# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Constant altitude curves on an ellipsoid**

**From:**Robin Stuart

**Date:**2015 Aug 11, 06:41 -0700

I have been working my way through the book “Geometry of Navigation” by Roy Williams (1998) and coming to terms with the intricacies of navigation on the surface of an ellipsoid. One question that I have had for some time is the effect of an ellipsoidal Earth on equal altitude circles of position. Williams shows that the cosine rule in geodetic coordinates (latitude defined by the direction of the observer’s vertical) on a ellipsoid takes the same form as the cosine rule on the sphere. Maybe there is an obvious reason for this but I don’t see it as yet. The implication of this surprising conclusion is that if I compute the latitude and longitude of points on an equal altitude circle of position on the surface of a sphere then these same latitude and latitude values fall on an equal altitude curve (not a circle) of position on the surface of an ellipsoid. This means that results such as double altitude formulas and the like derived on the sphere are exactly correct in geodetic coordinates on the surface of the ellipsoid and don't need any modification. To satisfy my incredulity I have worked some examples in 3D and it all seems to holds together.

Of course there are some things that do change. For example the intercept calculated as a difference in altitudes (or better zenithal distances) do not immediately translate into a distance along the surface of the Earth.

Williams also discusses the running fix on an ellipsoid pointing out the distortion that is produced by advancing the initial curve of position

http://fer3.com/arc/m2.aspx/Advancing-position-circle-Stuart-feb-2015-g30413

Robin Stuart