Orthographic map projection

The orthographic projection has been known since antiquity, with its cartographic uses being well documented. Hipparchus used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer Marcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions. Vitruvius also seems to have devised the term orthographic (from the Greek orthos (= “straight”) and graphē (= “drawing”)) for the projection. However, the name analemma, which also meant a sundial showing latitude and longitude, was the common name until François d'Aguilon of Antwerp promoted its present name in 1613. The earliest surviving maps on the projection appear as crude woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian). A highly-refined map, designed by Renaissance polymath Albrecht Dürer and executed by Johannes Stabius, appeared in 1515. Photographs of the Earth and other planets from spacecraft have inspired renewed interest in the orthographic projection in astronomy and planetary science. ==Mathematics==

The formulas for the spherical orthographic projection are derived using trigonometry. They are written in terms of longitude () and latitude () on the sphere. Define the radius of the sphere and the center point (and origin) of the projection . The equations for the orthographic projection onto the tangent plane reduce to the following: ==Orthographic projections onto cylinders==

In a wide sense, all projections with the point of perspective at infinity (and therefore parallel projecting lines) are considered as orthographic, regardless of the surface onto which they are projected. Such projections distort angles and areas close to the poles. An example of an orthographic projection onto a cylinder is the Lambert cylindrical equal-area projection. ==See also==