## 2 Calculation of the Length Correction from the Projection and Deformation  
There are several techniques for calculating the length correction. In the present case, we used the procedure in which, for a given distance in the S-JTSK coordinate system, the distance in the ETRS89 system on the GRS80 ellipsoid is calculated based on the method presented in [21] or [22]. First, we define two points A and B by their planar coordinates in the S-JTSK. The height of points A and B is assumed to be zero; the length will not be corrected from the convergence of the tangents due to the height (because we assume that the height correction for the lengths will be introduced in the case of construction work  
Figure 1: S-JTSK/05 transformation scheme [21]. in the field). The distance between points A and B in the S-JTSK will be \(d_{AB}\). The S-JTSK coordinates of points A and B will be converted to planar coordinates in the S-JTSK/05 system using the table of differences and, subsequently, by using the modified Krovak projection and transformations to obtain the geodetic coordinates B, L, H on the GRS80 ellipsoid within ETRF2000; see Figure 1. We performed the transformation using the program _ettr00-jtsk_v1801_[26], which uses the quasigeoid model _cr-2005_v1005.dat_ and the table of differences S-JTSK versus S-JTSK/05 _table_yx_3_v1710_TB-obd.dat_--both obtained from [26]. The distance between points A and B on the GRS80 ellipsoid is calculated by solving the second geodesic problem [27]. From the numerous solutions to this problem (e.g., ref. [28; 29; 30]), we used the sphere transition solution method, e.g., the method presented in [31], Section 3.3.4.-Solving the second geodesic problem. We calculated the length of the arc and the azimuths at the end points of the curve; however, the azimuths were not required. The radius of the auxiliary sphere is determined using the following formula:  
\[R=\sqrt{MN}, \tag{1}\]  
where \(M\) is the meridional radius of the curvature, and \(N\) is the transverse radius of the curvature at the center point between points A and B. The meridional radius of the curvature is calculated using the following formula:  
\[M=\frac{a\big{(}1-e^{2}\big{)}}{W^{3}}, \tag{2}\]  
where \(a\) is the major axis of the ellipsoid, \(e\) is the eccentricity of the ellipsoid, and \(W\) is the first geodesic function given for latitude \(\varphi\), determined by the following formula:  
\[W=\sqrt{1-e^{2}sin^{2}\varphi}. \tag{3}\]  
The eccentricity \(e\) is determined by the major \(a\) and minor \(b\) semi-axes of the respective ellipsoid:  
\[e^{2}=\frac{a^{2}-b^{2}}{a^{2}}. \tag{4}\]  
The transverse radius of curvature is calculated using the following formula:  
\[N=\frac{a}{W}. \tag{5}\]  
Both radii \(M\) and \(N\) depend on the parameters of the ellipsoid and the latitude used. For the GRS80 ellipsoid, the basic values of the constants are known, which are the lengths of the semi-axes of the ellipsoid. Therefore, \(a=6378137\ m\) and \(b=6356752.31425\ m\). For the Bessel ellipsoid, the lengths of the semi-axes are \(a=6377397.15508\ m\) and \(b=6356078.96290\ m\). The auxiliary sphere touches the ellipsoid at the center point \(P_{0}\); see Figure 2.  
The center point is chosen so that  
\[P_{0}(\varphi_{0})\ :\ \varphi_{0}=\frac{\varphi_{A}+\varphi_{B}}{2}. \tag{6}\]  
Next, we choose for (\(P_{0}\)):  
\[N(\varphi_{0})=N_{0}\,\ M(\varphi_{0})=M_{0} \tag{7}\]Figure 1.0  
Figure 1.1  
The last choice is as follows:  
Figure 1.2  
Adding and subtracting these two equations gives the following:  
Figure 1.3  
The problem is further solved as a problem on a sphere with the radius \(R\) and the coordinates of the points A and B lying on the sphere (\(\varphi_{A^{\prime}},\lambda_{A}\) a \(\varphi_{B^{\prime}},\lambda_{B}\)). The length of the arc with s\({}_{\rm AB}\), see Figure 3, is calculated using the cosine theorem on a sphere in the following form:  
Figure 1.4  
Figure 1.5 The second geodesic problem on a sphere [32].