# 2D Triangulation of Signals Source by Pole-Polar Geometric Models

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## Abstract

**:**

## 1. Introduction

#### 1.1. Considerations about Signal Data

- ${\rho}_{r}$ and ${\rho}_{e}$ are the receiver and the emitter power
- ${g}_{r}$ and ${g}_{e}$ are the receiver and emitter antennas gains
- $d$ is the radial distance between an emitter and a receiver in meters.
- $\lambda $ is the wavelength. For IEEE 802.11b and g to 2.4 GHz, $\lambda \approx 0.125\mathrm{m}$ and, to 5.7 GHz, $\lambda \approx 0.06\mathrm{m}$.

#### 1.2. Standard Methods and Considerations

#### 1.2.1. RSSI Approach

- $\rho \left(d\right)$ is the signal strength value (dB) expected by an AP located at a radial distance
- $d$ from the signal origin.
- ${\rho}_{0}$ is the signal strength value (dB) at some reference distance ${d}_{0}$.
- $\mathcal{L}$ indicates the rate at which the path loss increases with distance (empirical value).
- $w\left(nW,C\right)$ is the signal attenuation factor promoted by walls.
- $\lambda $ is the signal wavelength.
- ${x}_{\sigma}$ is a Gaussian noise with zero-mean and variance ${\sigma}^{2}$.

#### 1.2.2. Time-Based Approaches (ToA and TDoA)

#### 1.2.3. Generalization of 2D Location Function Range-Based

#### 1.2.4. How to Solve the System

- ${x}_{p}=\frac{-E+G}{H};\text{}{x}_{q}=\frac{-E-G}{H};\text{}{y}_{p}=\frac{A-B{x}_{p}}{C};\text{}{y}_{q}=\frac{A-B{x}_{q}}{C}$;
- $A={r}_{1}{}^{2}-{r}_{2}{}^{2}-{x}_{1}{}^{2}-{y}_{1}{}^{2}+{x}_{2}{}^{2}+{y}_{2}{}^{2}$; $B=-2\left({x}_{1}-{x}_{2}\right)$; $C=-2\left({y}_{1}-{y}_{2}\right)$; $L=\frac{B}{C}$; $D=1+{L}^{2}$;
- $E=2\left(L\left[1-A\right]-{x}_{1}\right)$; $F=L\left(L-2{y}_{1}\right)+{x}_{1}{}^{2}+{y}_{1}{}^{2}-{r}_{1}{}^{2};\text{}G=\sqrt{{E}^{2}-4DF}$; $H=2L$.

#### 1.3. Useful 2D Geometric Definitions

**(d-1)**- The Euclidean distance ${D}_{\overline{PQ}}$ between two points $P\left({x}_{1},{y}_{1}\right)$ and $Q\left({x}_{2},{y}_{2}\right)$ is given by$${D}_{\overline{PQ}}=\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}}.$$
**(d-2)**- For constants $A$, $B$, $C$ ($A$ and $B$ not both zero) all points $\left(x,y\right)$ satisfying the equation$Ax+By+C=0$ define the implicit line equation in the Cartesian plane. For two points $P\left({x}_{1},{y}_{1}\right)$ and $Q\left({x}_{2},{y}_{2}\right)$, a particular $\overleftrightarrow{PQ}$ line equation is obtained by$$A={y}_{1}-{y}_{2};B={x}_{2}-{x}_{1};C=-A{x}_{1}-B{y}_{1}.$$
**(d-3)**- Two particular lines ${A}_{1}x+{B}_{1}y+{C}_{1}=0$ and ${A}_{2}x+{B}_{2}y+{C}_{2}=0$ have an interception at the point $\left({x}_{+},{y}_{+}\right)$, if $d={A}_{1}{B}_{2}-{A}_{2}{B}_{1}\ne 0$, given by$$\left({x}_{+},{y}_{+}\right)=\left(\frac{{B}_{1}{C}_{2}-{B}_{2}{C}_{1}}{d},\frac{{A}_{2}{C}_{1}-{A}_{1}{C}_{2}}{d}\right),$$
**(d-4)**- The angle $\theta $ formed between two particular lines is given by$$\mathrm{tan}\theta =\frac{{A}_{1}{B}_{2}-{A}_{2}{B}_{1}}{{A}_{1}{A}_{2}+{B}_{1}{B}_{2}}.$$
**(d-5)**- The line equation ${A}_{p}x+{B}_{p}y+{C}_{p}=0$ that passes through point $P\left({x}_{p},{y}_{p}\right)$ and is perpendicular to the line $Ax+By+C=0$ is defined as$${A}_{p}=-B;{B}_{p}=A;{C}_{p}=A{x}_{p}-B{y}_{p}.$$Circle—In the Cartesian plane the equation ${\left(x-{x}_{c}\right)}^{2}+{\left(y-{y}_{c}\right)}^{2}={r}^{2}$ defines the implicit circle equation centered at the point $C\left({x}_{c},{y}_{c}\right)$ with radius $r$. Let $E\left({x}_{e},{y}_{e}\right)$ be an external point to a circle. By using $E$ we can obtain two tangent lines, ${t}_{1}$ and ${t}_{2}$, to the circle (Figure 1b), which pass through points $P{\left({x}_{1},{y}_{1}\right)}^{*}$ and $Q{\left({x}_{2},{y}_{2}\right)}^{*}$, respectively. Points $P$ and $Q$ can be computed by applying the geometric concept of pole-polar definition.Pole-Polar Geometry—Pole-point and polar-line are, respectively, a point and a line that have a unique reciprocal relationship with respect to a given conic section. If the point lies on the conic section, its polar-line is the tangent line to the conic section at that point [24]. If the pole-point is external to the conic section, the polar-line intercepts the conic section exactly at the points that allow passing tangent lines from this pole-point (Figure 1b). Our interest is to pass two tangent lines, ${t}_{1}$ and ${t}_{2}$, through a circle centered at point $C\left({x}_{c},{y}_{c}\right)$ with radius $r$. Moreover, these lines must pass through a known external point $E\left({x}_{e},{y}_{e}\right)$ (or pole-point) to this circle (Figure 1b). We need to locate the coordinates of the polar-points $P{\left({x}_{1},{y}_{1}\right)}^{*}$ and $Q{\left({x}_{2},{y}_{2}\right)}^{*}$, which define the polar-line $p$ and lies to the tangents lines ${t}_{1}$ and ${t}_{2}$. Additionally we must find the equation of the polar-line $p\left(Ax+By+C=0\right)$.
**(d-6)**- The general equation of a conic in the Cartesian coordinate system is given by ${a}_{xx}{x}^{2}+2{a}_{xy}xy+{a}_{yy}{y}^{2}+2{b}_{x}x+2{b}_{y}y+w=0$. We need the equation of the polar-line $p\left(Ax+By+C=0\right)$ that can be obtained by a known pole-point $E\left({x}_{e},{y}_{e}\right)$. The required coefficients of the respective polar-line $p$ is given by: $A={a}_{xx}{x}_{e}+{a}_{xy}{y}_{e}+{b}_{x}$; $B={a}_{xy}{x}_{e}+{a}_{yy}{y}_{e}+{b}_{y}$; $C={b}_{x}{x}_{e}+{b}_{y}{y}_{e}+w$. In this work, the expected conic section is a circle. For the circle case, the following simplifications are helpful: ${a}_{xx}=1$; ${a}_{xy}=0$; ${a}_{yy}=1$; ${b}_{x}=-{x}_{c}$; ${b}_{y}=-{y}_{c}$ and $w={x}_{c}^{2}+{y}_{c}^{2}-{r}^{2}$.The next step consists in placing points $P{\left({x}_{2},{y}_{2}\right)}^{*}$ and $Q{\left({x}_{2},{y}_{2}\right)}^{*}$, which are obtained by computing the intersection between the polar-line $p$ and the circle line (Figure 1b). To compute the intersection of a line, $Ax+By+C=0$, with a circle, ${\left(x-{x}_{c}\right)}^{2}+{\left(y-{y}_{c}\right)}^{2}={r}^{2}$, the following conditions must be considered: ${d}_{lc}=\frac{\left|A{x}_{c}+B{y}_{c}+C\right|}{\sqrt{{A}^{2}+{B}^{2}}}$ is the distance between the line and the circle center point $C\left({x}_{c},{y}_{c}\right)$,
- ○
- if ${d}_{lc}>r$, there is no intersection point;
- ○
- if ${d}_{lc}=r$, the line is tangent to the circle and has one intersection point;
- ○
- if ${d}_{lc}<r$, the line is secant to the circle and has two intersection points.

The algebraic solution for this intersection is an equation of degree two. Another way to solve this intersection is applying some geometric relationships, as follows. To find the intersections points $P{\left({x}_{1},{y}_{1}\right)}^{*}$ and $Q{\left({x}_{2},{y}_{2}\right)}^{*}$, which are the polar-points, we have first to drop a perpendicular line (by d-5) from the center $C\left({x}_{c},{y}_{c}\right)$ of the circle to the line $p$. Let $T\left({x}_{t},{y}_{t}\right)$ be the intersection point and $\overleftrightarrow{CT}$ be the line that passes through $C$ and $T$ (Figure 1b).The equation of line $p\left(Ax+By+C=0\right)$ is known (by d-6). Thus, the equation of line $\overleftrightarrow{CT}$ is $\overleftrightarrow{CT}\left(-Bx+Ay+A{x}_{c}-B{y}_{c}\right)$. This way, the point $T\left({x}_{t},{y}_{t}\right)$ can be computed by intersection between $p$ and $\overleftrightarrow{CT}$ lines (by d-3).${D}_{\overline{CT}}$ represents the Euclidean distance between points $C$ and $T$; ${D}_{\overline{PT}}$ refers to the distance between points $P$ and $T$; ${D}_{\overline{QT}}$ stands for the distance between points $Q$ and $T$; and ${D}_{\overline{CP}}={D}_{\overline{CQ}}=r$ (by d-1).The triangles $\Delta CPT$ and $\Delta CQT$ are right-angled, and hence we prove that$${D}_{\overline{CT}}^{2}+{D}_{\overline{PT}}^{2}={r}^{2},$$$${D}_{\overline{CT}}^{2}+{D}_{\overline{QT}}^{2}={r}^{2}.$$ **(d-7)**- So, ${D}_{\overline{QT}}={D}_{\overline{PT}}=h=\sqrt{{r}^{2}-{D}_{\overline{CT}}^{2}}$; now, if we translate the point $T$ by $h$ units in both directions along line $p$, the points $P$ and $Q$ are determined as follows$${\left({x}_{1},{y}_{1}\right)}^{*}=\left({x}_{t}-\frac{Bh}{\sqrt{{A}^{2}+{B}^{2}}},{y}_{t}+\frac{Ah}{\sqrt{{A}^{2}+{B}^{2}}}\right),$$$${\left({x}_{2},{y}_{2}\right)}^{*}=\left({x}_{t}+\frac{Bh}{\sqrt{{A}^{2}+{B}^{2}}},{y}_{t}-\frac{Ah}{\sqrt{{A}^{2}+{B}^{2}}}\right).$$The suggested geometric models that use the convex hull algorithm (CHC, PLI, TLI, and MAI) need to minimize the region of interest (ROI) and exclude bad points from final results. This minimization of ROI is obtained by obtaining a convex polygon defined on a set of previously computed points. A set $S$ is convex if $\forall x,y\in S\Rightarrow \overline{xy}\subseteq S$. Any region (polygon) with a “dent” is not convex [24]. The convex hull of a set of points is the smallest convex set containing these points [24,25,26].
**(d-8)**- The convex hull algorithm is used in this work to specify a Region of Interest (ROI).To illustrate the use of the convex hull we must consider the existence of three receivers centered at coordinates ${C}_{1}\ne {C}_{2}\ne {C}_{3}$ (and these coordinates cannot all be collinear) that collect a signal from a point with distance ${r}_{1},{r}_{2}$ and ${r}_{3}$ (radial distance), respectively. ${P}_{kj}$ and ${Q}_{kj}$, $k\ne j$, represents the polar-points that lie to the circle centered in ${C}_{j}$ that is obtained by the external point ${C}_{k}$ (center of another circle), as Figure 2a shows. Thus, the smallest convex polygon that contains all obtained polar-points is the convex hull for these points and this minimal polygon defines our ROI. This region in red-color lines is used to illustrate the ROI for the proposed geometric models presented below (always representing a convex hull to define a ROI).
**(d-9)**- The location estimation of the emitter, ${E}_{xy}\left({E}_{x},{E}_{y}\right)$, is based on a set $S$ that contains $n$ points, $\left(x,y\right)$, which are collected in a defined ROI. This location is given by the centroid point among all points in $S$, by$${E}_{x}=\frac{{\sum}_{x\in S}x}{n},\text{}{E}_{y}=\frac{{\sum}_{y\in S}y}{n}$$

## 2. The Proposed Geometric Models

#### 2.1. Accurate Data, Exact Result

#### 2.2. Polar-Points Centroid Model (PPC)

Algorithm 1. PPC—Polar-Points Centroid Model |

Data Input |

$m\ge 2$ is the number of receivers. ${C}_{k}\left({x}_{k},{y}_{k}\right)$, $k=1,2,\dots ,m$, is the planar position of each receiver. ${r}_{k},k=1,2,\dots ,m$, is the signal range of each receiver to an emitter. |

Procedure |

1: for each $k=1,2,3,\dots ,m$ |

2: $\forall k,j:1,2,\dots ,m/k\ne j$, by (d-6) and (d-7), for each receiver position ${C}_{k}$, used as pole-points, compute all combinations of polar-points ${P}_{kj}\left({x}_{pkj},{y}_{pkj}\right)$ and ${Q}_{kj}\left({x}_{qkj},{y}_{qkj}\right)$ with the respective receiver at position ${C}_{j}$ and signal range ${r}_{j}$. |

3: store the points ${P}_{kj}$ and ${Q}_{kj}$ in the set $\left(S\leftarrow S{\displaystyle \cup}\left\{{P}_{kj},{Q}_{kj}\right\}\right)$. |

4: end for |

5: Apply (d-9) in the set $S$, compute the location estimation, ${E}_{xy}\left({E}_{x},{E}_{y}\right)$, of the emitter. |

Information Output |

6: Emitter location estimation ${E}_{xy}\left({E}_{x},{E}_{y}\right)$. |

#### 2.3. Convex Hull Centroid Model (CHC)

Algorithm 2. CHC—Convex Hull Centroid Model Algorithm |

Data Input |

$m>2$ is the number of receivers. ${C}_{k}\left({x}_{k},{y}_{k}\right)$, $k=1,2,\dots ,m$, is the planar position of each receiver. ${r}_{k},k=1,2,\dots ,m$, is the signal range of each receiver to an emitter. |

Procedure |

1: Execute the steps 1 until 4 of algorithm Polar Points Centroid Model. |

2: Apply (d-8), find the convex hull polygon for all polar-points in $S$. The obtained polygon is the minimal convex polygon that involves all interest points in $S$. This polygon constitutes the ROI. |

3: Exclude all polar-points on the boundary of this convex polygon, called bad polar-points, from $S$. |

4: Apply (d-9) in $S$, compute the location estimation, ${E}_{xy}\left({E}_{x},{E}_{y}\right)$, of the emitter. |

Information Output |

5: Emitter location estimation ${E}_{xy}\left({E}_{x},{E}_{y}\right)$. |

#### 2.4. Polar Lines Intersections Model (PLI)

Algorithm 3. PLI—Polar Lines Intersections Model Algorithm |

Data Input |

$m\ge 3$ is the number of receivers. ${C}_{k}\left({x}_{k},{y}_{k}\right)$, $k=1,2,\dots ,m$, is the planar position of each receiver. ${r}_{k},k=1,2,\dots ,m$, is the signal range of each receiver to an emitter. |

Procedure |

1: for each $k=1,2,3,\dots ,m$ |

2: $\forall k,j:1,2,\dots ,m/k\ne j$, by (d-6) and (d-7), for each receiver position ${C}_{k}$, used as pole-points, compute all combinations of polar-points ${P}_{kj}\left({x}_{pkj},{y}_{pkj}\right)$ and ${Q}_{kj}\left({x}_{qkj},{y}_{qkj}\right)$ with the respective receiver at position ${C}_{j}$ and signal range ${r}_{j}$. |

3: Stores the corresponding points ${P}_{kj}$ and ${Q}_{kj}$ in the set $R$ $\left(R\leftarrow R{\displaystyle \cup}\left\{{P}_{kj},{Q}_{kj}\right\}\right)$. |

4: For each corresponding ${P}_{kj}$ and ${Q}_{kj}$ points, by (d-6), compute the ${p}_{kj}$ polar-line equation. |

5: end for |

6: For all ${p}_{kj}$ polar-lines, by (d-3), compute the intersections points, $\left({x}_{+},{y}_{+}\right)$, among all others polar-lines. Stores these intersections points in $S$. |

7: Apply (d-8), find the convex hull polygon for all polar-points in $R$. The obtained polygon is the minimal convex polygon that involves all interest points in $R$. This polygon constitutes our ROI. |

8: Exclude from $S$ all intersections points among all polar-lines on the boundary, or out, of the ROI, called bad intersections points. |

9: Apply (d-9), compute the location estimation, ${E}_{xy}\left({E}_{x},{E}_{y}\right)$, of the emitter. |

Information Output |

10: Emitter location estimation ${E}_{xy}\left({E}_{x},{E}_{y}\right)$. |

#### 2.5. Tangent Lines Intersections Model (TLI)

Algorithm 4. TLI—Tangent Lines Intersections Model Algorithm |

Data Input |

$m\ge 2$ is the number of receivers. ${C}_{k}\left({x}_{k},{y}_{k}\right)$, $k=1,2,\dots ,m$, is the planar position of each receiver. ${r}_{k},k=1,2,\dots ,m$, is the signal range of each receiver to an emitter. |

Procedure |

1: for each $k=1,2,3,\dots ,m$ |

2: $\forall k,j:1,2,\dots ,m/k\ne j$, by (d-6) and (d-7), for each receiver position ${C}_{k}$, used as pole-points, compute all combinations of polar-points ${P}_{kj}\left({x}_{pkj},{y}_{pkj}\right)$ and ${Q}_{kj}\left({x}_{qkj},{y}_{qkj}\right)$ with the respective receiver at position ${C}_{j}$ and signal range ${r}_{j}$. |

3: Stores the points ${P}_{kj}$ and ${Q}_{kj}$ in the set $R$ $\left(R\leftarrow R{\displaystyle \cup}\left\{{P}_{kj},{Q}_{kj}\right\}\right)$. |

4: For each corresponding ${P}_{kj}$ and ${Q}_{kj}$ polar-points, by (d-6), computes the respective two tangent lines equation ${t}_{Pkj}$ and ${t}_{Qkj}$ that passes by each ${C}_{k}$. |

5: end for |

6: For all ${t}_{Pkj}$ and ${t}_{Qkj}$ tangent-lines, by (d-3), computes the intersections points, $\left({x}_{+},{y}_{+}\right)$, among all tangent-lines. Store these intersections points in $S$. |

7: Apply (d-8), find the convex hull polygon for all polar-points in $R$. The obtained polygon is the minimal convex polygon that involves all interest points in $R$. This polygon constitutes our ROI. |

8: Exclude from $S$ all intersections points among all tangent-lines on the boundary, or out, of the ROI, called bad intersections points. |

9: Apply (d-9), compute the location estimation, ${E}_{xy}\left({E}_{x},{E}_{y}\right)$, of the emitter. |

Information Output |

10: Emitter location estimation ${E}_{xy}\left({E}_{x},{E}_{y}\right)$. |

#### 2.6. Tangent Lines with Minimal Angles Model (MAI)

Algorithm 5. MAI—Tangent Lines with Minimal Angles Intersections Model Algorithm |

Data Input |

$m\ge 2$ is the number of receivers. ${C}_{k}\left({x}_{k},{y}_{k}\right)$, $k=1,2,\dots ,m$, is the planar position of each receiver. ${r}_{k},k=1,2,\dots ,m$, is the signal range of each receiver to an emitter. |

Procedure |

1: for each $k=1,2,3,\dots ,m$ |

2: $\forall k,j:1,2,\dots ,m/k\ne j$, by (d-6) and (d-7), for each receiver position ${C}_{k}$, used as pole-points, compute all combinations of polar-points ${P}_{kj}\left({x}_{pkj},{y}_{pkj}\right)$ and ${Q}_{kj}\left({x}_{qkj},{y}_{qkj}\right)$ with the respective receiver at position ${C}_{j}$ and signal range ${r}_{j}$. |

3: Stores the points ${P}_{kj}$ and ${Q}_{kj}$ in the set $R$ $\left(R\leftarrow R{\displaystyle \cup}\left\{{P}_{kj},{Q}_{kj}\right\}\right)$. |

4: For each corresponding ${P}_{kj}$ and ${Q}_{kj}$ polar-points, by (d-6), compute the respective two tangent lines equation ${t}_{Pkj}$ and ${t}_{Qkj}$ that passes by ${C}_{k}$. |

5: end for |

6: For all ${t}_{Pkj}$ and ${t}_{Qkj}$ tangent-lines, by (d-3), compute the intersections points, $\left({x}_{+},{y}_{+}\right)$, among all tangent-lines. Stores these intersections points in $S$. |

7: Apply (d-8), find the convex hull polygon for all center circles points ${C}_{k}$. The obtained polygon is the minimal convex polygon that involves all interest points in $S$. This polygon constitutes our ROI. |

8: Exclude from $S$ all intersections points among all tangent-lines on the boundary, or out, of the ROI, called bad intersections points. |

9: Apply (d-9), compute the location estimation, ${E}_{xy}\left({E}_{x},{E}_{y}\right)$, of the emitter. |

Information Output |

10: Emitter location estimation ${E}_{xy}\left({E}_{x},{E}_{y}\right)$. |

## 3. Experimental Cases

#### 3.1. Methodology Applied to Real Data Acquisition

^{2}on the ground (Figure 14). Each AP is positioned at the Cartesian coordinate $\left(x,y\right)$ and the respective value (in meters) is registered alongside the respective AP (e.g., the AP5 is at $\left(9.60,-2.84\right)$—Figure 14). These Cartesian coordinate (2D) design the set of ${C}_{k}$ points (centers of range signals circles computed for the kth-AP) and defines the geometric arrangement of the covered area on the ground.

#### 3.2. Quality of Acquired Data

#### 3.3. Results and Analyses

- PPC—Polar Points Centroid Model (proposed).
- CHC—Convex Hull Centroid Model (proposed).
- PLI—Polar Lines Intersections Model (proposed).
- TLI—Tangent Lines Intersections Model (proposed).
- MAI—Tangent Lines with Minimal Angles Model (proposed).
- NRm—Newton–Rapson Method (for comparison).
- LSm—Least Square Method (for comparison).
- WLSm—Weighted Least Square Method (for comparison).

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Complementary Results

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**Figure 1.**(

**a**) Analytical solution to $\left(x,y\right)$ when there is an intersection between circles. (

**b**) Possible tangential straight lines ${t}_{1}$ and ${t}_{2}$ for the circle line obtained by an external point $E\left({x}_{e},{y}_{e}\right)$ (pole-point). Polar-line $p$ and its geometric relationship with the tangent lines and a given conic section (circle in this case).

**Figure 2.**(

**a**) Illustration of Convex Hull algorithm applied to a set of polar-points. (

**b**) An example of signal data and it respective signal ranges with its polar points. This case represents sets of signal data with satisfactory quality. The red dotted rectangle mark a region where a set of polar points enveloping the location of an emitter device.

**Figure 3.**Results produced by Polar-Points Centroid Model (PPC). (

**a**,

**b**) show the possible presence of noise in acquired data. (

**c**) Represents the best case for acquired data because the signal data are quasi-exact. The red dotted rectangle mark all (four) polar points that belong to all circle lines (signals range). The exact solution is one of these points (see Section 2.1).

**Figure 4.**(

**a**) Incoherence in data. The spatial position of a receiver is inside of range of another receiver. In this case, some polar-points are complex numbers (see the polar-points out of the line circles)—but their real parts belong to the region of interest (ROI). (

**b**) Accurate data. Exact solution produced by PPC using redundant data (four receivers). For this case, is better applying the exact solution providing by Section 2.1. (

**c**) Accurate data. A collinear arrangement among receivers (geometric arrangement not recommended) and the exact solution provided by PPC. The solution to this problem is impossible for Weighted Least Squares (WLSm), Least-Square (LSm), and Newton–Rapson (NRm).

**Figure 5.**Results produced by Convex Hull Centroid Model (CHC). (

**a**,

**b**) highlight the possible presence of noise in acquired data. (

**c**) Represents the best case for acquired data. For this case, is better applying the exact solution provided by Section 2.1.

**Figure 6.**Results produced by CHC using satisfactory redundant data (four receivers). For this case, is better applying the exact solution provided by Section 2.1.

**Figure 7.**Different cases and respective responses of the PPC and CHC models. (

**a**) Five receivers system. The data has some discrepancies. (

**b**) Four receivers system. The data has small inconsistencies. For both cases, is better applying the solution provided by Section 2.1.

**Figure 8.**Results produced by Polar Lines Intersections Model (PLI). (

**a**,

**b**) show the possible presence of noise in acquired data. (

**c**) Represents the best case for acquired data. For this case is better applying the exact solution provided by Section 2.1.

**Figure 9.**Result produced by PLI four receivers system. The non-symmetrical points are removed from the ROI by the use of Convex Hull. For this case is better applying the exact solution provided by Section 2.1.

**Figure 10.**Results produced by Tangent Lines Intersections (TLI) model. (

**a**,

**b**) show the possible presence of noise in acquired data. (

**c**,

**d**) represents the best case for acquired data. (

**d**) Use of four receivers (the legend box is omitted for best view). For cases (

**c**,

**d**) is better applying the exact solution provided by Section 2.1.

**Figure 11.**(

**a**) Illustrations that define the Tangent Lines with Minimal Angles Model (MAI). (

**b**) All intersections (red-star) with non-minimal angle tangent lines (orange-lines) must be eliminated. (

**c**) All points inside the ROI formed by the convex hull polygon among the circle centers are the required points (blue-star).

**Figure 12.**Results produced by MAI model. (

**a**,

**b**) show the possible presence of noise in acquired data. (

**c**,

**d**) represents the best case for acquired data. (

**d**) Application using four receivers (the legend box is omitted for best view). For the cases (

**c**,

**d**) is better applying the exact solution provided by Section 2.1.

**Figure 13.**Geometric incoherence. The receiver is placed inside the range of another receiver. The geometric models TLI and MAI are able to overcome this data inconsistency and produce a satisfactory result.

**Figure 14.**Structure composed of five Access Point (AP) devices and their respective position on an outdoor environment.

**Figure 15.**Acquired data quality (graphical units in meter). (

**a**,

**b**) worst quality data. (

**c**) Satisfactory data quality.

**Figure 17.**Some particular experimental results. (

**a**) The worst case. (

**b**) An intermediate case. (

**c**) The best case. The blue-dotted circle lines show the receivers/emitters that suffered severe interferences and promoted the acquisition of data with errors.

Methods | Magnitudes Errors (in Meters) | ||||||||
---|---|---|---|---|---|---|---|---|---|

Minimum Error | Maximum Error | Mean Error | |||||||

x-axis | y-axis | Distance | x-axis | y-axis | Distance | x-axis | y-axis | Distance | |

PPC * | 0.4 | 0.2 | 2.7 | 15.5 | 12.5 | 15.7 | 4.2 | 4.4 | 6.9 |

CHC * | 0.1 | 0.1 | 0.6 | 11.9 | 6.0 | 12.8 | 2.5 | 2.4 | 3.7 |

PLI * | 0.1 | 0.1 | 0.7 | 12.8 | 15.9 | 16.3 | 2.4 | 3.8 | 5.0 |

MAI * | 1.6 | 0.3 | 1.7 | 7.9 | 5.3 | 8.9 | 4.2 | 1.5 | 4.7 |

TLI * | 2.3 | 0.3 | 2.3 | 8.3 | 3.9 | 9.2 | 4.1 | 1.3 | 4.4 |

NRm | 0.1 | 1.1 | 1.3 | 14.7 | 12.2 | 15.2 | 2.8 | 5.1 | 6.5 |

LSm | 0.3 | 1.2 | 1.8 | 14.1 | 22.5 | 23.7 | 3.6 | 6.4 | 7.9 |

WLSm | 0.1 | 0.7 | 1.3 | 13.6 | 15.3 | 17.7 | 3.5 | 4.9 | 6.5 |

Mean Errors (in Meters) | |||
---|---|---|---|

x-axis | y-axis | Distance | |

Geometric Models | 3.7 | 2.9 | 5.3 |

NRm + LSm+ WLSm | 3.5 | 4.1 | 6.0 |

Methods | Standard Deviation of the Errors | Effective Variabilityof the Errors | ||
---|---|---|---|---|

x-axis | y-axis | Distance | ||

PPC * | 4.1 | 3.8 | 4.5 | 4.1 |

CHC * | 3.1 | 2.1 | 3.4 | 2.9 |

PLI * | 3.3 | 4.1 | 4.9 | 4.1 |

MAI * | 2.5 | 1.6 | 2.6 | 2.2 |

TLI * | 2.3 | 1.0 | 2.3 | 1.9 |

NRm | 4.0 | 4.0 | 4.9 | 4.3 |

LSm | 4.3 | 6.7 | 7.4 | 6.1 |

WLSm | 4.2 | 4.3 | 5.5 | 4.7 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Montanha, A.; Polidorio, A.M.; Dominguez-Mayo, F.J.; Escalona, M.J. 2D Triangulation of Signals Source by Pole-Polar Geometric Models. *Sensors* **2019**, *19*, 1020.
https://doi.org/10.3390/s19051020

**AMA Style**

Montanha A, Polidorio AM, Dominguez-Mayo FJ, Escalona MJ. 2D Triangulation of Signals Source by Pole-Polar Geometric Models. *Sensors*. 2019; 19(5):1020.
https://doi.org/10.3390/s19051020

**Chicago/Turabian Style**

Montanha, Aleksandro, Airton M. Polidorio, F. J. Dominguez-Mayo, and María J. Escalona. 2019. "2D Triangulation of Signals Source by Pole-Polar Geometric Models" *Sensors* 19, no. 5: 1020.
https://doi.org/10.3390/s19051020