Orthographic with Tissot's Indicatrices of Distortion

Orthographic projection in cartography has been used since antiquity. Like the

stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...

and gnomonic projection,

orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...

is a perspective (or azimuthal) projection in which the

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

is projected onto a

tangent plane In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

secant plane A secant plane is a plane containing a nontrivial section of a sphere or an ellipsoid, or such a plane that a sphere is projected onto. Secant planes are similar to tangent planes, which contact the sphere's surface at a point, while secant plane ...

. The ''point of perspective'' for the orthographic projection is at

infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...

distance. It depicts a

hemisphere Hemisphere refers to: * A half of a sphere As half of the Earth * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemisphere ** Land and water hemispheres * A half of the (geocentric) celes ...

of the

globe A globe is a spherical model of Earth, of some other celestial body, or of the celestial sphere. Globes serve purposes similar to maps, but unlike maps, they do not distort the surface that they portray except to scale it down. A model globe ...

as it appears from

outer space Outer space, commonly shortened to space, is the expanse that exists beyond Earth and its atmosphere and between celestial bodies. Outer space is not completely empty—it is a near-perfect vacuum containing a low density of particles, pred ...

, where the

horizon The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether i ...

is a

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...

. The shapes and areas are distorted, particularly near the edges.Snyder, John P. (1993). ''Flattening the Earth: Two Thousand Years of Map Projections'' pp. 16–18. Chicago and London: The University of Chicago Press. .

History

The

has been known since antiquity, with its cartographic uses being well documented.

Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos''; BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equi ...

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 Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...

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 In astronomy, an analemma (; ) is a diagram showing the position of the Sun in the sky as seen from a fixed location on Earth at the same mean solar time, as that position varies over the course of a year. The diagram will resemble a figure ...

'', which also meant a sundial showing latitude and longitude, was the common name until

François d'Aguilon François d'Aguilon (also d'Aguillon or in Latin Franciscus Aguilonius) (4 January 1567 – 20 March 1617) was a Jesuit, mathematician, physicist, and architect from the Spanish Netherlands. D'Aguilon was born in Brussels; his father was a secret ...

Antwerp Antwerp (; nl, Antwerpen ; french: Anvers ; es, Amberes) is the largest city in Belgium by area at and the capital of Antwerp Province in the Flemish Region. With a population of 520,504,

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 A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...

Albrecht Dürer Albrecht Dürer (; ; hu, Ajtósi Adalbert; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer (without an umlaut) or Due ...

and executed by

Johannes Stabius Johannes Stabius (Johann Stab) (1450–1522) was an Austrian cartographer and astronomer of Vienna who developed, around 1500, the heart-shape (cordiform) projection map later developed further by Johannes Werner. It is called the '' Werner map p ...

, appeared in 1515. Photographs of the

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

and other

planets A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a young ...

from spacecraft have inspired renewed interest in the orthographic projection in

astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

and planetary science.

Mathematics

The

formulas In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

for the spherical orthographic projection are derived using

trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...

. They are written in terms of

longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...

(''λ'') and

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

(''φ'') on the

. Define the

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

of the

''R'' and the ''center''

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

(and

origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

) of the projection (''λ''₀, ''φ''₀). The equations for the orthographic projection onto the (''x'', ''y'') tangent plane reduce to the following: :

\begin
x &= R\,\cos\varphi \sin\left(\lambda - \lambda_0\right) \\
y &= R\big(\cos\varphi_0 \sin\varphi - \sin\varphi_0 \cos\varphi \cos\left(\lambda - \lambda_0\right)\big)
\end

Latitudes beyond the range of the map should be clipped by calculating the

angular distance Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle between the two sightlines, or between two point objects as viewed from an observer. Angular distance appears in mathematics (in pa ...

''c'' from the ''center'' of the orthographic projection. This ensures that points on the opposite hemisphere are not plotted: :

\cos c = \sin\varphi_0 \sin\varphi + \cos\varphi_0 \cos\varphi \cos\left(\lambda - \lambda_0\right)\,

. The point should be clipped from the map if cos(''c'') is negative. That is, all points that are included in the mapping satisfy: :

-\frac < c < \frac

. The inverse formulas are given by: :

\begin
\varphi &= \arcsin\left(\cos c \sin\varphi_0 + \frac\right) \\
\lambda &= \lambda_0 + \arctan\left(\frac\right)
\end

where :

\begin
\rho &= \sqrt \\
   c &= \arcsin\frac
\end

For

computation Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm). Mechanical or electronic devices (or, historically, people) that perform computations are known as ''computers''. An es ...

of the inverse formulas the use of the two-argument atan2 form of the

inverse tangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Sp ...

function (as opposed to

atan Atan may refer to: Places * Atan, Armenia * Atan, Iran People * Atan Shansonga (born 1955), Zambian diplomat * Çağdaş Atan, Turkish footballer * Cem Atan, Turkish footballer Other * Attan, a Pashtun and Afghan traditional dance * arctang ...

) is recommended. This ensures that the

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

of the orthographic projection as written is correct in all

quadrants Quadrant may refer to: Companies * Quadrant Cycle Company, 1899 manufacturers in Britain of the Quadrant motorcar * Quadrant (motorcycles), one of the earliest British motorcycle manufacturers, established in Birmingham in 1901 * Quadrant Privat ...

. 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 In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...

. 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.

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 In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical equal-area projection. This projection is undistorted along the equator, which is its standard parallel, but distortion increa ...

References

External links

Orthographic Projection—from MathWorld
{{Map Projections Map projections

History

Mathematics

Orthographic projections onto cylinders

See also

References

External links