**5. The computing relations group with which it is achieved the determination of the target T geographic position in the horizontal plane of the referential ellipsoid**

At the computing of the linear distances, on the longitude and, respectively, on the latitude direction, between the video camera successive positions and, respectively, between the camera positions and the sighted target, it, also, takes account from the fact that this monocameral stereo-fotogrammetric system, permits the sighting of some objectives which are situated at distances of up to 200 – 300 meters from the lab vehicles. This way, it is possible to adopt the hypothesis consisting in the approximation of the terrestrial globe with an equivalent sphere with a radius R = 6.367.472 km., as it is presented in the Fig. 20.

The geographic position of the target T in the horizontal plane of the referential ellipsoid is achieved, by combining the determinations of the absolute angular coordinates, 111 , , *TTT* and respectively, 222 , , *TTT* , of the sighted target, for two different positions, 1 *L* and, respectively, 2 *L* , of the video camera, positions which are obtained as a result of the lab vehicle displacement with a distance in limits of which the target T is maintained in the video camera viewing field.

In the positioning scheme presented in Figure 20, the following notations were introduced:

GPS Positioning of Some Objectives Which are Situated

, which are delimited by two meridian

 

 

(9)

, the

(10)

(11)

(12)

(13)

at Great Distances from the Roads by Means of a "Mobile Slide Monitor – MSM" 175

; - the angular differences of latitude, between the target and,

1,2 2 1 

 

following form:

where: *r R* cos

segments, as follows:

following relation exists:

 ; 2,*T T* <sup>2</sup> 

min.

min.

*b meters R meters* 

*a meters r meters* 

.

Between two circular arcs on latitude,

respectively, the two positions of the lab vehicle.

circles and which are situated, the first at the latitude 0, and the other at the latitude *<sup>T</sup>*

 cos *<sup>T</sup>* 

*<sup>T</sup>*

Also on the basis of the scheme from the figure 20 which presents the positioning mode of a target on an equivalent sphere of the terrestrial globe, the linear distances can be calculated on the basis of angular coordinate differences by means of an equation set, with the

For the establishment, on this basis, of the computing relations for the geographic position absolute coordinates of the target T, it resorts to the positioning scheme presented in figure 21, taking account of the fact that due to the relative reduced dimensions of the sighting

By this, in plane projection of the sighting field, the circle arcs are replaced by linear

*T T aR R x*

1,2 <sup>1</sup> 2 ; 360 60 *b R* 

2, <sup>2</sup> 2 ; 360 60 *<sup>T</sup> b R* 

 

tan ;

 <sup>1</sup> 1 2 1,2 2, 1 1, 2 2,

cos

*T T*

 

 

*T*

(16)

 

 

> 

 1, 1, 1 2 2, 2 cos 2 cos ; 360 60 360 60

 1,2 1,2 2 2 2, 2 cos 2 cos ; 360 60 360 60 *T T aR R*

 

> 

 

 

(14)

(15)

field, its spherical curved surface is approximated by in plan projection of this field.

*T T*

 

 

These result in the following expressions for the azimuth angles:

*T*

*b b a*

 

2 for the distances on the longitude direction; 360 60

2 for the distances on the latitude direction; 360 60

 and *T* 

**Figure 20.** The Lab Vehicle and the sighted target positioning on the equivalent sphere of the terrestrial globe.

1 1 , and 2 2 , - the geographic coordinates of longitude and, respectively, of latitude, which are deduced for two sighting successive positions, 1 *L* and respectively 2 *L* , of the video camera from the determinations performed by the GPS-INS group, by introducing the offset corrections, which correspond to the distances between the emplacement location of this group in the lab vehicle and the camera objective:

, *T T* - the target geographic coordinates of longitude and, respectively, latitude;

1,*T T* <sup>1</sup> ; 1,2 1 2 ; - the angular differences of longitude;

1,2 2 1 ; 2,*T T* <sup>2</sup> ; - the angular differences of latitude, between the target and, respectively, the two positions of the lab vehicle.

Between two circular arcs on latitude, and *T* , which are delimited by two meridian circles and which are situated, the first at the latitude 0, and the other at the latitude *<sup>T</sup>* , the following relation exists:

$$
\Delta \hat{\mathcal{L}}\_{\phi\_{\Gamma}} = \Delta \hat{\mathcal{L}} \cdot \cos \phi\_{\Gamma} \tag{9}
$$

Also on the basis of the scheme from the figure 20 which presents the positioning mode of a target on an equivalent sphere of the terrestrial globe, the linear distances can be calculated on the basis of angular coordinate differences by means of an equation set, with the following form:

$$a\left[\text{meters}\right] = \frac{\Delta\hat{i}\left[\text{min.}\right]}{360 \times 60} \cdot 2\pi \cdot r\left[\text{meters}\right] - \text{for the distances on the longitude } \triangle \text{direction; (10)}$$

$$b\left[\text{meters}\right] = \frac{\Delta\hat{\rho}\left[\text{min.}\right]}{360 \times 60} \cdot 2\pi \cdot R\left[\text{meters}\right] - \text{ for the distances on the latitude } \varphi \text{ direction; (11)}$$

where: *r R* cos.

174 Cartography – A Tool for Spatial Analysis

1

2  ,2 *T* 

 2,1 

0 0

*<sup>T</sup> Rr*

*R*

*R*

*T*

*<sup>T</sup>* cos N

0

globe.

1 1 ,

, *T T* 

1,*T T* <sup>1</sup> 

 

 and 2 2 ,

this group in the lab vehicle and the camera objective:

 ; 1,2 1 2 

 

**Figure 20.** The Lab Vehicle and the sighted target positioning on the equivalent sphere of the terrestrial

S

which are deduced for two sighting successive positions, 1 *L* and respectively 2 *L* , of the video camera from the determinations performed by the GPS-INS group, by introducing the offset corrections, which correspond to the distances between the emplacement location of


; - the angular differences of longitude;


*L*2

T

 ,,1 

*T T*

*T*1

 ,1 *T* 

> *T*

1

*L*1

 2,1 

2 *T*

*<sup>T</sup>*<sup>2</sup>

 ,2,1 

> For the establishment, on this basis, of the computing relations for the geographic position absolute coordinates of the target T, it resorts to the positioning scheme presented in figure 21, taking account of the fact that due to the relative reduced dimensions of the sighting field, its spherical curved surface is approximated by in plan projection of this field.

> By this, in plane projection of the sighting field, the circle arcs are replaced by linear segments, as follows:

$$a\_1 = \frac{\Delta\hat{\lambda}\_{1,T}}{360 \times 60} \cdot 2\pi \cdot R \cdot \cos\rho\_T = \frac{\Delta\hat{\lambda}\_{1,T}}{360 \times 60} \cdot 2\pi \cdot R \cdot \cos\left(\rho\_2 + \Delta\rho\_{2,T}\right);\tag{12}$$

$$a\_2 = \frac{\Delta\hat{\lambda}\_{1,2}}{360 \times 60} \cdot 2\pi \cdot R \cdot \cos\varphi\_{\Gamma} = \frac{\Delta\hat{\lambda}\_{1,2}}{360 \times 60} \cdot 2\pi \cdot R \cdot \cos\left(\varphi\_2 + \Delta\varphi\_{2,\Gamma}\right);\tag{13}$$

$$b\_1 = \frac{\Lambda \hat{\phi}\_{1,2}}{360 \times 60} \cdot 2\pi \cdot \text{R};\tag{14}$$

$$b\_2 = \frac{\Delta\hat{\rho}\_{2,T}}{360 \times 60} \cdot 2\pi \cdot \text{R};\tag{15}$$

These result in the following expressions for the azimuth angles:

$$\tan\Psi\_{T\_1} = \frac{b\_1 + b\_2}{a\_1} = \frac{\Delta\varphi\_{1,2} + \Delta\varphi\_{2,T}}{\Delta\lambda\_{1,T} \cdot \cos\left(\varphi\_2 + \Delta\varphi\_{2,T}\right)};\tag{16}$$

$$\tan\Psi\_{T\_2} = \frac{b\_2}{a\_1 + a\_2} = \frac{\Delta\varphi\_{2,T}}{\left(\Delta\lambda\_{1,2} + \Delta\lambda\_{1,T}\right) \cdot \cos\left(\varphi\_2 + \Delta\varphi\_{2,T}\right)}.\tag{17}$$

From the first expression, we obtain:

$$
\Delta\beta\_{1,T} = \frac{\Delta\varphi\_{1,2} + \Delta\varphi\_{2,T}}{\tan\Psi\_{T\_1} \cdot \cos\left(\varphi\_2 + \Delta\varphi\_{2,T}\right)};\tag{18}
$$

GPS Positioning of Some Objectives Which are Situated

*T <sup>T</sup>* ,

at Great Distances from the Roads by Means of a "Mobile Slide Monitor – MSM" 177

*a*1

*T T*

 ,,1 

*<sup>T</sup>*

**Figure 21.** In plane projection of the target sighting field

min.

*<sup>T</sup> a R*

2 cos ; 360 60

2*a*

,2,1

*T*

*<sup>T</sup>*<sup>2</sup>

 <sup>222</sup> *L* ,

min. 2 ; 360 60 *b R*

the positions, 1 *L* and 2 *L* , of the two video cameras by means of the relations:

2

 <sup>111</sup>*L* ,

2

On this basis, it is, also, possible to calculate the direct distances between the target **T** and

1 1 12 *LT a b b* <sup>2</sup> <sup>2</sup>

Moreover, in conformity with the schema presented in figure 22, the height *Th* of the target T in the horizontal plane O.P. of the reference ellipsoid can also be calculated with one of the

1,2

2,

360 60 *<sup>T</sup> b R*

1 1 <sup>1</sup> sin or : *T L <sup>T</sup> h h LT* 2 2 <sup>2</sup> sin *T L <sup>T</sup> h h LT* (25)

min. 2 cos ; 360 60 *<sup>T</sup> a R*

*T*1

min.

2 .

2 12 2 and *LT a a b* (24)

(23)

1

*T*

1

<sup>2</sup> <sup>2</sup>

1

2*b*

 ,2 *T* 

1*b*

 2,1 

1,2

relations:

and from the second expression it results that:

$$
\Delta \lambda\_{1,T} = \frac{\Delta \rho\_{2,T} - \Delta \lambda\_{1,2} \cdot \cos \left(\rho\_2 + \Delta \rho\_{2,T}\right) \cdot \tan \Psi\_{T\_2}}{\cos \left(\rho\_2 + \Delta \rho\_{2,T}\right) \cdot \tan \Psi\_{T\_2}}; \tag{19}
$$

So by elimianatig the 1,*<sup>T</sup>* parameter, we obtain:

$$\begin{aligned} \frac{\Delta\varphi\_{1,2} + \Delta\varphi\_{2,T}}{\tan\Psi\_{T\_1}} &= \frac{\Delta\varphi\_{2,T} - \Delta\beta\_{1,2} \cdot \cos\left(\rho\_2 + \Delta\rho\_{2,T}\right) \cdot \tan\Psi\_{T\_2}}{\tan\Psi\_{T\_2}} \text{and} : \\\\ \left(\Delta\rho\_{1,2} + \Delta\rho\_{2,T}\right) \cdot \tan\Psi\_{T\_2} &= \Delta\rho\_{2,T} \cdot \tan\Psi\_{T\_1} - \Delta\beta\_{1,2} \cdot \cos\left(\rho\_2 + \Delta\rho\_{2,T}\right) \cdot \tan\Psi\_{T\_1} \cdot \tan\Psi\_{T\_2}; \end{aligned}$$

On this basis, the following implicit computing relation of the latitude angular difference 2,*<sup>T</sup>* is obtained:

$$\Delta\boldsymbol{\rho}\_{2,T} = \frac{\tan\Psi\_{T\_2}}{\tan\Psi\_{T\_1} - \tan\Psi\_{T\_2}} \cdot \left[\Delta\boldsymbol{\rho}\_{1,2} + \Delta\boldsymbol{\lambda}\_{1,2} \cdot \cos\left(\boldsymbol{\rho}\_2 + \Delta\boldsymbol{\rho}\_{2,T}\right) \cdot \tan\Psi\_{T\_1}\right] \tag{20}$$

and in continuation:

$$
\rho \mathfrak{o}\_{\Gamma} = \mathfrak{o} \mathfrak{o}\_{2} + \Delta \mathfrak{o}\_{2,\Gamma} \tag{21}
$$

With the determinated in this mode value of the angular difference 2,*<sup>T</sup>* , can be calculated in this phase and the value 1,*<sup>T</sup>* of the longitude angular difference by means of one of the two explicit relations (7.1) or (7.2).

In similar mode:

$$
\mathcal{Z}\_T = \mathcal{Z}\_1 + \Delta \mathcal{Z}\_{1,T} \tag{22}
$$

After obtaining, in the presented mode, of the target geographic coordinates, *T* and *<sup>T</sup>* , in continuation it is possible to calculate and the linear distances: *a*1 , *a*2 and *b*1 , *b*2 , on the longitude and respectively latitude directions, between the video camera successive positions and, respectively, between these positions and the sighted target, with the relations:

GPS Positioning of Some Objectives Which are Situated at Great Distances from the Roads by Means of a "Mobile Slide Monitor – MSM" 177

**Figure 21.** In plane projection of the target sighting field

176 Cartography – A Tool for Spatial Analysis

So by elimianatig the 1,*<sup>T</sup>*

 

in this phase and the value 1,*<sup>T</sup>*

two explicit relations (7.1) or (7.2).

is obtained:

and in continuation:

In similar mode:

relations:

2,*<sup>T</sup>* 

From the first expression, we obtain:

and from the second expression it results that:

*T*

*b a a*

1,

1,

 

 

*T*

*T*

 <sup>2</sup> 2 2,

 

tan cos

2, 1,2 2 2,

 

parameter, we obtain:

1 2

*T T*

tan tan

<sup>2</sup>

1 2

*T T*

*T*

tan

tan tan

 

1,2 2, 2, 1,2 2 2, cos tan

 

2, 1,2 1,2 2 2,

 

 

After obtaining, in the presented mode, of the target geographic coordinates, *T*

With the determinated in this mode value of the angular difference 2,*<sup>T</sup>*

 

2 2,

cos tan

 

*T T T T*

 

 

2 1 <sup>1</sup> <sup>2</sup> 1,2 2, 2, 1,2 2 2, tan tan cos tan tan ; *T T TT T TT*

On this basis, the following implicit computing relation of the latitude angular difference

*T T T*

*T T* 2 2,

*T T* 1 1,

continuation it is possible to calculate and the linear distances: *a*1 , *a*2 and *b*1 , *b*2 , on the longitude and respectively latitude directions, between the video camera successive positions and, respectively, between these positions and the sighted target, with the

 

 

 

1 2 1,2 1, 2 2, tan . cos

*T*

 <sup>1</sup> 1,2 2,

*T T*

*T T T*

 

2 2,

<sup>2</sup>

*T T*

cos tan

 

*T*

*T T*

(17)

;

2

<sup>2</sup>

 

(20)

cos tan

of the longitude angular difference by means of one of the

(18)

;

(19)

and :

1

, can be calculated

(21)

(22)

 and *<sup>T</sup>* , in

 

$$a\_1 = \frac{\Delta\lambda\_{17}\left[\text{min.}\right]}{360 \times 60} \cdot 2\pi \cdot R \cdot \cos\rho\_{\text{I'}} \cdot a\_2 = \frac{\Delta\lambda\_{1,2}\left[\text{min.}\right]}{360 \times 60} \cdot 2\pi \cdot R \cdot \cos\rho\_{\text{I'}};$$

$$b\_1 = \frac{\Delta\rho\_{1,2}\left[\text{min.}\right]}{360 \times 60} \cdot 2\pi \cdot R \cdot b\_2 = \frac{\Delta\rho\_{2,2}\left[\text{min.}\right]}{360 \times 60} \cdot 2\pi \cdot R. \tag{23}$$

On this basis, it is, also, possible to calculate the direct distances between the target **T** and the positions, 1 *L* and 2 *L* , of the two video cameras by means of the relations:

$$L\_1 T = \sqrt{a\_1^2 + \left(b\_1 + b\_2\right)^2} \quad \text{and} \ L\_2 T = \sqrt{\left(a\_1 + a\_2\right)^2 + b\_2^2} \tag{24}$$

Moreover, in conformity with the schema presented in figure 22, the height *Th* of the target T in the horizontal plane O.P. of the reference ellipsoid can also be calculated with one of the relations:

$$h\_T = h\_{L\_1} + L\_1 T \cdot \sin \Theta\_{T\_1} \\ \text{or} \\ \colon h\_T = h\_{L\_2} + L\_2 T \cdot \sin \Theta\_{T\_2} \tag{25}$$

GPS Positioning of Some Objectives Which are Situated

at Great Distances from the Roads by Means of a "Mobile Slide Monitor – MSM" 179

the dimensional width of the respective vehicle, which is around of 1.2 meters. Due to this fact, the two viewing lines are not intersected on the sighted target, but only with great errors and therefore the bi-cameral stereo photogrammetric systems cannot precisely identify the positions of the objectives that are at distances of more than approximatively 30

So by using a single video camera, the measurement basis of the applied stereo metric method is constituted from the distance interval of 20 – 30 meters, between two triggerings of the camera during the lab vehicle displacement, distance from which the camera sights

On this basis, at the returning at the computing center, it follows that from the obtained images to be selected by two images in which the interesting objective is evidenced in a corresponding mode, which will permit the selection of this objective in an electronic modality and the determination of the pixel coordinates, which achieves the objective displaying on the monitor screen. These coordinates together with the other data which accompany the two selected images, will permit to compute, using triangulation proceedings as well as methods for reporting to the spherical system of the terrestrial

The immediate result of this equipment functioning is represented by the obtaining of a series of digital documents structured in GIS format, documents which contain the data

Taking account of its conception, the "Slide Monitor" equipment can be installed not only on a terrestrial vehicle, but also on any kind of boats for the surveillance from an aquatic

Brzezinska D., Ron Li, Haal N. & Toth C., (2004), GPSVanTM Project: "From Mobile Mapping to Telegeoinformatics: Paradigm Shift in Geospatial Data Acquisition, Processing and

Photogrammetric Engineering and Remote Sensing, vol. 70, No.2, February 2004, pp. 197-

Caporali A. (2008) . System and Method For Monitoring and Surveying Movements of the Terrain, Large Infrastructures and Civil Building Works In General, Based Upon the Signals Transmitted by the GPS Navigation Satellite System. Patent application number:

coordinates, the geographic and elevation coordinates of the sighted objective.

registered in field, with the possibility to update them anytime.

*National Institute of R&D for Optoelectronics – INOE 2000, Romania* 

American Society for Photogrammetry and Remote Sensing

Axente Stoica, Dan Savastru and Marina Tautan

medium of some isolated objectives disposed on an inaccessible border.

meters from the equipment.

the same objective.

**Author details** 

**8. References** 

210.

Management,"

**Figure 22.** The diagram for the target height *Th* computing.
