## 1 Introduction  
In remote sensing surveying, it is necessary to use map projection to display remote sensing images more intuitively and scientifically on a plane [1; 2]. Precise measurements are required to achieve accurate navigation and positioning, and tools such as maps and geographic coordinates are used to intuitively express the information obtained [3; 4; 5]. In surveying and mapping, we often calculate the six common latitudes and their transformations: geodetic latitude, geocentric latitude, reduced latitude, rectifying latitude, conformal latitude, and authialic latitude [6; 7]. Given the continued application and research of map projections, more stringent requirements are required for accurate and efficient transformations between different projections. The key is to complete mutual transformation of important variables, such as common latitude, more efficiently to ensure high accuracy [8; 9; 10; 11; 12]. Much in-depth research has been conducted on this, leading to theories of latitude change. For example, [PERSON] et al. [3; 13; 14] derived and calculated the relations between common latitudes and the positive and negative solutions of common latitude. Some of these calculations involve an elliptic integral of the second kind, which cannot be obtained analytically, and some are expressed in the form of practical expansions [15]. Because of historical factors and manual calculation, the formulas still have errors. With the development of computer algebra, [PERSON] et al. [16] derived power series expansions of the common latitude with geodetic latitude as a variable. They obtained more uniform, symbolic expressions in form, which are suitable for the Earth's ellipsoid. [PERSON] et al. [17; 18] derived the symbolic expressions of the positive and negative solutions of the common latitude with geocentric latitude as a variable and extended the power series of coefficients in the formula to more directly represent the relationship between common latitude and geocentric latitude. [PERSON] et al. [19; 20; 21] studied and derived the power series expansions of the common latitude with naturalized latitude as a variable, as well as power series expressions of auxiliary latitude functions. To solve the problems of elliptic integration of the second kind in latitude transformation and make the forms neat and unified, the coefficients in the commonly used latitude expansion formula are often expressed as the power series form of the first eccentricity. To ensure the accuracy of latitude transformation, it is often expanded to \(e^{10}\)[22; 23]. In this form, the coefficients of the expansion formula are complex, and the convergence speed is slow. In addition, the formula is long and cannot express the relations between latitudes succinctly, which is not good for propagation. Although some achievements have been made in previous studies on latitude transformation, it is still incomplete and has some shortcomings: 1. Most power series expansions are long and inefficient; 2. Part of the formulas involve iterative calculation and lack theoretical support; 3. Some of the formulas are numerical forms, which are only applicable to a specific earth ellipsoid and lack universality [24]; 4. Some formulas have approximate substitution, some errors, and occasionally, coefficient errors. In general, the traditional latitude transformation formulas cannot meet the needs of the rapid development of modern map projection theory, and there is still a large space for improvement. What is more, in recent years, the third flattening \(n\) has been applied to ellipsoidal geodesy, which has solved problems related to the calculation of meridian arc length, various auxiliary latitudes, and projection transformation [25; 26]. Given this, with the help of the symbolic computation program Mathematica, we can derive the common latitude expansions with geodetic latitude and geocentric latitude as variables. The coefficients in the formulas are expressed as a power series of ellipsoidal eccentricity \(e\) and the third flattening \(n\).