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Cayley-Menger coordinates.

M J Sippl, H A Scheraga

    Proceedings of the National Academy of Sciences of the United States of America
    |April 1, 1986
    PubMed
    Summary
    This summary is machine-generated.

    This study simplifies complex Cayley-Menger determinants for distance geometry. New Cayley-Menger coordinates offer a natural norm for embedding incomplete distance data into Euclidean spaces.

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    Area of Science:

    • Mathematics
    • Computational Geometry

    Background:

    • Distance geometry techniques often face challenges due to the analytical complexity of Cayley-Menger determinants.
    • These determinants are crucial for characterizing Euclidean spaces based on inter-point distances.

    Purpose of the Study:

    • To simplify the application of distance geometry by addressing the complexity of Cayley-Menger determinants.
    • To introduce a new coordinate system, Cayley-Menger coordinates, for representing points in Euclidean space.
    • To provide a natural norm for solving the incomplete embedding problem.

    Main Methods:

    • Utilizing a theorem by Jacobi to replace complex Cayley-Menger determinants with simpler ones.
    • Deriving the concept of Cayley-Menger coordinates based on distances to reference points.

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  • Developing tools for completing incomplete distance matrices.

    • Main Results:

    • Demonstrated that complex Cayley-Menger determinants can be simplified using Jacobi's theorem.
    • Introduced Cayley-Menger coordinates, simplifying point characterization in Euclidean spaces.
    • Established Cayley-Menger coordinates as a natural norm for the incomplete embedding problem.

    Conclusions:

    • The simplification of Cayley-Menger determinants facilitates distance geometry applications.
    • Cayley-Menger coordinates offer a novel and effective approach to embedding incomplete distance data.
    • This work provides essential tools for reconstructing and embedding geometric structures from partial distance information.