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Stereographic projection

An azimuthal projection that is conformal (The correct shapes of small features are preserved. In other words, shapes are not distorted.). Size distortions become large at the edges.
stereographic1

Gnomic projection

All great circles (e.g. the equator, lines of longitude, the day/night terminator) are straight lines. Imagine a light source at the center of the earth projecting an image of the earth onto a plane. Distortions are extreme at the edges of the map. No more than one hemisphere can be seen at one time.
gnomonic1

Perspective projection

The earth as it would appear if viewed from a finite distance. No more than one hemisphere can be viewed at one time.
perspective1

Mercator projection

Many world maps use this projection. Compass directions (loxodromes or rhumb lines) between locations are correct. Size distortions are extreme at the top and bottom of the map.
mercator1

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