Stereographic map projection

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Stereographic projection of the world north of 30degS. 15deg graticule. Stereographic projection SW.JPG
Stereographic projection of the world north of 30°S. 15° graticule.
The stereographic projection with Tissot's indicatrix of deformation. Stereographic with Tissot's Indicatrices of Distortion.svg
The stereographic projection with Tissot's indicatrix of deformation.

The stereographic projection, also known as the planisphere projection or the azimuthal conformal projection, is a conformal map projection whose use dates back to antiquity. Like the orthographic projection and gnomonic projection, the stereographic projection is an azimuthal projection, and when on a sphere, also a perspective projection.

Contents

On an ellipsoid, the perspective definition of the stereographic projection is not conformal, and adjustments must be made to preserve its azimuthal and conformal properties. The universal polar stereographic coordinate system uses one such ellipsoidal implementation.

History

World map made by Rumold Mercator in 1587, using two equatorial aspects of the stereographic projection. Mercator World Map.jpg
World map made by Rumold Mercator in 1587, using two equatorial aspects of the stereographic projection.

The stereographic projection was likely known in its polar aspect to the ancient Egyptians, though its invention is often credited to Hipparchus, who was the first Greek to use it.[ citation needed ] Its oblique aspect was used by Greek Mathematician Theon of Alexandria in the fourth century, and its equatorial aspect was used by Arab astronomer Al-Zarkali in the eleventh century. The earliest written description of it is Ptolemy's Planisphaerium , which calls it the "planisphere projection".

The stereographic projection was exclusively used for star charts until 1507, when Walther Ludd of St. Dié, Lorraine created the first known instance of a stereographic projection of the Earth's surface. Its popularity in cartography increased after Rumold Mercator used its equatorial aspect for his 1595 atlas. [1] It subsequently saw frequent use throughout the seventeenth century with its equatorial aspect being used for maps of the Eastern and Western hemispheres. [2]

In 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. [3] He used the recently established tools of calculus, invented by his friend Isaac Newton.

Formulae

The spherical form of the stereographic projection is usually expressed in polar coordinates:

where is the radius of the sphere, and and are the latitude and longitude, respectively.

The sphere is normally chosen to model the Earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. For maps of smaller regions, an ellipsoidal model must be chosen if greater accuracy is required. [1]

The ellipsoidal form of the polar ellipsoidal projection uses conformal latitude. There are various forms of transverse or oblique stereographic projections of ellipsoids. One method uses double projection via a conformal sphere, while other methods do not.

Examples of transverse or oblique stereographic projections include the Miller Oblated Stereographic [4] and the Roussilhe oblique stereographic projection. [2]

Properties

As an azimuthal projection, the stereographic projection faithfully represents the relative directions of all great circles passing through its center point. As a conformal projection, it faithfully represents angles everywhere. In addition, in its spherical form, the stereographic projection is the only map projection that renders all small circles as circles.

3D illustration of the geometric construction of the stereographic projection. Stereographic projection in 3D.svg
3D illustration of the geometric construction of the stereographic projection.

The spherical form of the stereographic projection is equivalent to a perspective projection where the point of perspective is on the point on the globe opposite the center point of the map.

Because the expression for diverges as approaches , the stereographic projection is infinitely large, and showing the South Pole (for a map centered on the North Pole) is impossible. However, it is possible to show points arbitrarily close to the South Pole as long as the boundaries of the map are extended far enough. [1]

Derived projections

The parallels on the Gall stereographic projection are distributed with the same spacing as those on the central meridian of the transverse stereographic projection.

The GS50 projection is formed by mapping the oblique stereographic projection to the complex plane and then transforming points on it via a tenth-order polynomial.

Comparison azimuthal projections.svg
Comparison of the Stereographic map projection and some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii. (click for detail)

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<span class="mw-page-title-main">Spherical coordinate system</span> 3-dimensional coordinate system

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<span class="mw-page-title-main">Azimuth</span> Horizontal angle from north or other reference cardinal direction

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<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

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<span class="mw-page-title-main">Earth radius</span> Distance from the Earth surface to a point near its center

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<span class="mw-page-title-main">Transverse Mercator projection</span> Adaptation of the standard Mercator projection

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<span class="mw-page-title-main">Azimuthal equidistant projection</span> Azimuthal equidistant map projection

The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection.

<span class="mw-page-title-main">Equirectangular projection</span> Cylindrical equidistant map projection

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<span class="mw-page-title-main">Tissot's indicatrix</span> Characterization of distortion in map protections

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

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<span class="mw-page-title-main">Cassini projection</span> Cylindrical equidistant map projection

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In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, then their images on a map with a conformal projection cross at a 39° angle.

<span class="mw-page-title-main">Gall stereographic projection</span> Cylindrical compromise map projection

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<span class="mw-page-title-main">Web Mercator projection</span> Mercator variant map projection

Web Mercator, Google Web Mercator, Spherical Mercator, WGS 84 Web Mercator or WGS 84/Pseudo-Mercator is a variant of the Mercator map projection and is the de facto standard for Web mapping applications. It rose to prominence when Google Maps adopted it in 2005. It is used by virtually all major online map providers, including Google Maps, CARTO, Mapbox, Bing Maps, OpenStreetMap, Mapquest, Esri, and many others. Its official EPSG identifier is EPSG:3857, although others have been used historically.

References

  1. 1 2 3 Snyder, John P. 1987. "Map Projections---A Working Manual". Professional Paper. United States Geological Survey. 1395: 154--163. ISBN   0-226-76746-9.
  2. 1 2 Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections p.~169. Chicago and London: The University of Chicago Press. ISBN   0-226-76746-9.
  3. Timothy Feeman. 2002. "Portraits of the Earth: A Mathematician Looks at Maps". American Mathematical Society.
  4. Sprinsky, William H.; Snyder, John P. (1986). "The Miller Oblated Stereographic Projection for Africa, Europe, Asia and Australasia". The American Cartographer. 13 (3): 253–261. doi:10.1559/152304086783899908.