Space-oblique Mercator projection is a

map projection In cartography, a map projection is any of a broad set of Transformation (function) , transformations employed to represent the curved two-dimensional Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In a map projection, ...

devised in the 1970s for preparing maps from Earth-survey

satellite A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scient ...

data. It is a generalization of the oblique Mercator projection that incorporates the time evolution of a given satellite ground track to optimize its representation on the map. The oblique Mercator projection, on the other hand, optimizes for a given

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

History

The space-oblique Mercator projection (SOM) was developed by John P. Snyder, Alden Partridge Colvocoresses and John L. Junkins in 1976. Snyder had an interest in maps dating back to his childhood; he regularly attended

cartography Cartography (; from , 'papyrus, sheet of paper, map'; and , 'write') is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an imagined reality) can ...

conferences whilst on vacation. In 1972, the

United States Geological Survey The United States Geological Survey (USGS), founded as the Geological Survey, is an agency of the U.S. Department of the Interior whose work spans the disciplines of biology, geography, geology, and hydrology. The agency was founded on Mar ...

(USGS) needed to develop a system for reducing the amount of distortion caused when

pictures of the ellipsoidal Earth were printed on a flat page. Colvocoresses, the head of the USGS's national mapping program, asked attendees of a geodetic sciences conferences for help solving the projection problem in 1976. Snyder work on the problem with his newly purchased pocket calculator and devised the mathematical formulas needed to solve the problem. After submitting his calculations to Waldo Tobler for review, Snyder submitted these to the USGS at no charge. Impressed with his work, USGS officials offered Snyder a job, and he promptly accepted. His formulas were then used to produce maps from

Landsat 4 Landsat 4 is the fourth satellite of the Landsat program. It was launched on July 16, 1982, with the primary goal of providing a global archive of satellite imagery. Although the Landsat Program is managed by NASA, data from Landsat 4 was coll ...

, which launched in the summer of 1978 .

Projection description

The space-oblique Mercator projection provides continual, nearly

conformal map In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...

ping of the swath sensed by a satellite. Scale is true along the ground track, varying 0.01 percent within the normal sensing range of the satellite. Conformality is correct within a few parts per million for the sensing range. Distortion is essentially constant along lines of constant distance parallel to the ground track. The space-oblique Mercator is the only projection which takes the rotation of Earth into account.

Equations

The forward equations for the Space-oblique Mercator projection for the sphere are as follows: :

\begin
\frac &= \int_^ \fracd\lambda' - \frac\ln\tan\left(\frac+\frac\right) \\
\frac &= \left(H+1\right) \int_^ \fracd\lambda' + \frac\ln\tan\left(\frac+\frac\right) \\
S &= \tfrac \sin i \cos \lambda' \\
H &= 1 - \tfrac \cos i \\
\tan\lambda' &= \cos i \tan \lambda_ + \frac \\
\sin\varphi' &= \cos i \sin \varphi - \sin i \cos \varphi \sin \lambda_ \\
\lambda_ &= \lambda + \tfrac \lambda'. \\
\varphi &= \text \\
\lambda &= \text \\
P_ &= \text \\
P_ &= \text \\
i &= \text \\
R &= \text \\
x,y &= \text
\end

References

*John Hessler, ''Projecting Time: John Parr Snyder and the Development of the Space Oblique Mercator Projection'', Library of Congress, 2003
Snyder's 1981 Paper Detailing the Projection's Derivation
Map projections {{Cartography-stub