Earth section paths

Earth section paths are useful as approximate solutions for geodetic problems, the direct and inverse calculation of geographic distances.

The rigorous solution of geodetic problems involves skew curves known as geodesics.

and also find the departure and arrival azimuths (angle from true north) of that curve,

This problem is best solved using analytic geometry in earth-centered, earth-fixed (ECEF) Cartesian coordinates.

The above referenced paper provides a derivation for an arc length formula involving the central angle and powers of

In other words, the tangent takes the place of the chord since the destination is unknown.

The inverse of the central angle arc length series above may be found on page 8a of Rapp, Vol.

[3] An alternative to using the inverse series is using Newton's method of successive approximations to

The great ellipse is the curve formed by intersecting the ellipsoid with a plane through its center.

A normal section connecting the two points will be closer to the geodesic than the great ellipse, unless the path touches the equator.

To illustrate the dependence on section type for the direct problem, let the departure azimuth and trip distance be those of the geodesic above, and use the great ellipse to define the direct problem.

Of course using the departure azimuth and distance from the great ellipse indirect problem will properly locate the destination,

[7] The importance of normal sections in surveying as well as a discussion of the meaning of the term line in such a context is given in the paper by Deakin, Sheppard and Ross.

To illustrate the dependence on section type for the direct problem, let the departure azimuth and trip distance be those of the geodesic above, and use the surface normal at NY to define the direct problem.

Of course, using the departure azimuth and distance from the normal section indirect problem will properly locate the destination in Paris.

Using the surface normal at Reykjavik (while still using the departure azimuth and trip distance of the geodesic to Paris) will have you arriving about 347 nm from Paris, while the normal at Zürich brings you to within 5.5 nm.

Continuing the example from New York to Paris on WGS84 gives the following results for the mean normal section:

This path is only slightly closer to the geodesic that the mean normal section.

Finishing the example from New York to Paris on WGS84 gives the following results for the geodesic midpoint normal section:

In the third and fourth charts the terminal point was defined using the direct algorithm for the geodesic with the given distance and initial azimuth.

The alternative of finding the corresponding point on the section path and computing geodesic distances would produce slightly different results.

The first chart is typical of mid-latitude cases where the great ellipse is the outlier.

The normal section associated with the point farthest from the equator is a good choice for these cases.

The second example is longer and is typical of equator crossing cases, where the great ellipse beats the normal sections.

The worst case deviation for normal sections of 5000 nautical miles length is about 2.8 nm and occurs at initial geodesic azimuth of 132° from 18° north latitude (48° azimuth for south latitude).

Consequently the fourth chart shows only 7 distinct lines out of the 24 with 15 degree spacing.

As the departure point moves north the lines at azimuths 90 and 270 are no longer geodesics, and other coincident lines separate and fan out until 18° latitude where the maximum deviation is attained.

Find where a section from New York to Paris, intersects the Greenwich meridian.

The results are as follows: The maximum (or minimum) latitude is where the section ellipse intersections a parallel at a single point.

The simpler approach is to compute the end points of the minor and major axes of the section ellipse using