Orthographic projection (cartography)
Encyclopedia
An orthographic projection is a map projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

 of cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

. Like the stereographic projection
Stereographic projection
The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...

 and gnomonic projection
Gnomonic projection
A gnomonic map projection displays all great circles as straight lines. Thus the shortest route between two locations in reality corresponds to that on the map. This is achieved by projecting, with respect to the center of the Earth , the Earth's surface onto a tangent plane. The least distortion...

, orthographic projection is a perspective (or azimuthal) projection, in which the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 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
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 of the globe
Globe
A globe is a three-dimensional scale model of Earth or other spheroid celestial body such as a planet, star, or moon...

 as it appears from outer space
Outer space
Outer space is the void that exists between celestial bodies, including the Earth. It is not completely empty, but consists of a hard vacuum containing a low density of particles: predominantly a plasma of hydrogen and helium, as well as electromagnetic radiation, magnetic fields, and neutrinos....

, where the horizon
Horizon
The horizon is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not. At many locations, the true horizon is obscured by trees, buildings, mountains, etc., and the resulting...

 is a great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

. The shapes and areas are distorted, particularly near the edges, but distances are preserved along parallel
Circle of latitude
A circle of latitude, on the Earth, is an imaginary east-west circle connecting all locations that share a given latitude...

s.

History

The orthographic projection has been known since antiquity, with its cartographic uses being well documented. Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...

 used the projection in the 2nd century B.C. to determine the places of star-rise and star-set. In about 14 B.C., Roman engineer Marcus Vitruvius Pollio
Vitruvius
Marcus Vitruvius Pollio was a Roman writer, architect and engineer, active in the 1st century BC. He is best known as the author of the multi-volume work De Architectura ....

 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
Analemma
In astronomy, an analemma is a curve representing the angular offset of a celestial body from its mean position on the celestial sphere as viewed from another celestial body relative to the viewing body's celestial equator...

, which also meant a sundial showing latitude and longitude, was the common name until François d'Aguilon
François d'Aguilon
François d'Aguilon , was a Belgian Jesuit mathematician, physicist and architect....

 of Antwerp promoted its present name in 1613.

The earliest surviving maps on the projection appear as woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian). These were crude. A highly refined map designed by Renaissance polymath
Polymath
A polymath is a person whose expertise spans a significant number of different subject areas. In less formal terms, a polymath may simply be someone who is very knowledgeable...

 Albrecht Dürer
Albrecht Dürer
Albrecht Dürer was a German painter, printmaker, engraver, mathematician, and theorist from Nuremberg. His prints established his reputation across Europe when he was still in his twenties, and he has been conventionally regarded as the greatest artist of the Northern Renaissance ever since...

 and executed by Johannes Stabius
Johannes Stabius
Johannes Stabius was an Austrian cartographer of Vienna who developed, around 1500, the heart-shape projection map later developed further by Johannes Werner. It is called the Werner map projection, but also the Stabius-Werner or the Stab-Werner projection...

 appeared in 1515.

Photographs of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

 and other planets from spacecraft have inspired renewed interest in the orthographic projection in astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

 and planetary science
Planetary science
Planetary science is the scientific study of planets , moons, and planetary systems, in particular those of the Solar System and the processes that form them. It studies objects ranging in size from micrometeoroids to gas giants, aiming to determine their composition, dynamics, formation,...

.

Mathematics

The formulas for the spherical orthographic projection are derived using trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

. They are written in terms of longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

 (λ) and latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

 (φ) on the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. Define the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

 of the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 R and the center point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

 (and origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

) of the projection (λ0, φ0). The equations for the orthographic projection onto the (x, y) tangent plane reduce to the following:


Latitudes beyond the range of the map should be clipped by calculating the distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

  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 is negative.

The inverse formulas are given by:


where


For computation
Computation
Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...

 of the inverse formulas (e.g., using C
C (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....

/C++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

, Fortran
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...

, or other programming language
Programming language
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....

), the use of the two-argument atan2
Atan2
In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...

 form of the inverse tangent function (as opposed to atan) is recommended. This ensures that the sign
Sign (mathematics)
In mathematics, the word sign refers to the property of being positive or negative. Every nonzero real number is either positive or negative, and therefore has a sign. Zero itself is signless, although in some contexts it makes sense to consider a signed zero...

 of the orthographic projection as written is correct in all quadrants
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

.

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
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

 and numerical integration
Numerical integration
In numerical analysis, numerical integration constitutes 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. This article focuses on calculation of...

. 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. These kinds of 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
Lambert cylindrical equal-area projection
In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is acylindrical, equal area map projection...

.

External links

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
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