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Principles of Cartography The Earth and Its Coordinate System Map Projections Selecting a Map Projection Scale and Generalization http://montgomerycollege.edu/Departments/planet/M_AS102/coordinates/EarthLatLong.gif http://maic.jmu.edu/sic/images/projections.gif http://go.owu.edu/~jbkrygie/krygier_html/geog_222/geog_222_lo/geog_222_lo04_gr/scale.jp g Geodesy The science of earth measurement Approximations of the shape of the earth o Sphere o Ellipsoid Approximate shape of earth o Geoid Most precise shape Flattened at the pole Extended at the equator Includes surface irregularities http://jaeger.earthsci.unimelb.edu.au/ImageLibrary/Favourites/Page s/monim_1300.html https://www.e-education.psu.edu/files/geog482/image/ellipsoid_diagram.jpg http://go.owu.edu/~jbkrygie/krygier_html/geog_353/geog_353_lo/geog_353_lo05_gr/projprocess.jpg Datums A starting point Gives context to locations on the earth’s surface Defines the size and shape of earth (such as a reference ellipsoid) and some sort of “tie point” that fixes the ellipsoid to the to surface of the earth or the center of the earth http://www.colorado.edu/geography/gcraft/notes/coordsys/coordsys_f.html Not everyone uses the same ellipsoid. Airy is used in the UK, North America uses Clarke, but everyone uses the WGS (Worldwide, Global Positioning System). http://www.colorado.edu/geography/gcraft/notes/coordsys/coordsys_f.html Geodesic datums and coordinate reference systems Geodetic datums define the reference systems that describe the size and shape of the earth. Hundreds of different datums have been used to frame position descriptions since the first estimates of the earth's size were made by Aristotle. Datums have evolved from those describing a spherical earth to ellipsoidal models derived from years of satellite measurements. Modern geodetic datums range from flat-earth models used for plane surveying to complex models used for international applications which completely describe the size, shape, orientation, gravity field, and angular velocity of the earth. While cartography, surveying, navigation, and astronomy all make use of geodetic datums, the science of geodesy is the central discipline for the topic. Referencing geodetic coordinates to the wrong datum can result in position errors of hundreds of meters. Different nations and agencies use different datums as the basis for coordinate systems used to identify positions in geographic information systems, precise positioning systems, and navigation systems. The diversity of datums in use today and the technological advancements that have made possible global positioning measurements with submeter accuracies requires careful datum selection and careful conversion between coordinates in different datums. Geodetic datums and the coordinate reference systems based on them were developed to describe geographic positions for surveying, mapping, and navigation. Through a long history, the "figure of the earth" was refined from flat-earth models to spherical models of sufficient accuracy to allow global exploration, navigation and mapping. True geodetic datums were employed only after the late 1700s when measurements showed that the earth was ellipsoidal in shape. Ha ha! “In the realm of thematic mapping, the focus is on examining spatial patterns and distributions, and so knowledge of the precise direction, distance and area is not critical.” Given the scale of most thematic maps issues of geodesy rarely enter into our world. In thematic mapping the assumption is that the earth is a perfect sphere which drastically simplifies our lives in terms of mathematics and creating or selecting a map projection. Basic Coordinate Systems There are many basic coordinate systems. These systems can represent points in two-dimensional or three-dimensional space. René Descartes (1596-1650) introduced systems of coordinates based on orthogonal (right angle) coordinates. These two and three-dimensional systems used in analytic geometry are often referred to as Cartesian systems. Similar systems based on angles from baselines are often referred to as polar systems. http://www.ncgia.ucsb.edu/education/curricula/giscc/units/u014/figures/figure01.gif http://upload.wikimedia.org/wikipedia/commons/thumb/2/2c/3D_coordinate_system .svg/487px-3D_coordinate_system.svg.png http://eusoils.jrc.ec.europa.eu/gisco_dbm/dbm/img/ch3-5_fig2.jpg Global Systems Latitude and Longitude o The most commonly used coordinate system today is the latitude, longitude, and height system. o The Prime Meridian and the Equator are the reference planes used to define latitude and longitude. Latitude and Longitude combine aspects of planar coordinate systems (Cartesian) with angular measurement to accommodate the spherical shape of the earth. http://www.colorado.edu/geography/gcraft/notes/coordsys/coordsys_f.html http://www.colorado.edu/geography/gcraft/notes/coordsys/coordsys_f.html http://media.tiscali.co.uk/images/feeds/hutchinson/ency/c01522.jpg Latitude The location on the earth’s surface between the equator and either the North or South Pole. Latitude is a function of the angle between the horizon and the North Star (or some other fixed star). As you travel towards the pole the angle increases. Latitude is designated in angular degrees N or S from 0° at the equator to 90° at the poles. The surface designation for each degree of latitude is about 69.2 miles Latitude lines are parallel to the equator and to each other. The equator is a great circle, dividing the earth into two equal parts. All other lines of latitude are small circles. Longitude Earth rotates on its axis every 24 hours therefore any point on the earth moves through 360 angular degrees in a day’s time or 15° in each hour. In order to measure longitude you need a fixed point of reference and accurate time keeping to determine the difference in time between the local time and the point of reference. This time can be converted into degrees and thus position. The base line or reference for longitude is the Prime Meridian (as of 1884) and has an angular designation of 0° Longitude position is designated as 0° to 180° east or west of the prime meridian for a total of 360° The 180° meridian is the international date line Lines of longitude are not equally spaced At the equator each degree is 69.2 miles, but the distance narrows towards the poles until the lines converge. This is called the convergence of meridians. Why do we care? Familiarity with the spherical geographic grid and the characteristics of the arrangement of meridians and parallels is important in estimating graticule distortion on the flat map. http://ian.macky.net/pat/gallery/scale.gif Map Projections A map projection is a way to represent the curved surface of the Earth on the flat surface of a map. A good globe can provide the most accurate representation of the Earth. However, a globe isn't practical for many of the functions for which we require maps. Map projections allow us to represent some or all of the Earth's surface, at a wide variety of scales, on a flat, easily transportable surface, such as a sheet of paper. Map projections also apply to digital map data, which can be presented on a computer screen. There are hundreds of different map projections. The process of transferring information from the Earth to a map causes every projection to distort at least one aspect of the real world – either shape, area, distance, or direction. Each map projection has advantages and disadvantages; the appropriate projection for a map depends on the scale of the map, and on the purposes for which it will be used. For example, a projection may have unacceptable distortions if used to map the entire country, but may be an excellent choice for a large-scale (detailed) map of a county. The properties of a map projection may also influence some of the design features of the map. Some projections are good for small areas, some are good for mapping areas with a large east-west extent, and some are better for mapping areas with a large north-south extent. http://www-atlas.usgs.gov/articles/mapping/a_projections.html Basic Types of Projections Projections are developed from “developable surfaces.” Developable surfaces are surfaces that can be flattened to for a plane without compressing or tearing any part of it. Three commonly used developable surfaces for map projections are the cylinder, cone and plane. Developable surfaces may be described as tangent or secant case Tangent Case: Describes a developable surface that intersects the reference globe along one line, usually a parallel line of latitude. Secant Case: Describes a developable surface that intersects the reference globe along two separate lines, usually two parallel lines of latitude. The CONE, CYLINDER and PLANE are developable geometric shapes. The curved surface of the Earth can be projected on to these shapes that can be unrolled to make a flat map. http://www.geography.ccsu.edu/kyem/GEOG256/Map_projections/Projection_2_files/image001.gif http://www.innovativegis.com/basis/pfprimer/Topic7/Topic7-3.gif Map Projection Properties A map projection is a method of portraying the two-dimensional curved surface of the Earth on a flat planar surface. Projections are created to preserve one or several measurements of the following qualities: Area Shape Direction Bearing Distance Scale Each projection handles the conversion of these properties from the curved surface of a globe to the flat surface of map differently. The purpose of the map is of primary importance in choosing a projection to illustrate spatial patterns of Earth phenomena. For instance, the Mercator projection was long used for navigation or maps of equatorial regions. The cylindrical Mercator projection projects the globe onto a cylinder tangent to the Equator. Large areas become distorted which increases toward away from the Equator. Distances are true only along the Equator; special scales are provided for other latitudes for measurement. The Robinson projection uses tabular coordinates rather than mathematical formulas to make earth features look the "right" size and shape. A better balance of size and shape result is a more accurate picture of high-latitude lands like Russia, Soviet and Canada. Greenland is truer to size but compressed. http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/essentials/maps.html Comparing distortions… Lambert Conformal Conic Projection Mercator Projection Tissot Indicatrix The two maps above illustrate the effect that choosing a projection can have on how features are displayed on a map. The first map uses the Mercator projection and the second the Lambert Conformal Conic projection. Both projections preserve conformality, which is illustrated by the circular shape of the distortion ellipses on the maps. In the Lambert projection there is no distortion along the standard parallels; distortion increases as you move away from the standard parallels. The area distortion is much greater in the Mercator projection than in the Lambert as evident by the increasing size of the distortion ellipses as you move away from the equator. Area distortion in the Lambert projection is minimal due to the placement of the standard parallels. Differences in the height and width of the distortion ellipses can be used to identify the distance distortion. Distortion of distance in the Lambert projection appears minimal. In the Mercator projection, distances increase as you move north as evident by the increasing height and width of the distortion ellipses. http://www.personal.psu.edu/kwa107/projects/p01_report.htm Tissot’s Indicatricies A Tissot indicatrix ("Tiss-oh") displays properties of a map projection at a point. A projection cannot preserve all the geometric properties of space at every point: length, area, angle. The Tissot indicatrix is a figure that shows how a projection changes the geometry. Interpreting Tissot indicatrices A conformal projection preserves local angles. The indicatrix will be a circle. It might be a different size than the reference circle. Conversely, wherever the indicatrix is circular, the projection is conformal. Tissot indicatrices for the Mercator projection, showing conformality. The large distortion at extreme southern latitudes is evident in the discrepancies between the gray (reference) circles and the Tissot indicatrices (blue circles). An area-preserving transformation preserves local areas. The indicatrix may be an ellipse. In that case, the expansion along the major axis will exactly compensate for the contraction along the minor axis in order to preserve area. (The area of an ellipse is proportional to the product of its major and minor axis lengths.) Tissot indicatrices for the Peters projection, showing its equal-area property. The blue ellipses maintain constant areas, but the amount, degree, and direction of scale distortion varies dramatically from north to south, illustrating the large distortions present in this projection. An equidistant transformation preserves distances along some path, often a parallel or meridian. Along this path, the indicatrix and the reference circle will intersect. Tissot indicatrices for an azimuthal equidistant projection from the south pole. The blue ellipses meet the reference circles at their northern and southern tips, showing how scale is consistently preserved along meridians. The indicatrices make the increased eastwest distortion to the north apparent. http://www.quantdec.com/tissot/index.htm Preserving Qualities in the Map Projection Equivalent map projections preserve landmasses in their true proportions, as found on the earth’s surface. Alber’s Equivalent Conic Projection - Two standard lines at 30° N and 45° N http://www.personal.psu.edu/jwr206/albers3.png Conformal Projections preserve angular relationships around a point by preserving scale relations about that point in all directions. Conformal projections do not preserve shapes per se but preserve the angular relationships so that the scale factor of the map changes along a path between to points at the same rate. http://2.bp.blogspot.com/_R4hC30cg9Z4/S-kNj4Vvz8I/AAAAAAAAACA/v7ezl_-nzaE/s1600/conformal.jpg Equidistant Projections Preserve the principal scale from two points on the map to any other point on the map (example: if the two points are the poles, then all the meridians are straight lines that have the same principal scale). http://matplotlib.sourceforge.net/basemap/doc/html/_images/aeqd_fulldisk.png http://www.mgaqua.net/AquaDoc/Projections/img/Equidistant%20Cylindrical.jpg Azimuthal Projections Directions or azimuths are preserved from the center of the map to any other point on the map. All straight lines drawn or measured to distant points represent great circle routes http://matplotlib.sourceforge.net/basemap/doc/html/_images/nplaea.png Selecting a Map Projection Select a class of map… In theory, the selection of a map projection for a particular area can be made on the basis of: the shape of the area, the location (and orientation) of the area, and the purpose of the map. Brazil on an Azimuthal Kazakhstan on a Conic Tunisia on a Cylindrical \http://www.geo.hunter.cuny.edu/~jochen/GTECH201/ Lectures/Lec6concepts/Map%20coordinate%20system s/How%20to%20choose%20a%20projection_files/ima ge012.gif Ideally, the general shape of the mapping area should match with the distortion pattern of a specific projection. If an area is approximately circular it is possible to create a map that minimizes distortion for that area on the basis of an azimuthal projection. The cylindrical projection is best for a rectangular area and a conic projection for a triangular area (figure below). The choice of the map projection class (cylindrical, conical or azimuthal) depends largely on the general shape of the mapping area. http://gisremote.blogspot.com/2008_02_10_archive.html Select your aspect… Map aspect refers to the placement of a projection’s center with respect to the earth’s surface: common aspects are equatorial, polar, and oblique. http://www.mathworks.com/help/toolbox/map/cyl-aspect-types.gif The choice of the aspect of a map projection depends largely on the location (and orientation) of the geographic area to be mapped. Optimal is when the projection centre coincides with centre of the area, or when the projection plane is located along the main axis of the area to be mapped. Once the class and aspect of the map projection have been selected, the distortion property of the map projection has to be chosen. The most appropriate type of distortion property for a map depends largely on the purpose for which it will be used. Consider the properties of the map projection… For example… Map projections with a conformal distortion property represent angles and local shapes correctly, but as the region becomes larger, they show considerable area distortions. An example is the Mercator projection. Although Greenland is only one-eighth the size of South America, Greenland appears to be larger. Maps used for the measurement of angles (e.g. aeronautical charts, topographic maps) often make use of a conformal map projection. The Mercator projection is a cylindrical map projection with a conformal property. The area distortions are significant towards the polar regions. An example, Greenland appears to be larger but is only one-eighth the size of South America. Map projections with a equal-area distortion property on the other hand, represent areas correctly, but as the region becomes larger, it shows considerable distortions of angles and consequently shapes. Maps which are to be used for measuring areas (e.g. distribution maps) often make use of an equal-area map projection. The cylindrical equal-area projection after Lambert is a cylindrical map projection with an equal-area property. The shape distortions are significant towards the polar regions. The equidistant distortion property is achievable only to a limited degree. That is, true distances can be shown only from one or two points to any other point on the map or in certain directions. If a map is true to scale along the meridians (i.e. no distortion in NorthSouth direction) the map is equidistant along the meridians (e.g. the equidistant cylindrical projection in the figure below). If a map is true to scale along all parallels we say the map is equidistant along the parallels (i.e. no distortion in East-West direction). Maps which require correct distances measured from the centre of the map to any point (e.g. air-route, radio or seismic maps) or maps which require reasonable area and angle distortions (several thematic maps) often make use of an equidistant map projection. The equidistant cylindrical projection (also called Plate Carrée projection) is a cylindrical map projection with an equidistant property. The map is equidistant (true to scale) along the meridians (in NorthSouth direction). Both shape and area are reasonably well preserved with the exception of the polar regions. Once the class and aspect of a map projection have been selected, the choice of the property of a map projection has to be made on the basis of the purpose of the map. The ideal map projection for any country would either be an azimuthal, cylindrical, or conic projection, depending on the shape of the area, with a secant projection plane located along the main axis of the country or the area of interest. The selected distortion property depends largely on the purpose of the map. http://www.kartografie.nl/geometrics/map%20projections/body.htm Some simplified guidelines… If you want to show a country or region on or near the equator use a cylindrical projection Africa on a Cylindrical Equal-Area projection with true-scale standard parallels at 20° N and S http://www.gis.psu.edu/projection/chap5figs.html If you want to show a country or region in the temperate zones, use a conic projection. A Lambert Conformal Conic projection of the U.S. based on truescale standard parallels at 33° and 45° N. http://www.gis.psu.edu/projection/chap5figs.html If you want show a country or a region near the poles, use an azimuthal projection. The azimuthal Equidistant projection is also known as the Postel, and the Zenithal Equidistant. If you want to show a country or region with minimal distortion in the map projection, change the aspect of the map projection so that the area of interest is in the center of the map. http://www.kartografie.nl/geometrics/map%20projections/body.htm When working with world maps, o If you want to make a statistical map of the world an equalarea pseudo cylindrical projection is a good choice. Pseudo-cylindrical projections are projections in which the parallels are represented by parallel straight lines and the meridians by curves. The central meridian is the only meridian that is straight. Robinson's pseudocylindrical projection. Shapes and areas are reasonable well preserved. http://www.kartografie.nl/geometrics/map%20projections/body.htm o If you want to make a world map that others can take off right off a web page and use in a GIS program select the Plate Carrée or equi-rectangular projection. http://www.gisanalyst.com/projects/labs/export/3040_lab6_plate_carree.jpg Now that you are familiar with the basic classes of maps and some of the advantages and disadvantages of different types of projections, before you select a projection think very hard about the purpose of the map. For your purpose is it more important to show accurate area, shape, direction, distance or scale? Or do any of these matter? The following is a relatively short list of types of projections, their characteristics and uses. This may be of help to you in selecting a map projection. Also review “A Summary of Map Projections in ArcView,” which may further help narrow your decision. You may access this document through the webpage or directly through the following link. http://info.wlu.ca/~wwwgeog/special/geomatics/html/arcprojections.htm Map Projections Grouped By Class Projection Description Cylindrical Central cylindrical Map projection is perspective but not conformal nor equal area. Projected perspectively from the center of the Earth onto a cylinder tangent to the equator. Only used for teaching purposes. Equidistant cylindrical Also known as simple cylindrical or Plate Carrée. The projection is equidistant in the direction of the meridians. Parallels and meridians (half as long as the parallels) are equally spaced straight lines forming square blocks. This projection maps longitude and latitude directly into x and y, hence is sometimes called the latitude-longitude projection. In Google Earth used for display of imagery. The transverse version is known as the Cassini projection. Equirectangular Also known as Plate rectangle, a variant of Plate Carrée. Used for raster maps which store information of the whole world: each pixel represents a rectangular block of latitudelongitude coordinates. Gall-Peters Similar to Lambert's cylindrical equal-area projection, but with standard parallels at 45 degrees North and South. Lambert cylindrical equal-area It is of little use for world maps because of the distortions. Mainly used for educational purposes. Miller cylindrical Modified Mercator projection proposed by O.M. Miller. Compromise between Mercator and other cylindrical projections. Shape, area and scale distortion increases moderately away from the equator. Used in numerous world maps. Mollweide Pseudo-cylindrical projection. Map is equal area. Occasionally used in thematic world maps. Mercator Conformal map projection. Designed for navigational use; standard for marine charts. Recommended use for conformal mapping of regions predominantly bordering the equator. Often inappropriately used as a world map. Trasverse Mercator Also called Gauss Conformal, or Gauss Krüger. Transverse form (transverse cylinder) of the Mercator projection. Used for topographic maps at scales from 1: 20,000 to 1: 250,000. Recommended for conformal mapping of regions that are predominantly north-south in extent. Universal Transverse A version of the Transverse Mercator, but one with a secant map surface. It divides the world into 60 narrow longitudinal zones of 6 degrees. Widely used standard for topographic Mercator (UTM) maps and military maps. Azimuthal Azimuthal equidistant Distances measured from the centre of the map to any point are correct and the bearing of any point from the center is correct (this applies to all azimuthal maps). Commonly used in the polar aspect for maps of polar regions and the Northern and Southern hemispheres. The oblique aspect is frequently used for world or air-route maps centered on important cities and occasionally for maps of continents. Gnomonic Map is perspective and neither conformal nor equal area. Area, shape, distance and direction distortions are extreme. It is used to show great circle paths as straight lines and thus to assist navigators and aviators. Hammer-Aitoff A variant of Lambert azimuthal equal-area. Used for thematic maps of the whole world. Lambert azimuthal equal-area Used for maps of continents and hemispheres. Also suited for regions extending equally in all directions from a center point, such as Asia and the Pacific Ocean. Recommended to the European Commission for statistical analysis and display. Orthographic Known by Egyptians and Greeks 2000 years ago. Map is perspective and neither conformal nor equal area. Only one hemisphere can be shown. The Earth appears as it would on a photograph from space. Stereographic Apparently invented by Hipparchus (2nd century bc). Used in combination with UTM projection as Universal Polar Stereographic (UPS) for mapping poles and in navigation charts for latitudes above 80°. Recommended for conformal mapping of regions that are approximately circular in shape; a modified version of the stereographic projection is used in the Netherlands for large-scale and topographic maps. Conical Albers equal area conic It is equal to Lambert's equal area conic, but has two standard parallels (secant cone). Excellent for mid-latitude distribution maps. The projection does not contain the noticeable distortions of the Lambert projections. Frequently used for maps of the United States, for thematic maps and for world atlases. Lambert conformal conic Lambert conformal conic, also called conical orthomorphic (Lambert, 1972). Extensively used for large-scale mapping of regions predominantly east-west in extent. Further widely used for topographic maps. Polyconic or American polyconic (Hassler, ± 1820). Map is neither conformal nor equal area, but each parallel is true to scale. The sole projection used for large scale mapping of the United States by the USGS until the 1950's. Simple conic Also known as equidistant conic. Meridians are true to scale (i.e. no distortion in northsouth direction). The most common projection in atlases for small countries. Other projections Sinusoidal Used since 16th century. Also called Sanson-Flamsteed or Mercator equal area projection. Pseudo-cylindrical projection. Map is equal area. Used in atlas maps of South America and Africa. Occasionally used for world maps. Modifications are called sinusoidal interrupted and sinusoidal 3x interrupted. Van der Grinten Shows the entire Earth within one circle. All areas, shapes and angles are greatly distorted. Winkel Tripel Used in several atlases. A triple compromise of reduced shape, area and distance distortion. Selected by the National Geographic Society (NGS) for its new reference world map, in place of the Robinson projection. http://www.kartografie.nl/geometrics/map%20projections/body.htm Scale and Generalization Scale provides an indication of the amount of reduction that has taken place on a map. Generalization refers to the process of reducing the information content of maps due to scale change, map purpose, intended audience or technical constraints. Cartographic Generalization Conceptual Objectives (Why to generalize) Cartometric Evaluation (When to generalize) Fundamental Operations (How to generalize) Why Generalization? Reduce complexity Maintain spatial accuracy of critical components Maintain attribute accuracy of critical components Maintain aesthetic qualities Maintain a logical hierarchy When to Generalize… When there is congestion- too many objects that compressed into too small a space When there is coalescence-when features collide When there is conflict-when there is inconsistency among features When there is complication-when features become unduly complicated How to Generalize… Simplification- weed out unnecessary data http://gis.unbc.ca/courses/geog205/labs/lab2/images/image027.jpg Smoothing-shift position of points to improve the appearance of a feature http://www.gis.unbc.ca/courses/geog205/lectures/generalization/dougpoik.jpg Aggregation-merge point features and display them as areal units http://www.gis.unbc.ca/courses/geog205/lectures/generalization/gen2.jpg Amalgamation-fuse neighboring polygons http://www.satprints.com/ProductImages/Minnesota.jpg http://www.worldmapsinfo.com/mapimage/minnesota.jpg Collapse-Convert the geometry http://www.gis.unbc.ca/courses/geog205/lectures/generalization/index.php Refine-reduce multiple sets of features Merge-fuse groups of lines http://1.bp.blogspot.com/_zwaRuUhzrcM/SZbRXWRLGhI/AAAAAAAAAr0/0RI4rslenlc/s320/Picture 6.png http://www.zonu.com/imapa/inmonacional/images/Satellite_Image_Photo_Sierra_Segura_Sierra_Filabr es_Sierra_Nevada_Spain.jpg Exaggerate-amplify a part of a feature to make it visible http://www.dcda.org.uk/Cartography/3detailed.html Enhance-change the symbolization to emphasize importance http://www.lib.utexas.edu/maps/usgs_ref/usgs2.jpg Displace-pull features apart to prevent coalescence http://www.geocomputation.org/2000/GC034/gc034_100.jpg Issues of generalization increase with reductions in scale. The smaller the scale, the more generalization will be needed. The level of generalization is also part and parcel part of the design process dictated by the purpose of the map and the intended audience. Whereas the majority of these issues are addressed through the use feature specific algorithms, fortunately for you, many are now integrated into GIS software. The issue becomes how to apply them, and even after you have applied them how much manual readjustment you may be required to make.