Oblique Stereographic Projection

 

Oblique Stereographic Projection

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An azimuthal, conformal, polyconic (general) perspective projection that is visually similar to the ordinary Stereographic.  Used most conveniently with a single hemisphere. This version of the Stereographic projection also is called the Double Stereographic because it is really two projections: the surface of the utilized ellipsoid is first conformally projected to a sphere and then the sphere is stereographically projected to a plane. It therefore provides both a perspective view while also providing conformal mapping of the original ellipsoid onto a plane.

 

Scale

 

True only where the central latitude crosses the central meridian or, alternatively, along a circle concentric about the projection center (or a parallel on the polar aspect). Scale is constant along any circle having its center at the projection center, but scale increases rapidly with distance from the center within a hemisphere.

 

Distortion

 

Only the center or the circle of true scale (if not the center) is free from all distortion. Areas grow greater the farther from the center, albeit in a conformal manner.

 

Usage

 

Similar to that of the ordinary Stereographic. Commonly used in the polar aspect for topographic maps of polar regions. The Equatorial aspect was used regularly for maps of the Eastern and Western hemispheres in the 17th and 18th centuries. Oblique aspects are used to show paths of solar eclipses.

 

Recommended for conformal mapping of regions approximately circular in extent. For example, in 1997 it was used as the standard projection for the Fermilab Main Injector project, which involved injecting protons and antiprotons into a large, circular Tevatron ring. By centering the Double Stereographic projection at the center of the Tevatron, scale throughout the circular tunnels hosting the Tevatron could be kept accurate.

 

Limitations

 

The Double Stereographic projection must not be used to map the entire world’s surface at once: at least the point directly opposite to the projection origin must be excluded. This limitation arises because the Double Stereographic projection maps the point opposite the projection origin to infinity, causing numeric overflows. For example, if the North Pole is used as the projection origin, the South Pole and region immediately about the South Pole should not be included in the map.

 

Origin

 

Apparently developed in polar aspect by Egyptians and Greeks by the 2nd Century BC

 

Options

 

Specify the latitude origin and longitude origin to center the map projection to the area to be mapped. Specifying a non-Equatorial or non-polar origin causes an oblique projection.