# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 2 The Proposed Geometric Models
### Accurate Data, Exact Result  
Before presenting the five proposed geometric models to solve the problem of locating a signal source, we present now the perfect solution, if the acquired signal data are accurate and if the spatial geometric arrangement among the receivers be symmetric along the \(x\)-axis and \(y\)-axis. This exact solution has complexity \(O(n)\). All the geometric models proposed in this work are based on the calculation of polar points, which belong to a circle line (signal range). If it is certain that the data acquired from the signal are accurate, the solution for the location of the signal source is immediate. According to ISO 5725-1, the term \"accuracy\" is used to describe the closeness of a measurement to the actual value. The Figure 2b, Figure 3c, Figure 4b,c illustrate some cases that involve accurate signal data and therefore must have exact solutions. The Figure 3c is the perfect case using a system composed by \(m=3\) receivers. The Figure 4b,c show perfect cases using a system with \(m=4\) receivers. In fact, the case presented in Figure 3c is almost perfect. The signal data has been purposely shifted from perfection to being able to see that in the red dotted rectangle there is more than one polar point (there are four). If the data were accurate, these four polar points would self-overlap at the intersection point among the lines of these three circles. See in Figure 3c that the solution pointed out by the geometric model PPC (green rectangle) is not exact. The reason for this divergence is explained in the description and analysis of the PPC model (Section 2.2). For the case illustrated in Figure 3c, the exact solution is obtained by the set of polar points that overlap the point of intersection among the lines of the circles. If the signal data are accurate, the exact solution is given by a single point, among all the polar points, that having the property to belongs at the same time to all circles lines (signal ranges). The formal proof for this exact solution is obtained by construction, or proof by example, which requires the construction of a concrete example with exact solution to show that some polar point is the exact solution because have the property to belongs to all circles lines. To verify this proof, use the proposed PPC method (Section 2.2) provided in the simulator system (see Section 4). Using the simulator, you can construct examples of ideal situations (all lines of circles must intersect at a single point and the geometric arrangement of the receivers in the plane must respect aspects of non-collinearity and neither all them can be located in the range of another receiver). If these conditions are met, the PPC method will determine one or more polar-points over that single point of intersection among all circle lines and/or determine the solution over that intersection. To obtain the exact solution we must discover at least one polar point that has the property of belonging to all circles lines. The number of polar points that have this property depends on the geometric arrangement among the position of the receivers. For example, the arrangement presented in the Figure 3c has four polar points with this property. For Figure 4b, there are eight polar points. For Figure 4c, although the data is perfect, there is no polar point that has this property. Regardless of the number of receivers \(m\) used in the system, the maximum number of polar points that can have this property is eight. The Figure 2b shows a system composed of four receivers. As you can see the lines of all circles do not intersect at a single point. In this case, there is some error associated with the signal data. Even so, we can admit that the data quality is satisfactory. If the geometric arrangement among the position of the receivers is favorable and if the quality of the signal data is high there will always be an expressive set of polar points involving the location of the signal-emitting device.