Geography
Map Projections
Map projections are methods used to represent the three-dimensional surface of the Earth on a two-dimensional map. They involve transforming the Earth's curved surface into a flat plane, which can cause distortions in size, shape, distance, or direction. Different map projections are used for different purposes, and no single projection can perfectly represent the entire Earth's surface.
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eBook - PDFSpatial Mathematics
Theory and Practice through Mapping
- Sandra Lach Arlinghaus, Joseph J. Kerski(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
197 Map Projections Keywords: Projection(s) (133), map (66), globe (26), plane (24), paral-lels (20) A map is the greatest of all epic poems. Its lines and colors show the realization of great dreams. Gilbert H. Grosvenor, Editor National Geographic Magazine, 1899–1954 9.1 Introduction The only commonly used, dimensionally true representation of the Earth, free of distortion, is a globe. Maps, whether on paper or on a computer screen, are flat. The transformation by which geographic locations (latitude and 9 198 Spatial Mathematics longitude) are sent from the curved, two-dimensional, surface of a sphere to a flat, two-dimensional, map is called “projection.” The transformation pre-serves two dimensionality. Imagine the Earth as an inflated balloon. Cut it open and flatten it. It will be stretched in some places and shrunk in others. Distortion is inevitable. Since the transformation from a bounded, curved two-dimensional surface to an unbounded, flat two-dimensional surface is not a smooth one, every map pro-jection is distorted in at least three, and sometimes in four, of the following properties: Shape, area, distance, and direction. Thus, the challenge to the map-per in choosing appropriate projections, as in selecting colors, data sets, and a host of other variables, becomes, “but does it fit or is it appropriate?” Readers will have a chance to put theory into practice at the end of this chapter: Look for example at links associated with the Mollweide projection in the theory section and then practice selecting it, or other projections, in the later practice activities. Readers with even deeper interests might wish to delve into the vast literature on the topic of Map Projections and cartography, some of which is cited at the end of this book. Let us see why map projection selection might (or might not) matter, beginning with a real-world story with some serious implications.
eBook - PDF- Narayan Panigrahi(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
A small globe is useless for referring to a small country or landscape because it distorts the smaller land surfaces and depicts the land surfaces inappropriately. So for practical purposes a globe is least useful or helpful in the field. Moreover it is neither easy to compare in detail different regions of the Earth over the globe, nor is it convenient to measure distances over it. Hence maps were devised to overcome such diffi-culties. A map is a two-dimensional representation of a globe drawn on paper map which is convenient to fold and carry in the field and easy to compare and locate different parts of the Earth. Locating a known feature, guiding and navigating from one position to another and comparing two different re-gions over a map are convenient and easy. Transforming a three-dimensional globe to a two-dimensional paper map is accomplished using map projection. Topographical maps of different scales, atlases and wall maps are prepared us-ing Map Projections. Thus map projection plays a crucial role in preparation of different types of maps with different scales, coordinate systems and themes. 4.2 Mathematical Definition of Map Projection A map projection is defined as a mathematical function or formula which projects any point ( φ, λ ) on the spherical surface of Earth to the two-dimensional point ( x, y ) on a plane surface. The forward map projection is given by ( x, y ) = f ( φ, λ ) (4.1) Often the geographic data obtained in Cartesian coordinate needs to be transformed to spherical coordinates. This necessitates the reverse process which is known as inverse map projection. The inverse map projection is given by ( φ, λ ) = f -1 ( x, y ) (4.2) Thus, the mathematical function, which realizes the map projection, essen-tially projects the 3D spatial features onto 2D map surfaces. Invaluable re-
No longer available |Learn more- James Petersen, Dorothy Sack, Robert Gabler(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Although maps are not actually made this way, certain projections can be demonstrated by putting a light inside a trans- parent globe so that the grid lines are projected onto a flat surface (plane), a cone, a cylinder, or some other geo- metric forms that are flat or can be cut and flat- tened out (● Fig. 2.18). Today, Map Projections are developed mathematically, using computers to fit the geographic grid to a surface, and there are hundreds of different kinds. Map Projections will always distort certain characteristics displayed on the map, such as the shape of mapped features, accurate area relationships, true compass direc- tions, uniform distance relationships, or some combination of these factors. Planar Projections Projecting the grid lines onto a plane, or flat surface, produces a map called a planar projection (Fig. 2.18a). These maps are most often used to show the polar regions, with the pole centrally located on the circular map, which displays one hemisphere. Maps of the Arctic Ocean with its surrounding polar region, as well as maps of Antarctica most commonly use this kind of projection. Maps and Map Projections Maps are extremely versatile—they can be reproduced easily, can depict the entire Earth or show a small area in great detail, are easy to handle and transport, and can be displayed on a computer monitor, cell phone, or tablet. Yet, it is not possible for one map to fit all uses. The many different varieties of maps have quali- ties that can be either advantageous or problematic, depending on the application. Knowing some basic concepts concerning the characteristics of maps and cartography greatly enhances a person’s ability to effectively use a map and to select the right one for a particular task. Advantages of Maps If a picture is worth a thousand words, then a map is worth a mil- lion.
eBook - ePubGIS Cartography
A Guide to Effective Map Design, Third Edition
- Gretchen N. Peterson(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
(Gren-itch) Meridian since it passes through the Royal Observatory in Greenwich, England. The zero point doesn't have to be at the Greenwich Meridian, but it is what most authorities use. An elevation might also be specified.A projected coordinate system, also known as a map projection, is the mathematical transformation of a geographic coordinate system onto a two-dimensional (2D) surface. In other words, when you need to place real-world features onto a flat page or device and, naturally, scale them down, you need a projection. This is, in fact, the most fundamental definition of a projection and it can extend to anything that you are “projecting” onto a flat surface, such as a movie image or in our case, maps. The math behind projecting the global map image onto a flat surface is both interesting and complicated. Many, many attempts have been made through history to better the math behind projecting maps, with varying success, accuracy, and popularity. In fact, thousands of Map Projections exist. The projection is the foundation for mapping. Without it, we'd be plotting all our data onto globes.Arthur H. Robinson, who was a University of Wisconsin geography professor from 1947 to 1980, made one such attempt at creating a projection in the early 1960s when the company Rand McNally asked him to identify a projection that minimizes certain distortions, such as area and distance. Robinson was unable to find a projection that comported with the McNally requirements, so he built his own. He went about it a little differently than other projection pioneers in that he started with an idea of how he wanted it to look and behave, used trial and error with various formulae, and then came up with the overarching formula to make it work. Most projections are created by starting with a mathematical concept first, and then basically accepting how it looks and behaves once it is plotted out. The Robinson Projection wound up striking a nice balance among certain desirable features. One of its great accomplishments was minimizing area and distance distortion in the places where we need it the most: in the regions with the highest populations, which correspond to the Earth's two temperate zones (see Figure 8.3
eBook - PDF- James Petersen, Dorothy Sack, Robert Gabler, , James Petersen, Dorothy Sack, Robert Gabler(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
By being aware of these map characteristics, we can make accurate comparisons and measurements on maps and better understand the information that a map conveys. Examples of Map Projections Transferring a spherical grid onto a flat surface produces a map projection . Although maps are not actually made this way, cer-tain projections can be demonstrated by putting a light inside a transparent globe so that the grid lines are projected onto a flat surface (plane), a cone, a cylinder, or other geometric forms that are flat or can be cut and flattened out ( ■ Fig. 2.14). Today, Map Projections are developed mathematically, using computers to fit the geographic grid to a surface. Map Projections always distort ■ FIGURE 2.13 Lunar geography. A detailed map shows a major moon crater that is 120 km in diameter (75 mi). Even the side of the moon that never faces Earth has been mapped in considerable detail. How were we able to map the moon in such detail? NASA (a) Planar projection (c) Cylindrical projection (b) Conical projection ■ FIGURE 2.14 The concept behind the development of (a) planar, (b) conic, and (c) cylindrical projections. Although projections are not actually produced this way, they can be demonstrated by projecting light from a transparent globe. Why do we use different Map Projections? © Cengage Learning Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C H A P T E R 2 • R E P R E S E N TAT I O N S O F E A R T H 32 are at the equator ( ■ Fig. 2.15). The spacing of parallels on a Mercator projection is also not equal, as they are on Earth.
eBook - PDF- Aylmer Johnson(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Thus, all such projections involve some degree of distortion on the resulting map, except at certain points or along particular lines. By varying the exact method of projection, it is possible to manipulate the changes in scale so as to reduce or avoid some aspects of distortion on a map, but usually at the expense of causing increased distor-tion in other respects. In particular, there are three important properties which maps may have, as follows: 1. A conformal projection manages the scaling effects such that, at any point on the projection, the scale in all directions is the same. * Such maps are also orthomorphic , which means that small shapes on the ellipsoidal surface (e.g., buildings or fields) are shown as the same shape on the map. One result of this is that the angle at which any two horizontal lines cross on the earth’s surface is preserved exactly on the map. However, the shortest distance over the ellipsoid between two points on its surface (i.e., a geodesic) does not in gen-eral plot as a straight line on the map. In fact, since meridians and the equator are the only geodesics which lie in a plane, they are also the only geodesics which can ever appear as exactly straight lines on any standard projection. 2. An equal area projection manages scaling such that, if the scale at a point is unavoidably increased in one direction, then it is corre-spondingly reduced in the orthogonal direction. Thus, the area of any feature (e.g., a country) is exactly preserved on an equal area map, subject to the quoted scale of the map. However, the shape of the area will not be preserved exactly; and at any point on the map, the scale in one direction (e.g., north–south) will generally differ from the scale in any other direction (e.g., east–west). Small circles drawn on the surface of the earth would thus plot on the projection as ellipses with the same area, but with greater eccentricities in places where the distortion of the projection is higher.
eBook - PDF- Peter Dale(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
223 11 Map Projections 11.1 Map Projections In this chapter we will consider ways in which the location of points on the surface of the Earth can be represented in what are commonly referred to as Map Projections . The assumption behind the transformations considered in Chapter 10 has been that we have been dealing with straight lines or flat surfaces, in effect, a flat Earth. When we wish to represent the curved surface of the Earth, we need to take a different approach. Consider a point A on the Earth’s surface with latitude ø A and longitude λ (Figure 11.1). The length of the arc from the equator ( Q ) to A = R ø A where R is the radius of the Earth, assumed here to be a sphere, and ø is measured in radians. Let C be a point with latitude ø C and longitude λ C . Let the difference in longitude between A and C = Δλ = λ C – λ A and the difference in latitude = Δ ø = ø C – ø A . Let B be on the same parallel of latitude as C and the same meridian of longitude as A . In triangle ABC then in terms of physical length: AB = R Δ ø. Since the radius of the circle for the parallel of latitude ø = PB and PB = PC = R cos ø, then BC = PB Δλ = R cos ø C Δλ These quantities form the basis for plotting on a flat surface with a rectangular grid (see Figure 11.2). AB on the sphere becomes A ′ B ′ = Δ N and BC becomes B ′ C ′ = Δ E , the differences in northings and eastings between A ′ and C ′ . In the simplest map projection, known as the simple cylindrical or plate carrée projection, the distance north is plotted as N = R ø and the eastings as R λ so that there is a rectangular grid. This means, in effect, that the distances north–south are treated as correct but distances east–west must be stretched from ( R cos ø) λ on the globe to R λ on the grid. East–west distances on the globe must be increased by a scale factor of (1/cos ø) or sec ø in order to plot them on the rectangular grid.
eBook - PDF- D.H. Maling(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
C H A P T E R 7 The appearance, classification and naming of Map Projections Viewed in this light the projections of M. Tissot assume a new aspect, and it is clearly necessary to study them anew, and to master his rather repellent ter-minology, that seems so superfluously different from that of his compatriot Germain. A. R. Hinks, GeographicalJournal, 1921 Introduction Examination of the illustrations of different Map Projections which appear in this book indicates the great variety in the shape and detailed appearance of them. Some of the world maps are rectangular in outline, others are bounded by ellipses or more complicated curves. Some pro-jections have rectilinear parallels or meridians; others have various com-binations of curved graticule lines. In this chapter we introduce some of the terms which are commonly used to describe the appearance of Map Projections. These may be used in conjunction with distortion theory to select and describe suitable Map Projections for particular purposes, or to recognise the projection used for a particular map. If the cartographer has not done his job properly, and has failed to indicate this information, or has described the projection in unfamihar terms, the critical user has to make a reasoned guess about what projection has been used. The cartographer can communicate with the map user if both understand the same technical terms, but confusion and mis-interpretation result if they do not. The subject of Map Projections is embarrassingly rich in words which mean the same thing. Therefore the beginner who is already struggling to understand many new concepts is also confronted with and confused by dupHcate terms. Some of these are synonymous, such as the words 'autogonal' and 'orthomorphic' to mean conformal, or the use of 'authalic' or 'orthembadic' instead of equal-area.
eBook - PDF- James Petersen, Dorothy Sack, Robert Gabler, , James Petersen, James Petersen, Dorothy Sack, Robert Gabler(Authors)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
If we use a park map while hiking, the distortion will be too small to affect us. Knowledgeable map users know which properties a certain kind of map depicts accurately, which attributes it distorts, and for what purpose a type of map is best suited. If we are aware of these map characteristics, we can make accurate comparisons and better understand the information that a map conveys. 2-3b Map Projections Transferring a spherical grid onto a flat surface produces a map projection. Maps are not actually made this way, but certain 2-3a Maps Have Advantages and Limitations Map Advantages If a picture is worth a thousand words, a map is worth a million. Because they are graphic representations and typically use symbols along with words, maps are an efficient medium for communicating spatial relationships and portraying geographic information. As visual representations, maps can sup- ply an enormous amount of information that would take many pages to describe, usually less successfully, in words. Imagine trying to verbally explain all of the information that a map of your city, county, state, or campus provides: sizes, areas, distances, directions, street patterns, railroads, bus routes, hospitals, schools, highways, business districts, residential areas, and so forth. Maps can display directions for navigation and the shapes and sizes of Earth features. They can be used for measuring distances or areas and can show the best route from one place to another. The poten- tial applications of maps are practically infinite, even “out of this world.” Detailed maps of the moon, Mars, and many other extrater- restrial features have been developed for space programs and ● FIGURE 2.15 An intern in Wyoming helps the U.S. Bureau of Land Management with field surveying by collecting accurate locations with a global positioning system receiver. Many geography students serve in summer intern positions with federal government agencies.
eBook - PDF- Judith A. Tyner(Author)
- 2017(Publication Date)
- The Guilford Press(Publisher)
The Earth’s Graticule and Projections 123 projection from those available. Even if the computer program provides a list of suit- able projections for a given task, the cartographer must be able to make an intelligent choice among them. While two or more projections may be suitable, probably not all will illustrate or communicate the given situation equally well. A well-chosen projec- tion can enhance the communicative value of the map; a poorly chosen projection may even mislead the map reader. To make the best choice, it is necessary to consider the purpose of the map, the subject of the map, the size and shape of the subject area, the location of the subject area, the audience, and the size and shape of the page. Other considerations are the appearance of the graticule, the general attractiveness of the projection, and the avail- ability of the projection within the chosen software program. FIGURE 6.36. Goode’s homolosine projection is a combination of the sinusoidal and Moll- weide (homolographic) projections. It is almost always in interrupted form. FIGURE 6.37. An interrupted projection can be condensed to gain scale. 124 THE GEOGRAPHIC AND CARTOGRAPHIC FRAMEWORK Subject and Purpose of the Map The subject and purpose of the map are the most critical factors for picking a projec- tion. For distribution maps, equal-area projections should be used; for maps show- ing world distributions, a nonequivalent projection will usually be misleading since quantity and area are related (Figure 6.38). When the distribution is illustrated by dots, the impression of the relative density of the distribution will be distorted if non- equivalent projections are used. For navigation, maps that show angles or azimuths properly are needed; still other needs might dictate equidistance or conformality. Appendix A will aid in the selection of a suitable projection. The USGS has created a poster of Map Projections and their properties (see Appendix B).
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