Computing the length of elliptic arcs with polar coordinates is a little more challenging. To determine a point on an ellipse we need to know the point's distance from the origin and the direction to it. Using differential geometry allows us to derive the formula for the differential arc length in terms of the geocentric polar angle or colatitude, θ.
The radial distance of the point can be solved for using the formula for an ellipse in Cartesian coordinates and once this is known we can find the rate of its change with respect to the polar angle.
We can integrate the formula for the differential arc length using the functions derived for polar coordinates and compare with the previous results found using the parametric polar angle. In this calculation a=1.
The results show that the semi-elliptic arc length is only dependent on the ratio of the semi-minor to the semi-major axes, ε, and not the way in which the arc length is calculated.
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