## 5 Truncation Error and Accuracy Analysis  
The common latitude formulas with geodetic latitude and geocentric latitude as variables are expanded into a power series in terms of \(e\) up to \(e^{10}\) and in terms of \(n\) up to \(n^{5}\), which are symbolic expressions appropriate for ellipsoids with different parameters. To verify the accuracy and reliability of these formulas, the calculation error of each expansion formula is analyzed by using CGCS2000 (China Geodetic Coordinate System 2000) and the reference ellipsoid constants \(a=6378137\)\(m\) and \(1/f=298.25722101\)\(m\). Taking authalic latitude as an example, we chose the determined geodetic latitude \(B_{0}\) and inserted it into Equation (1) to obtain the theoretical value for authalic latitude \(\vartheta_{0}(B)\). Inserting the determined geodetic latitude \(B_{0}\) into Equation (5) gives the authalic latitude \(\vartheta_{1}(B)\) in terms of the power series expansion of eccentricity \(e\). The difference between \(\vartheta_{1}(B)\) and \(\vartheta_{0}(B)\) gives the error of Equation (5). Similarly, when the geodetic latitude \begin{table}
\begin{tabular}{l c} \hline \hline \multicolumn{1}{c}{**Expand to \(e^{10}\) Based on the First**} & **Expand to \(n^{5}\) Based on the Third** \\ \multicolumn{1}{c}{**Eccentricity \(e\)**} & **Flattening \(n\)** \\ \hline \(\left\{\begin{array}{l}d_{2}=\frac{1}{3}e^{2}+\frac{1}{3}e^{4}+\frac{5}{64}e^ {6}+\frac{7}{128}e^{8}+\frac{21}{512}e^{10}\\ d_{4}=\frac{1}{3}e^{4}+\frac{1}{31}e^{6}+\frac{7}{32}e^{8}+\frac{3}{1028}e^{1 0}\\ d_{6}=\frac{1}{12}e^{6}+\frac{1}{128}e^{8}+\frac{1}{1028}e^{10}\\ d_{8}=\frac{1}{120}e^{10}\\ \hline\end{array}\right.\) & \(\left\{\begin{array}{l}d_{2}=n\\ d_{4}=\frac{1}{2}n^{2}\\ d_{6}=\frac{1}{4}n^{3}\\ d_{8}=\frac{1}{4}n^{4}\\ d_{10}=\frac{1}{5}n^{5}\\ \hline\end{array}\right.\) \\ \hline \end{tabular}
\end{table} 
Table 10: Coefficients in Equation (30). \begin{table}
\begin{tabular}{l c} \hline \hline \multicolumn{1}{c}{**Expand to \(e^{10}\) Based on the First**} & **Expand to \(n^{5}\) Based on the Third** \\ \multicolumn{1}{c}{**Eccentricity \(e\)**} & **Flattening \(n\)** \\ \hline \(\left\{\begin{array}{l}d_{2}=\frac{1}{3}e^{2}+\frac{1}{3}e^{4}+\frac{1}{64}e^ {6}+\frac{7}{128}e^{8}+\frac{21}{512}e^{10}\\ d_{4}=\frac{1}{3}e^{4}+\frac{1}{31}e^{6}+\frac{7}{32}e^{8}+\frac{3}{1028}e^{1 0}\\ d_{6}=\frac{1}{12}e^{6}+\frac{1}{128}e^{8}+\frac{1}{1028}e^{10}\\ d_{8}=\frac{1}{120}e^{8}+\frac{1}{5120}e^{10}\\ d_{10}=\frac{1}{5120}e^{10}\\ \hline\end{array}\right.\) & \(\left\{\begin{array}{l}d_{2}=n\\ d_{4}=\frac{1}{2}n^{2}\\ d_{6}=\frac{1}{4}n^{3}\\ d_{8}=\frac{1}{4}n^{4}\\ d_{10}=\frac{1}{5}n^{5}\\ \hline\end{array}\right.\) \\ \hline \end{tabular}
\end{table} 
Table 9: Coefficients in Equation (29). Similarly, when geocentric latitude \(\phi\) is the variable, we obtain the calculation errors of the derived expressions of geodetic latitude, reduced latitude, rectifying latitude, conformal latitude, and authalic latitude, and the calculation error \(\Delta\) varies with geodetic geocentric latitude \(\phi\) as shown in Figure 2. When expanded to \(e^{10}\), the calculation errors of the expressions are shown as Figure 2a-e, and when expanding to \(n^{5}\), the calculation errors of the expressions are shown as Figure 2f-j, respectively. Figures 1 and 2 show that when the power series of the common latitude formulas with geodetic latitude and geocentric latitude as variables are expanded in terms of \(e\) up to \(e^{10}\) or in terms of \(n\) up to \(n^{5}\), the calculation errors oscillate as a function of geodetic latitude or geocentric latitude. The maximum calculation error of common latitude is less than \(1.0\times 10^{-8}\) arc seconds, which is greater than the accuracy required by geodesy. However, the corresponding power series formulas are complicated. The calculation accuracy of the expansion formula is related to the expansion order.