Analytical Expressions of Differences Between Commonly Used Latitudes and Reduced Latitude

  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 $\mathrm{\pi }/\mathrm{4}$, 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.