## 2 Altimeter (Analog Instrument and Computer Simulation)  
The altimeter shows the aircraft's altitude above sea level. The analog instrument measures the difference between the atmospheric pressure and the pressure in a stack of aneroid capsules placed inside the mechanism. It can be adjusted to local barometric pressure. The instrument takes the information directly from the pitot tube and works as follows. If the aircraft ascends, the capsule expands and the static pressure drops because of the wind force applied; then, the needles on the instrument's dial indicate a higher altitude. The opposite effect occurs if the aircraft descends. Nowadays, there are updated versions of the altimeter, such as the radar altimeter, which measures the altitude by directly calculating the time it takes for a radio signal reflected off the Earth's surface to return to the aircraft [4; 5; 6; 35; 36; 37; 38; 39; 40; 41; 42]. In this work, the altimeter's dial is represented by a square region with a central circular dial; the 360\({}^{\circ}\) circumference is divided into 10 main divisions, numbered 0 to 9. Additionally, commands for setting 10 secondary divisions are included, and these are placed between each main division to improve the visual precision for the pilot, as shown in Figure 1a,b. The secondary divisions are shorter than the main divisions so they can be easily identified. These figures represent an example of a partial segmentation performed in a rounded dial. Figure 1a indicates the angular division of the circumference; the shaded areas indicate where the indicators will move. Figure 1b shows the dial of the instrument simulated with the indicators as displayed on the computer screen [4; 5; 6; 27; 28; 29; 30; 36; 37; 38; 39; 40; 41]. The altimeter values are taken from the variable dzt + \(\Delta\)t, which corresponds to the aircraft's displacement along the vertical axis (z). Equations (1)-(3) are solved to calculate the displacement for the z-axis as a function of the aircraft's vertical speed; here, the super-indexes (n and n - 1) refer to the actual and previous maneuvers. Then, the numerical values are transformed into angular values using Equation (4). Moreover, the values can be adjusted with the information placed on the right screen if the pilot decides to show them on a small numerical display. In addition, \(\Delta\)t is the step time, calculated as a function of the maneuver time (t\({}_{\text{n}}\)) and the number of steps used in the simulation (ns). Equations (1) and (2) are physical expressions, where (v\({}_{\text{s}}\)) is the real vertical aircraft speed; conversely, Equation (3)can be programmed and nested in a computational loop. Then, Equation (4) transforms the displacement into a representative angular value in order to display it on the instrument. The altimeter displays two indicators (needles) on the dial to indicate the altitude value. One is long, measuring the altitude in meters [1; 2; 3; 7; 8; 17; 18; 20; 21; 24]. Each value in the main division is equal to 100 m. Consequently, one revolution equals 1000 m. The second indicator is shorter and moves in a perimetral path. This needle indicates the altitude using a 1000 m scale, which is why one revolution of the large indicator will move the short needle by only one number on the main dial scale. Equation (5) calculates the angular position of the short needle. \[d_{z}^{t+\Delta t}=d_{z}^{m-1}+d_{z}^{\Delta t} \tag{1}\]  
\[d_{z}^{t+\Delta t}=d_{z}^{n-1}+(v_{z}^{n}-v_{z}^{n-1})\Delta t \tag{2}\]  
\[d_{z}^{t+\Delta t}=d_{z}^{n-1}+(v_{z}^{n}-v_{z}^{n-1})\bigg{(}\frac{t_{n}}{ns} \bigg{)} \tag{3}\]  
\[\alpha_{altimeter}=d_{z}^{t+\Delta t}\bigg{(}\frac{360^{\circ}}{1000m}\bigg{)} \tag{4}\]  
\[\alpha_{altimeter2}=\frac{\alpha_{altimeter}}{10} \tag{5}\]  
The following assumptions are accounted for in the altimeter's movement:  
1. The indicators in the altimeter can move clockwise or counterclockwise: 1. If the altimeter indicator moves clockwise, the aircraft is ascending. 2.