Integral Expressions of Earth Ellipsoid Radius Vector and Mean Radius of Curvature

  Objectives  The Earth's radial diameter and mean radius of curvature are the basic parameters commonly used in measurement and Earth science calculation.According to the requirements of Earth scien-ce and space science and some requirements, the Earth's radial diameter, mean radius of curvature, mean radius of sphere, equal distance radius of sphere, equal area radius of sphere and equal volume radius of sphere are commonly used. The concept of integral mean values of ellipsoid radius vector and mean radius of curvature is introduced.With the application and development of space technology and computer technology in geodesy and cartography, it is of more important practical value to study the relationship between the Earth's radial diameter and the common Earth radius.

  Methods  The symbolic expressions of them are deduced by computer algebraic system and are expressed as the power series of eccentricity.We use the method that comparing the radial vector integral mean and the radius integral mean with the four common sphere radius respectively.

  Results  The results shows that the difference between the radial integral mean and the four common sphere radii is smaller. Since the Earth is a rotating ellipsoid, the radial and the radius of curvature deviate. When the radial vector is the largest, the radius of curvature is the smallest. When the radial is the smallest, the radius of curvature is the largest.

  Conclusions  The radius of curvature considered by traditional thinking cannot accurately represent the radius of the earth average values, to a certain extent, which means that integral mean value of ellipsoid radius vector is more representative of the average of the Earth's radius. These research results can provide theoretical basis for Earth science, space science and navigation and positioning.