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The Algebro-geometric Study of Range Maps

Abstract

Localizing a radiant source is a problem of great interest to many scientific and technological research areas. Localization based on range measurements is at the core of technologies such as radar, sonar and wireless sensor networks. In this manuscript, we offer an in-depth study of the model for source localization based on range measurements obtained from the source signal, from the point of view of algebraic geometry. In the case of three receivers, we find unexpected connections between this problem and the geometry of Kummer’s and Cayley’s surfaces. Our work also gives new insights into the localization based on range differences.

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Notes

  1. 1.

    For a picture of the surface, see, for example, Wikipedia.

  2. 2.

    According to our choice of the reference receiver, we change to three every subscript 0 appearing in Compagnoni et al. (2014b). As a consequence, we have to correct signs in formulas whenever necessary.

  3. 3.

    In order to avoid confusion with the range notation, we call \(B_i(\varvec{\tau })\) the set that in Compagnoni et al. (2014b) is called \(A_i(\varvec{\tau })\).

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Acknowledgments

The authors would like to thank R. Sacco for useful discussions and suggestions during the preparation of this work and the anonymous referees for the valuable remarks on the preliminary version of the paper.

Author information

Correspondence to Marco Compagnoni.

Additional information

Communicated by Melvin Leok.

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Compagnoni, M., Notari, R., Ruggiu, A.A. et al. The Algebro-geometric Study of Range Maps. J Nonlinear Sci 27, 99–157 (2017). https://doi.org/10.1007/s00332-016-9327-4

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Keywords

  • Source localization model
  • Range maps
  • Kummer’s surfaces
  • Applied algebraic geometry

Mathematics Subject Classification

  • 14J99
  • 14P10
  • 51K99
  • 94A12