Orthogonal-Projection Localization for Polar Sensing Systems

Abstract

Greg Passmore

The geometry of polar sensing systems contains sufficient information to reveal the position of the emitter or receiver that pro- duced the data. Each beam or return defines a directional path between its near and far samples, and the intersection of these paths identifies the origin from which the energy was transmitted or received. This point is found by minimizing the summed orthogonal distances between a candidate origin and all ray directions, a relationship that emerges naturally from the spatial arrangement of the data. The same structure is present in radar, LiDAR, and sonar observations, where energy follows radial trajectories through a common geometric framework. The process also applies to hidden or non-cooperative emitters detected only through intermediate reflections or interference patterns, where indirect observations carry sufficient directional evidence to infer the source position. When timing, Doppler, or received-power information is available, these can be incorporated as natural weighting terms within the same least squares balance. The result exposes an underlying geometric law that links mea- sured returns, their directional vectors, and the origin implied by their geometric intersection.

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