Exploring Meridian Arc Length Calculation on Sphere and Ellipsoid

What are the formulas and methods used to calculate the meridian arc length between points on a sphere and an ellipsoid?

Meridian on a Sphere:

For a sphere with a radius equal to the major semi-axis of the ellipsoid:

Formula: The meridian distance on the sphere is given by d = r * ∆φ, where ∆φ = φB - φA.

Using the given values, the meridian arc length between points A and B on a sphere is calculated to be 353,306.39 meters.

Meridian on an Ellipsoid (Helmert's Formulae):

The meridian arc length on an ellipsoid is calculated using Helmert's formulae, which involves the ellipsoid's parameters and latitudes of points A and B.

The calculated meridian arc length on the ellipsoid using the formula is 6,713,642.4 meters.

The difference in the meridian arc length for the sphere and ellipsoid from Helmert's formulae is approximately -6.4 kilometers.

Meridian on an Ellipsoid (Numerical Integration):

By applying numerical integration with different angular intervals, we compute the meridian arc length, radius of curvature, and segment lengths for each interval.

The differences in the meridian arc lengths between Helmert's formulae and numerical integration range from -35.85 meters to -707.30 meters for various angular intervals.