# Orthographic map projection

*https://en.wikipedia.org/wiki/Orthographic_projection_(cartography)*

**Orthographic projection in cartography** has been used since antiquity. Like the
stereographic projection and
gnomonic projection,
orthographic projection is a
perspective (or azimuthal) projection in which the
sphere is projected onto a
tangent plane or
secant plane. The *point of perspective* for the orthographic projection is at
infinite distance. It depicts a
hemisphere of the
globe as it appears from
outer space, where the
horizon is a
great circle. The shapes and areas are
distorted, particularly near the edges.^{
[1]}^{
[2]}

## History

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.^{
[2]}

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.^{
[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 Renaissance
polymath
Albrecht Dürer and executed by
Johannes Stabius, appeared in 1515.^{
[2]}

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 (*λ*) and
latitude (*φ*) on the
sphere. Define the
radius of the
sphere *R* and the *center*
point (and
origin) of the projection (*λ*_{0}, *φ*_{0}). The
equations 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 the
distance *c* from the *center* 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.

The inverse formulas are given by:

where

For computation of the inverse formulas the use of the two-argument atan2 form of the inverse tangent function (as opposed to atan) is recommended. This ensures that the sign of the orthographic projection as written is correct in all quadrants.

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 for
plotting and
numerical 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]}

## 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.^{[
clarification needed]}

An example of an orthographic projection onto a cylinder is the Lambert cylindrical equal-area projection.

## See also

## References

- ^
^{a}^{b}Snyder, J. P. (1987).*Map Projections—A Working Manual (US Geologic Survey Professional Paper 1395)*. Washington, D.C.: US Government Printing Office. pp. 145–153. - ^
^{a}^{b}^{c}^{d}Snyder, John P. (1993).*Flattening the Earth: Two Thousand Years of Map Projections*pp. 16–18. Chicago and London: The University of Chicago Press. ISBN 9780226767475. **^**Zinn, Noel (June 2011). "Ellipsoidal Orthographic Projection via ECEF and Topocentric (ENU)" (PDF). Retrieved 2011-11-11.

## External links

Wikimedia Commons has media related to Orthographic projection (cartography). |