## 3 Attitude Indicator (Analogue Instrument and Computer Simulation)
\[\beta^{t+\Delta t}=\beta^{n-1}+\Big{(}\beta^{n}-\beta^{n-1}\Big{)}\bigg{(} \frac{t_{n}}{np}\bigg{)} \tag{4}\]  
\[\chi^{t+\Delta t}=\chi^{n-1}+\Big{(}\chi^{n}-\chi^{n-1}\Big{)}\bigg{(}\frac{t_{ n}}{np}\bigg{)} \tag{5}\]  
The rounded dial is divided into segments to represent rolling; here, an equator and parallel divisions are used to represent the pitching movement, as is shown in Figure 3. Figure 4 illustrates the representation of the attitude indicator, in concordance with some rolling and pitching movements. Then, the rolling movement is represented in the attitude indicator with a movement around the circumference and according to pilot position. These movements have with the following meanings:  
1. If the right wing moves up, the rolling is positive, and the real position of the terrain will go down on the same side of the instrument; so, the instrument movement is counterclockwise. 2. If the left wing moves up, the rolling is negative, and the real position of the terrain will go down in the same side of the instrument; so, the instrument movement is clockwise. Although the aircraft of commercial airlines do not roll as strongly as acrobatic aircraft, the aircraft can roll over 360\({}^{\circ}\) without restrictions, as is illustrated in Figure 4a, in a clockwise direction or against it; thus, the algorithm must be capable of representing all these conditions [1; 24; 25; 26; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47]. Moreover, the attitude indicator can move as a function of the pitching angle, as is shown in Figure 4b. This is in accordance with the following assumptions:  
1. If the aircraft is climbing up, the real terrain position will go down, and so it is considered that the aircraft is ascending with a positive angle value. 2. If the aircraft is descending, the angle is negative, and the real terrain position will go up. Then, these two behaviors must be represented with the algorithm. Furthermore, if there are changes in rolling and pitching angles during the same maneuver, the position of sky and terrain will be moved from the instrument's central position simultaneously. Then, the graphical positions used to represent the instantaneous pitching (\(\beta\)) and rolling (\(\chi\)) movements in the attitude indicator are obtained as a function of a line moving in a circle. The rolling angles (\(\chi\)) are represented in Figure 5a, considering an original position (x\({}_{0}\) = 0, y\({}_{0}\) = 0) in the middle point, and so the border line between the ground and sky and the inclination is calculated using Equations (6) to (9); here, (\(r_{AI}\)) is the radius of the instrument on the computer screen [8; 9; 10; 11; 24; 25; 34; 35; 36; 37; 38; 39]. \[x_{1}{}^{t+\Delta t} =x_{0}{}^{t+\Delta t}+r_{AI}\cos\left(\chi^{t+\Delta t}\right) \tag{6}\] \[x_{2}{}^{t+\Delta t} =x_{0}{}^{t+\Delta t}-r_{AI}\cos\left(\chi^{t+\Delta t}\right)\] (7) \[y_{1}{}^{t+\Delta t} =y_{0}{}^{t+\Delta t}+r_{AI}\sin\left(\chi^{t+\Delta t}\right)\] (8) \[y_{2}{}^{t+\Delta t} =x_{0}{}^{t+\Delta t}-r_{AI}\sin\left(\chi^{t+\Delta t}\right) \tag{9}\]  
The variation in the attitude indicator when pitching (\(\beta\)) is positive is illustrated in Figure 5b; here, a new value for the original radius is required (r\({}_{\beta}\)) due to the border line being shorter in comparison with that of a horizontal flight, and so Equation (10) is used. Then, Figure 5c is illustrates the combination of the pitching (\(\beta\)) and rolling (\(\chi\)) movements. Here, the calculation of the points (x\({}_{1}\),y\({}_{1}\)) and (x\({}_{2}\),y\({}_{2}\)) results from the sum of the influence of both movements. Finally, Figure 6 shows the computational representation of the attitude indicator BEING developed. This is one of the most complicated instruments that will be displayed due to the combination of two different aircraft conditions. It must be properly represented.