# A Simplified Pointing Model for Alt-Az Telescopes
## 2 Pointing Model Principle  
The PAST telescope employs an alt-az mount (as shown in Figure 1), primarily consisting of an azimuth axis and an elevation axis. Precise angle sensors are installed on each axis, allowing the telescope to rotate by specific angles based on input commands, thereby achieving the directional tracking of targets. Adjustments to the telescope's azimuth and elevation angles are made with reference to a defined horizontal plane, namely, the geodetic horizontal coordinate system. Prior to leaving the factory, the azimuth and elevation axes undergo an accurate calibration of angle precision and their relationship to the optical axis. This calibration effectively accounts for inherent system errors in the telescope's pointing model, which are generally of a minor magnitude.  
The pointing of the telescope is based on the local geodetic horizontal coordinate system. The key parameters used to determine the coordinate system are the local horizontal plane and the local north. There are a steel plates with a thickness of 30 mm embedded into the top surface of the concrete pier on which the telescope is mounted. Although the relationship between the zero azimuth of the telescope and the mounting hole is determined, the levelness and rotation of the steel plate cannot be guaranteed by the construction process. Therefore, the pointing errors of the telescope primarily stem from deviations in the installation plane (as illustrated in Figure 2), namely the horizontal level errors of the telescope's pedestal and rotational angle errors. The accurate measurement of both of these errors is generally challenging, especially the rotational angles on the horizontal plane. Determining the true east and north orientations of the pedestal installation plane also presents significant difficulty and high costs. In practical applications, a common approach involves measuring targets across different directions in the entire celestial sphere, constructing an error matrix, and fitting it with a mathematical model in order to derive a pointing error correction function. This process concurrently corrects the inherent system errors of the telescope, thereby significantly enhancing the system's pointing precision.  
Figure 1: Illustration of PAST.  
Figure 2: The coordinate system and error illustration for PAST. E: Represents the true east direction in the local horizontal plane of the telescope’s location. N: Corresponds to the true north direction in the local horizontal plane of the telescope’s location. Z: Denotes the zenith direction in the local horizontal plane of the telescope’s location. E1: Signifies the projection of E onto the telescope’s installation plane. N1: Represents the projection of N onto the actual installation plane of the telescope. Z1: Depicts the normal direction of the actual installation plane’s zenith. T: Denotes the position of the telescope’s azimuth angle 0 and elevation angle 0 in the actual installation plane. ENZ: Constitutes the genuine local geodetic horizontal coordinate system, formed by the combination of E, N, and Z. E1N1Z1: Forms the horizontal coordinate system of the actual installation plane, composed of E1, N1, and Z1. The azimuth (\(A\)) and elevation (\(E\)) angles of the telescope itself are defined within the E1N1Z1 coordinate system, while the azimuth (\(A_{\text{T}}\)) and elevation (\(E_{\text{T}}\)) angles of the observed targets are defined within the ENZ coordinate system. The discrepancy between these two sets of angles represents the pointing error of the telescope, and the distribution of this error is a function of the true azimuth and true elevation angles.