Space-oblique Mercator projection is a

map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longit ...

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

satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioiso ...

data. It is a generalization of the

oblique Mercator projection The oblique Mercator map projection is an adaptation of the standard Mercator projection. The oblique version is sometimes used in national mapping systems. When paired with a suitable geodetic datum, the oblique Mercator delivers high accuracy in ...

that incorporates the time evolution of a given satellite

ground track A ground track or ground trace is the path on the surface of a planet directly below an aircraft's or satellite's trajectory. In the case of satellites, it is also known as a suborbital track, and is the vertical projection of the satellite' ...

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 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 connection. ...

History

The space-oblique Mercator projection (SOM) was developed by

John P. Snyder John Parr Snyder (12 April 1926 – 28 April 1997) was an American cartographer most known for his work on map projections for the United States Geological Survey (USGS). Educated at Purdue and MIT as a chemical engineer, he had a lifetime interest ...

, Alden Partridge Colvocoresses and John L. Junkins in 1976. Snyder had an interest in maps, originating back to his childhood and he regularly attended

cartography Cartography (; from grc, χάρτης , "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 ...

conferences while on vacation. In 1972, the

United States Geological Survey The United States Geological Survey (USGS), formerly simply known as the Geological Survey, is a scientific agency of the United States government. The scientists of the USGS study the landscape of the United States, its natural resources, ...

(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 attended the conference and became motivated to work on the problem armed with his newly purchased pocket calculator and devised the mathematical formulas needed to solve the problem. After submitting his calculations to

Waldo Tobler Waldo Rudolph Tobler (November 16, 1930 – February 20, 2018) was an American-Swiss geographer and cartographer. Tobler's idea that "Everything is related to everything else, but near things are more related than distant things" is referred to ...

for review, Snyder submitted these to the USGS at no charge. Impressed with his work, USGS officials offered Snyder a job with the organization, which he accepted. His formulas were 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 ...

images 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 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-preserving) at a point u_0\in ...

ping of the swath sensed by a satellite.

Scale Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number ...

is true along the

, 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 presented that 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