## 2 Calculation Methods
### Cylindrical Mirror Error Calculation  
Solar trough collector systems employing cylindrical mirrors instead of convectional trough mirrors can reduce manufacturing costs. Additionally, the cylindrical mirrors can be fixed in place, with solar position tracking achieved by changing the position of the receiver. This simplifies the structure of the solar trough collector and further reduces costs. However, this approach has limitations, such as reduced intercept factor and optical efficiency. In particular, the fixed installation of cylindrical mirrors results in a small cosine factor and very low efficiency. As a result, it is rarely employed in practical situations. One factor contributing to the low efficiency is the inadequate selection of design parameters, which leads to a low light concentration ratio and hence, low efficiency. We, therefore, propose a new type of solar trough collector that uses multiple cylindrical mirrors spliced to concentrate sunlight, with each cylindrical mirror's center located on the original parabola and having the same normal vector. When employing cylindrical mirrors as replacements for parabolic mirrors, the critical issue to address is the spherical aberration, for example, of a circular mirror because of its constant curvature, that is, light rays hitting different parts of the mirror do not focus on the same point, rays converge on the focal plane instead of the focal line resulting in aberration. Aberration theory states that, in [PERSON]'s \"Research and Inspection of Optical Systems\" [30], in order to limit the aberration, the relative aperture of a spherical reflector as the primary mirror should not exceed 1:10.8, which means that the ratio of the reflector aperture to the focal length is \(\leqslant\)1/10.8, which is then converted to the ratio of the reflector aperture to the spherical radius \(\leqslant\)1/21.6. In the field of solar energy concentration, optical systems are mainly used for concentration, and the requirements are much lower than those for optical imaging. As seen in Figure 1, it is assumed that the central solar ray reflected by point \(P\) on the mirror reaches the focal point \(F\), corresponding to rim angle \(\varphi\). For another reflection point \(P^{\prime}\), if its rim angle is \(\varphi^{\prime}\) and the angle between the solar ray and the central solar ray is \(\theta\). the ray reflected by \(P^{\prime}\) will intersect the focal plane at \(G\). The ideal design occurs when the spherical aberration of the central solar ray reflected at the rim angle of the cylindrical surface is equal to the maximum spherical aberration at point \(P\), resulting in the lowest aberration. Suppose the angle of incidence is \(\lambda\) to any point \(P\). The coordinates are  
\[[{\it rsin}(\varphi/2),{\it r}(1-cos(\varphi/2)),0] \tag{17}\]  
The normal vector is  
\[n=[-sin(\varphi/2),cos(\varphi/2),0] \tag{18}\]  
The central ray of the sun is  
\[i=[0,cos\lambda,sin\lambda] \tag{19}\]  
The vector of incident rays in any direction is  
\[i=[sin\theta,cos\lambda cos\theta,cos\theta sin\lambda] \tag{20}\]  
The reflected ray vector is  
\[r=-[sin\theta,cos\lambda cos\theta,cos\theta sin\lambda]\] \[+2(-sin\theta sin(\varphi/2)+cos\lambda cos\theta cos(\varphi/2) )[-sin(\varphi/2),cos(\varphi/2),0] \tag{21}\] \[=[-cos\varphi sin\theta-sin\varphi cos\lambda cos\theta,-sin\varphi sin \theta+cos\varphi cos\lambda cos\theta,-cos\theta sin\lambda]\]  
The equation for reflected rays is  
Figure 1: Concentrating sunlight with the trough collector that uses multiple cylindrical spliced mirrors. \[\frac{x-rsin(\varphi/2)}{-cos\varphi sin\theta-sin\varphi cos\lambda cos\theta}=\frac{y- (1-cos(\varphi/2))r}{-sin\varphi sin\theta+cos\varphi cos\lambda cos\theta}=\frac{z} {-cos\theta sin\lambda} \tag{22}\]  
When the interception angle \(\theta\) = 0, and light rays reach the focal point \(f\), the focal length can be obtained:  
\[f=OF=r(1-\frac{1}{2 cos(\varphi_{0}/2)}) \tag{23}\]  
where \(\varphi_{0}\) is used to represent a special rim angle.