## 5 Truncation Error and Accuracy Analysis
The results in Tables 11 and 12 show that the precision of commonly used latitude power series expansions is related to the expansion order. The greater the expansion order, the greater the corresponding precision, and the longer the expression. The accuracy of the expansion based on the third flatness is greater than that of the accuracy based on the common latitude power series expansion to \(n^{5}\) and \(e^{10}\), \(n^{4}\) and \(e^{8}\), \(n^{3}\) and \(e^{6}\), or \(n^{2}\) and \(e^{4}\), indicating the superiority of the third flattening for the power series expansion of latitude transformation theory. An analysis and comparison allow us to conclude that when the expansion is based on the ellipsoidal eccentricity power series to \(e^{6}\) or on the third flattening to \(n^{3}\), the error of the expansions of the common latitude is less than \(10^{-4}\), which not only meets the requirements for geodesy but also greatly simplifies the expression. The practical expansions of the common latitude with geodetic latitude and geocentric latitude as variables are thus obtained. When the geodetic latitude \(B\) is taken as the variable, the practical, simplified expansion for rectifying latitude \(\psi\), authalic latitude \(\vartheta\), conformal latitude \(\varphi\), reduced latitude \(u\), and geocentric latitude \(\phi\) are  
\[\psi(B) =B+\left(-\frac{3}{8}e^{2}-\frac{3}{16}e^{4}-\frac{111}{1024}e^{6 }\right)\sin 2B+\left(\frac{15}{256}e^{4}+\frac{15}{256}e^{6}\right)\sin 4B- \frac{35}{3072}e^{6}\sin 6B \tag{31}\] \[=B+\left(-\frac{3}{2}n+\frac{9}{16}n^{3}\right)\sin 2B+\frac{15}{ 16}n^{2}\sin 4B-\frac{35}{48}n^{3}\sin 6B,\]  
\[ \begin{array}{rl}\vartheta(B)&=B+\left(-\frac{1}{3}e^{2}-\frac{31}{180}e^{4 }-\frac{59}{560}e^{6}\right)\sin 2B+\left(\frac{17}{360}e^{4}+\frac{61}{1260}e^{6 }\right)\sin 4B-\frac{383}{45360}e^{6}\sin 6B\\ &=B+\left(-\frac{4}{3}n-\frac{4}{45}n^{2}+\frac{88}{315}n^{3}\right)\sin 2B +\left(\frac{34}{85}n^{2}+\frac{8}{105}n^{3}\right)\sin 4B-\frac{1532}{2335}n^{3} \sin 6B,\end{array}  \tag{32}\]  
\[ \begin{array}{rl}\varphi(B)&=B+\left(-\frac{1}{2}e^{2}-\frac{5}{24}e^{4}- \frac{3}{32}e^{6}\right)\sin 2B+\left(\frac{5}{48}e^{4}+\frac{7}{80}e^{6} \right)\sin 4B-\frac{13}{480}e^{6}\sin 6B\\ &=B+\left(-2n+\frac{2}{3}n^{2}+\frac{3}{4}n^{3}\right)\sin 2B+\left(\frac{5}{ 3}n^{2}-\frac{16}{15}n^{3}\right)\sin 4B-\frac{26}{15}n^{3}\sin 6B,\end{array}  \tag{33}\]  
\[ \begin{array}{rl}u(B)&=B+\left(-\frac{1}{4}e^{2}-\frac{1}{8}e^{4}-\frac{5}{ 64}e^{6}\right)\sin 2B+\left(\frac{1}{32}e^{4}+\frac{1}{32}e^{6}\right) \sin 4B-\frac{1}{192}e^{6}\sin 6B\\ &=B-n\sin 2B+\frac{1}{2}n^{2}\sin 4B-\frac{1}{3}n^{3}\sin 6B,\end{array}  \tag{34}\]  
\[ \begin{array}{rl}\phi(B)&=B+\left(-\frac{1}{2}e^{2}-\frac{1}{4}e^{4}-\frac{ 1}{8}e^{6}\right)\sin 2B+\left(\frac{1}{8}e^{4}+\frac{1}{8}e^{6}\right)\sin 4B- \frac{1}{24}e^{6}\sin 6B\\ &=B+\left(-2n+2n^{3}\right)\sin 2B+2n^{2}\sin 4B-\frac{5}{3}n^{3}\sin 6B. \end{array} \]  
When the geocentric latitude \(\phi\) is taken as the variable, the practical, simplified expansion for rectifying latitude \(\psi\), authalic latitude \(\vartheta\), conformal latitude \(\varphi\), reduced latitude \(u\), and geodetic latitude \(B\) are  
\[ \begin{array}{rl}\psi(\phi)&=\phi+\left(\frac{1}{8}e^{2}+\frac{1}{16}e^{4}+ \frac{53}{1024}e^{6}\right)\sin 2\phi+\left(-\frac{1}{256}e^{4}-\frac{1}{256}e^{6} \right)\sin 4\phi-\frac{5}{1024}e^{6}\sin 6\phi\\ &=\phi+\left(\frac{1}{2}n+\frac{13}{16}n^{3}\right)\sin 2\phi-\frac{1}{16}n^{2} \sin 4\phi-\frac{5}{16}n^{3}\sin 6\phi,\end{array}  \tag{35}\]