The 2D point location problem has applications in several areas, such as geographic information systems, navigation systems, motion planning, mapping, military strategy, location and tracking moves. We aim to present a new approach that expands upon current techniques and methods to locate the 2D position of a signal source sent by an emitter device. This new approach is based only on the geometric relationship between an emitter device and a system composed of m≥2 signal receiving devices. Current approaches applied to locate an emitter can be deterministic, statistical or machine-learning methods. We propose to perform this triangulation by geometric models that exploit elements of pole-polar geometry. For this purpose, we are presenting five geometric models to solve the point location problem: (1) based on centroid of points of pole-polar geometry, PPC; (2) based on convex hull region among pole-points, CHC; (3) based on centroid of points obtained by polar-lines intersections, PLI; (4) based on centroid of points obtained by tangent lines intersections, TLI; (5) based on centroid of points obtained by tangent lines intersections with minimal angles, MAI. The first one has computational cost On and whereas has the computational cost Onlognwhere n is the number of points of interest.

In the broader context of smart cities, to ensure mobility of people regardless of their physical or sensory condition becomes a complex and difficult challenge to be treated. All papers referenced in this work are presented as a solution to equip the blind people with devices and sensors (controlled by a computational system) with the ability to capture environmental structure data and somehow describe it to the understanding of the blind people. Our work explores another side of this problem: how the environment can transmit data about itself to safely-help guide blind orientation in this environment? In other words, from our view, the environment must report data on its structure as opposed to make the blind person try to extract these data from this environment. So, here we propose to use an intelligent semaphore system (traffic lights) to communicate with a mobile system carried by the blind person and by the coherent processing of the signals sent and received between the mobile device and the intelligent semaphore, to conduct the blind in the streets crossing the crosswalk safely.

Several technological solutions have emerged over the last several months to support proximity contact tracing to fight the COVID-19 pandemic. For this reason, today more than ever, accurate signal location is needed, even in indoor public areas (supermarkets, public transport, etc.). In a previous work, we proposed five methods to solve the problem of signal localization using elements of pole-polar geometry. The proposals were innovative, since they solved a geometric problem (locating a point in a coordinate system) only by applying concepts of geometry. Among these developed methods, the PPC (Pole-Polar Centroid model) was also presented. Although the PPC solves the problem of locating a device with better precision than conventional methods (based on numerical or optimization methods), its accuracy was found to be the worst among the five proposed geometric methods. In this context, this work proposes an extension to our PPC method, called the weighted Pole-Polar Centroid method (wPPC), which improves the accuracy of the previous PPC results. Such an extension does not change the complexity O( m 2 ) or the minimum dimensionality ( m = 2) of nodes, which integrate a location network to perform the triangulation of such signals. Moreover, this extension estimates a device's location coordinates by means of the interaction, via signals, of this device with the network nodes distributed in any coordinate system. An IEEE 802.11 network infrastructure is used to accomplish the experiments. Errors in signal data are common, and our new proposed method, the wPPC, can mitigate the influence of these errors, produce more accurate results than the PPC, and outperform some of the other four proposed geometric methods and current numeric methods. Despite the use of an IEEE 802.11 network infrastructure for testing here, this range-based method for signal triangulation can be applied to any signal type (such as Wi-Fi, Bluetooth, and light and sound propagation).

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