Orthographic projection in cartography has been used since antiquity. Like thestereographic projection andgnomonic projection,orthographic projection is aperspective projection in which thesphere is projected onto atangent plane orsecant plane. Thepoint of perspective for the orthographic projection is atinfinite distance. It depicts ahemisphere of theglobe as it appears fromouter space, where thehorizon is agreat circle. The shapes and areas aredistorted, particularly near the edges.[1][2]
Theorthographic 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 engineerMarcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions.[2]
Vitruvius also seems to have devised the term orthographic (from the Greekorthos (= “straight”) and graphē (= “drawing”)) for the projection. However, the nameanalemma, which also meant a sundial showing latitude and longitude, was the common name untilFrançois d'Aguilon ofAntwerp promoted its present name in 1613.[2]
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 RenaissancepolymathAlbrecht Dürer and executed byJohannes Stabius, appeared in 1515.[2]
Photographs of theEarth and otherplanets from spacecraft have inspired renewed interest in the orthographic projection inastronomy andplanetary science.
Theformulas for the spherical orthographic projection are derived usingtrigonometry. They are written in terms oflongitude (λ) andlatitude (φ) on thesphere. Define theradius of thesphereR and thecenterpoint (andorigin) of the projection (λ0,φ0). Theequations for the orthographic projection onto the (x,y) tangent plane reduce to the following:[1]
Latitudes beyond the range of the map should be clipped by calculating theangular distancec from thecenter of the orthographic projection. This ensures that points on the opposite hemisphere are not plotted:
The point should be clipped from the map if cos(c) is negative. That is, all points that are included in the mapping satisfy:
The inverse formulas are given by:
where
Forcomputation of the inverse formulas the use of the two-argumentatan2 form of theinverse tangent function (as opposed toatan) is recommended. This ensures that thesign of the orthographic projection as written is correct in allquadrants.
The inverse formulas are particularly useful when trying to project a variable defined on a (λ,φ) grid onto a rectilinear grid in (x,y). Direct application of the orthographic projection yields scattered points in (x,y), which creates problems forplotting andnumerical integration. One solution is to start from the (x,y) projection plane and construct the image from the values defined in (λ,φ) by using the inverse formulas of the orthographic projection.
See References for an ellipsoidal version of the orthographic map projection.[3]
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.[clarification needed]
An example of an orthographic projection onto a cylinder is theLambert cylindrical equal-area projection.
