## 5 Truncation Error and Accuracy Analysis
We analyzed the maximum error of the common latitude expansions with geodetic latitude as variables based on the ellipsoidal eccentricity and the third flattening, respectively. The results are given in Table 11.  
In the same way, the maximum error of the common latitude expansions with geocentric latitude as variables based on the ellipsoidal eccentricity and the third flattening, respectively is given in Table 12. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\Delta\phi\)(B) & \(\Delta u\)(B) & \(\Delta\psi\)(B) & \(\Delta\vartheta\)(B) & \(\Delta\varphi\)(B) \\ \hline \(e^{10}\) & \(2.56114\times 10^{-9}\) & \(9.31323\times 10^{-10}\) & \(1.42609\times 10^{-9}\) & \(1.76893\times 10^{-8}\) & \(1.45519\times 10^{-9}\) \\ \(e^{8}\) & \(4.52972\times 10^{-8}\) & \(1.63272\times 10^{-7}\) & \(2.44094\times 10^{-7}\) & \(2.2928\times 10^{-7}\) & \(2.65311\times 10^{-7}\) \\ \(e^{6}\) & \(8.10159\times 10^{-5}\) & \(3.01542\times 10^{-5}\) & \(4.58977\times 10^{-5}\) & \(4.24962\times 10^{-5}\) & \(1.47941\times 10^{-6}\) \\ \(e^{4}\) & \(0.0151037\) & \(0.00590163\) & \(0.00913985\) & \(0.00833335\) & \(0.0107551\) \\ \(n^{5}\) & \(1.60071\times 10^{-10}\) & \(1.16415\times 10^{-10}\) & \(1.16415\times 10^{-10}\) & \(1.76369\times 10^{-8}\) & \(1.45519\times 10^{-10}\) \\ \(n^{4}\) & \(4.51109\times 10^{-8}\) & \(6.1118\times 10^{-10}\) & \(2.76486\times 10^{-9}\) & \(1.27475\times 10^{-8}\) & \(2.08966\times 10^{-8}\) \\ \(n^{3}\) & \(1.15433\times 10^{-5}\) & \(4.10277\times 10^{-7}\) & \(1.58537\times 10^{-6}\) & \(1.11121\times 10^{-6}\) & \(8.84886\times 10^{-6}\) \\ \(n^{2}\) & \(0.00455778\) & \(0.000325557\) & \(0.00126153\) & \(0.000803024\) & \(0.00312175\) \\ \hline \end{tabular}
\end{table} 
Table 11: Truncation error of common latitude expanded to different orders with geodetic latitude as the variable (units: arc seconds). Figure 1: The calculation error of the derived expressions (geodetic latitude as the variable). **(a)** Calculation error \(\Delta\phi(B)\) of geocentric latitude (expanded to \(e^{10}\)). **(b)** Calculation error \(\Delta u(B)\) of reduced latitude (expanded to \(e^{10}\)). **(c)** Calculation error \(\Delta\psi(B)\) of rectifying latitude (expanded to \(e^{10}\)). **(d)** Calculation error \(\Delta\vartheta(B)\) of authalic latitude (expanded to \(e^{10}\)). **(e)** Calculation error \(\Delta\varphi(B)\) of conformal latitude (expanded to \(e^{10}\)). **(f)** Calculation error \(\Delta\varphi(B)\) of geocentric latitude (expanded to \(n^{5}\)). **(g)** Calculation error \(\Delta u(B)\) of reduced latitude (expanded to \(n^{5}\)). **(h)** Calculation error \(\Delta\psi(B)\) of rectifying latitude (expanded to \(n^{5}\)). **(i)** Calculation error \(\Delta\vartheta(B)\) of rectifying latitude (expanded to \(n^{5}\)). **(j)** Calculation error \(\Delta\varphi(B)\) of conformal latitude (expanded to \(n^{5}\)). Figure 2: The calculation error of the derived expressions (geocentric latitude as the variable). (**a**) Calculation error \(\Delta\text{B}(\phi)\) of geodetic latitude (expanded to \(e^{10}\)). (**b**) Calculation error \(\Delta\text{u}(\phi)\) of reduced latitude (expanded to \(e^{10}\)). (**c**) Calculation error \(\Delta\text{u}(\phi)\) of rectifying latitude (expanded to \(e^{10}\)). (**d**) Calculation error \(\Delta\text{\theta}(\phi)\) of authalic latitude (expanded to \(e^{10}\)). (**e**) Calculation error \(\Delta\text{u}(\phi)\) of conformal latitude (expanded to \(e^{10}\)). (**f**) Calculation error \(\Delta\text{B}(\phi)\) of geodetic latitude (expanded to \(n^{5}\)). (**g**) Calculation error \(\Delta\text{u}(\phi)\) of reduced latitude (expanded to \(n^{5}\)). (**h**) Calculation error \(\Delta\text{u}(\phi)\) of rectifying latitude (expanded to \(n^{5}\)). (**i**) Calculation error \(\Delta\text{\theta}(\phi)\) of rectifying latitude (expanded to \(n^{5}\)). (**j**) Calculation error \(\Delta\text{\varphi}(\phi)\) of conformal latitude (expanded to \(n^{5}\)).