38 (2022).

    MathSciNet  MATH  Google Scholar 

  3. Alexandrov V.A., “Recognition of affine-equivalent polyhedra by their natural developments,” Sib. Math. J., vol. 64, no. 2, 269–286 (2023).

    Article  MathSciNet  MATH  Google Scholar 

  4. Alexandrov A.D., Convex Polyhedra, Springer, Berlin etc. (2005).

    MATH  Google Scholar 

  5. Blumenthal L.M., Theory and Applications of Distance Geometry. 2nd ed., Chelsea, New York (1970).

    MATH  Google Scholar 

  6. Havel T. and Li H., “From molecular distance geometry to conformal geometric algebra,” in: Handbook of Geometric Constraint Systems Principles, CRC, Boca Raton (2019), 107–137.

  7. Marker D., Model Theory: An Introduction, Springer, New York (2002) (Graduate Texts in Mathematics; vol. 217).

    MATH  Google Scholar 

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Funding

The research was carried out within the State Task to the Sobolev Institute of Mathematics (Project FWNF–2022–0006).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    V. A. Alexandrov

  2. Department of Physics, Novosibirsk State University, Novosibirsk, Russia

    V. A. Alexandrov

Authors
  1. V. A. Alexandrov
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Correspondence to V. A. Alexandrov.

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As author of this work, I declare that I have no conflicts of interest.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 6, pp. 1131–1137. https://doi.org/10.33048/smzh.2023.64.602

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Alexandrov, V.A. On the Existence of Two Affine-Equivalent Frameworks with Prescribed Edge Lengths in Euclidean \( d \)-Space. Sib Math J 64, 1273–1278 (2023). https://doi.org/10.1134/S0037446623060022

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  • DOI: https://doi.org/10.1134/S0037446623060022

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