The algebro-geometric study of range maps

TL;DR

This paper explores the algebraic geometric structure of source localization using range measurements, revealing unexpected connections with classical algebraic surfaces and providing new insights into localization methods.

Contribution

It introduces a novel algebraic geometric framework for analyzing range-based source localization, uncovering links to Kummer's and Cayley's surfaces.

Findings

Connections between localization models and classical algebraic surfaces

New geometric insights into range difference localization

Potential for improved localization algorithms based on algebraic geometry

Abstract

Localizing a radiant source is a widespread problem to many scientific and technological research areas. E.g. localization based on range measurements stays at the core of technologies like radar, sonar and wireless sensors networks. In this manuscript we study in depth 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 gives new insights also on the localization based on range differences.