Orthographic projection in cartography

The use of orthographic projection in cartography dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective 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 near the edges. The orthographic projection has been known since antiquity, with its cartographic uses being well documented. Hipparchus used the projection in the 2nd century B. C. to determine the places of star-set. In about 14 B. C. Roman engineer Marcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions. Vitruvius seems to have devised the term orthographic for the projection. However, the name analemma, which 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 woodcut drawings of terrestrial globes of 1509, 1533 and 1551, 1524 and 1551. These were crude. A 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; the formulas for the spherical orthographic projection are derived using trigonometry. They are written in terms of latitude on the sphere. Define the radius of the sphere R and the center point of the projection; the equations for the orthographic projection onto the tangent plane reduce to the following: x = R cos φ sin y = R 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: cos c = sin φ 0 sin φ + cos φ 0 cos φ cos .

The point should be clipped from the map. The inverse formulas are given by: φ = arcsin λ = λ 0 + arctan where ρ = x 2 + y 2 c = arcsin ρ R For computation of the inverse formulas, the use of the two-argument atan2 form of the inverse tangent function

Werner projection

The Werner projection is a pseudoconic equal-area map projection sometimes called the Stab-Werner or Stabius-Werner projection. Like other heart-shaped projections, it is categorized as cordiform. Stab-Werner refers to two originators: Johannes Werner, a parish priest in Nuremberg and promoted this projection, developed earlier by Johannes Stabius of Vienna around 1500; the projection is a limiting form of the Bonne projection, having its standard parallel at one of the poles. Distances along each parallel and along the central meridian are correct, as are all distances from the north pole. List of map projections Media related to Maps with Stab-Werner projection at Wikimedia Commons Table of examples and properties of all common projections, Radical Cartography

Polyconic projection class

Polyconic can refer either to a class of map projections or to a specific projection known less ambiguously as the American polyconic projection. Polyconic as a class refers to those projections whose parallels are all non-concentric circular arcs, except for a straight equator, the centers of these circles lie along a central axis; this description applies to projections in equatorial aspect. Some of the projections that fall into the polyconic class are: American polyconic projection Latitudinally equal-differential polyconic projection Rectangular polyconic projection Van der Grinten projectionA series of polyconic projections, each in a circle, was presented by Hans Mauer in 1922, who presented an equal-area polyconic in 1935. Another series by Georgiy Aleksandrovich Ginzburg appeared starting in 1949. List of map projections Table of examples and properties of all common projections, from radicalcartography.net

Gall stereographic projection

The Gall stereographic projection, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection; the projection is conventionally defined as: x = R λ 2. It is a perspective projection if the point of projection is allowed to vary with longitude: the point of projection being on the equator on the opposite side of the earth from the point being mapped and with the projective surface being a cylinder secant to the sphere at 45°N and 45°S. Gall called the projection "stereographic" because the spacing of the parallels is the same as the spacing of the parallels along the central meridian of the equatorial stereographic projection; the reverse projection is defined as: λ = x 2 R. This yields a projection tangent to the sphere, its formula is: x = R λ. Washington: Government Printing Office. Gall in proj4 Progonos.com

Hobo–Dyer projection

The Hobo–Dyer map projection is a cylindrical equal-area projection, with standard parallels at 37.5° north and south of the equator. The map was commissioned in 2002 by Bob Abramms and Howard Bronstein of ODT Inc. and drafted by cartographer Mick Dyer, as a modification of the 1910 Behrmann projection. The name Hobo–Dyer is derived from Bronstein and Abramms' first names and Dyer's surname; the original ODT map is printed on two sides, one side with north upwards and the other, south upwards. This, together with its equal-area presentation, is intended to present a different perspective compared with more common non-equal area, north-up maps; this goal is similar to that of other equal-area projections, but the Hobo-Dyer is billed by the publisher as "more visually satisfying". To this end, the map stretches the low latitudes vertically less than Peters, but at the price of greater compression near the poles. In 2002, the Carter Center used the Hobo–Dyer projection in a map of its global locations that it circulated to mark its founder Jimmy Carter's receipt of the Nobel Peace Prize.

List of map projections "THE WORLD TURNED UPSIDE DOWN". Archived from the original on August 13, 2012. Retrieved February 17, 2006. "The Upsidedown Map Page". Retrieved February 17, 2006

General Perspective projection

The General Perspective projection is a map projection. When the Earth is photographed from space, the camera records the view as a perspective projection; when the camera is aimed toward the center of the Earth, the resulting projection is called Vertical Perspective. When aimed in other directions, the resulting projection is called a Tilted Perspective; the Vertical Perspective is related to the stereographic projection, gnomonic projection, orthographic projection. These are all true perspective projections, meaning that they result from viewing the globe from some vantage point, they are azimuthal projections, meaning that the projection surface is a plane tangent to the sphere. This results in correct directions from the center to all other points; the point of perspective, or vantage point, for the General Perspective Projection is at a finite distance. It depicts the earth as it appears from some short distance above the surface a few hundred to a few tens of thousands of kilometers; when tilted, the General Perspective projection is not azimuthal.

Tilted perspectives are common from aerial and low orbit photography taken from at a height measured in kilometers to hundreds of kilometers, rather than the hundreds or thousands of kilometers typical of a vertical perspective. Some forms of the projection were known to the Egyptians 2,000 years ago, it was studied by several British scientists in the 18th and 19th centuries. However, the projection had little practical value at that time. Space exploration led to a renewed interest in the perspective projection. Now the concern was for a pictorial view from space, not for minimal distortion. A picture taken with a hand-held camera from the window of a spacecraft has a tilted vertical perspective, so the manned Gemini and Apollo space missions sparked interest in this projection; some prominent Internet mapping tools use the tilted perspective projection. For example, Google Earth and NASA World Wind show the globe; these applications permit a wide variety of interactive pan and zoom operations, including fly-through simulations, mimicking pictures or movies taken with a hand-held camera from an airplane or spacecraft.

List of map projections Map projection

Bonne projection

The Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre, modified Flamsteed, or a Sylvanus projection. Although named after Rigobert Bonne, the projection was in use prior to his birth, in 1511 by Sylvano, Honter in 1561, De l'Isle before 1700 and Coronelli in 1696. Both Sylvano and Honter’s usages were approximate, it is not clear they intended to be the same projection; the Bonne projection maintains accurate shapes areas along the central meridian and the standard parallel, but progressively distorts away from those regions. Thus, it best maps "t"-shaped regions, it has been used extensively for maps of Asia. The projection is defined as: x = ρ sin E y = cot φ 1 − ρ cos E where ρ = cot φ 1 + φ 1 − φ E = cos φ ρ and φ is the latitude, λ is the longitude, λ0 is the longitude of the central meridian, φ1 is the standard parallel of the projection. Parallels of latitude are concentric circular arcs, the scale is true along these arcs. On the central meridian and the standard latitude shapes are not distorted.

The inverse projection is given by: φ = cot φ 1 + φ 1 − ρ λ = λ 0 + ρ cos φ arctan where ρ = ± x 2 + 2 taking the sign of φ1. Special cases of the Bonne projection include the sinusoidal projection, when φ1 is zero, the Werner projection, when φ1 is 90°; the Bonne projection can be seen as an intermediate projection in the unwinding of a Werner projection into a Sinusoidal projection. List of map projections Cybergeo article Bonne Map Projection Table of examples and properties of all common projections, from radicalcartography.net An interactive Java Applet to study the metric deformations of the Bonne Projection Bonne Projection