引用本文: 叶彤, 李厚朴, 钟业勋, 金立新. 常用纬度与归化纬度差异极值分析表达式[J]. 武汉大学学报 ( 信息科学版), 2022, 47(3): 473-480.
YE Tong, LI Houpu, ZHONG Yexun, JIN Lixin. Analytical Expressions of Differences Between Commonly Used Latitudes and Reduced Latitude[J]. Geomatics and Information Science of Wuhan University, 2022, 47(3): 473-480.
 Citation: YE Tong, LI Houpu, ZHONG Yexun, JIN Lixin. Analytical Expressions of Differences Between Commonly Used Latitudes and Reduced Latitude[J]. Geomatics and Information Science of Wuhan University, 2022, 47(3): 473-480.

## Analytical Expressions of Differences Between Commonly Used Latitudes and Reduced Latitude

• 摘要: 从常用纬度与归化纬度的定义出发，借助于计算机代数分析，对常用纬度与归化纬度差值进行了系统的分析，推导出了常用纬度与归化纬度的差异极值分析表达式，并将式中系数展开为椭球第一偏心率$e$和第三扁率n的幂级数形式。研究结果表明，常用纬度与归化纬度差值极值点均位于45$°$附近，基于n展开的分析表达式比基于$e$展开的分析表达式形式上更加简洁，该分析表达式适用于任何椭球参数，具有通用性，一定程度上丰富了椭球几何学常用纬度分析理论。

Abstract:
Objectives  The reduced latitude can simplify many complex problems in ellipsoid geodesy. In addition to the reduced latitude, geodetic, geocentric, rectifying, authalic, and conformal latitudes are commonly used in map projection transformation. The differences between commonly used latitudes and reduced latitude are generally expressed in numerical form, and there are no corresponding analytical expressions. Analytical expressions of differences between commonly used and reduced latitudes can visually display the differences between latitudes.
Methods  Starting from the definition of commonly used and reduced latitudes, with the aid of computer algebra analysis methods, the function relations between commonly used latitudes and reduced latitude are derived, then the differences between commonly used and reduced latitudes can be obtained by finding the extreme points. Analytical expressions of differences between them are derived and their coefficients are expressed in a power series of the eccentricity e, and the third flattening n.
Results  Theoretical and numerical analysis results show that the analytical expressions in terms of n are more concise than those in terms of e , the absolute value of the differences between commonly used latitudes and reduced latitude first increase and then decrease, and the extreme points are all near $π/4$, the reduced latitude is closer to commonly used latitudes in numerical values.
Conclusions  Expressions of differences between commonly used latitudes and reduced latitude are more intuitively and applicable to any referenced ellipsoid, to some extent, they could enrich the analytical theory of the various commonly used latitudes.

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