Meridian Arc - Meridian Distance On The Ellipsoid

Meridian Distance On The Ellipsoid

The determination of the meridian distance, that is the distance from the equator to a point at a latitude on the ellipsoid is an important problem in the theory of map projections, particularly the Transverse Mercator projection. Ellipsoids are normally specified in terms of the parameters defined above, but in theoretical work it is useful to define extra parameters, particularly the eccentricity, and the third flattening . Only two of these parameters are independent and there are many relations between them:

$\begin{align} f&=\frac{a-b}{a}, \qquad e^2=f(2-f), \qquad n=\frac{a-b}{a+b}=\frac{f}{2-f}\\ b&=a(1-f)=a(1-e^2)^{1/2},\qquad e^2=\frac{4n}{(1+n)^2}. \end{align}$

The radius of curvature is defined as

so that the arc length of an infinitesimal element of the meridian is (with in radians). Therefore the meridian distance from the equator to latitude is

$\begin{align} m(\varphi) &=\int_0^\varphi M(\varphi) \, d\varphi = a(1 - e^2)\int_0^\varphi \left (1 - e^2 \sin^2 \varphi \right )^{-3/2} \, d\varphi. \end{align}$

The distance from the equator to the pole, the polar distance, is

$m_p = m(\pi/2).\,$

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