United States Department of Transportation
i
Geometric analysis of an observer on a spherical earth and an aircraft or satellite
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2013-09-30
[PDF-2.08 MB]
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Alternative Title:Project Memorandum - September 2013
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Abstract:This memorandum contains a large amount of technical detail. However, in significant contrast,
it addresses an easily-understood and fundamental need in surveillance and navigation systems
analysis — quantifying the geometry of two locations relative to each other and to a spherical
earth. Here, geometry simply means distances and angles. Sometimes, distances are the lengths
of straight lines; in other cases they are the lengths of arcs on the earth’s surface. Similarly, angles may be measured between lines on a plane or between lines on a spherical surface.
Because the earth has an established latitude/longitude coordinate system, the approach that first
comes to mind is to address this situation as a three-dimensional problem and use vector analysis.
However, the approach preferred here is that, to simplify and clarify the analysis process, the
three-dimensional problem should be re-cast as two separate two-dimensional problems:
Vertical Plane Formulation (Section 1.2 and Chapter 3)*— This analysis considers
the vertical plane containing the two locations of interest and the center of the earth.
The two locations are unconstrained vertically, although one altitude must be known.
Plane trigonometry is the natural analysis tool for this problem. Latitudes and
longitudes are not involved, which is its biggest limitation.
Spherical Surface Formulation (Section 1.3 and Chapter 4)— This analysis—
which is sometimes called great-circle navigation —only considers two locations on
the surface of a spherical earth. Spherical trigonometry is a natural analysis tool in
this setting, and latitudes and longitudes are inherent in this method. A significant
limitation of this analysis is that altitudes cannot be accounted for.
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