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Map Projection types Coordinate system Geodesy Geodetic Datum Geodesy Datum (Geodesy) Map projection Map projection Map Projections: From Spherical Earth to Flat Map Cartographical Map Projections flat-Earth and round-Earth representations Specifying Grids, Ellipsoids, and Map Projections From Spherical Earth to Flat Map Map Projections Normal Earth Shape: Sphere, Ellipsoid Map Projections - album GIS map projection Projections Tissots index how the circle will be deformed on map projection dependet from its latitude and longitude. Pink Schapes shows how the circle in 500 miles will be deformed (size and form). Mercator Projection THE MERCATOR PROJECTION IS CONFORMAL NOT EQUAL AREA (FOR LARGE POLYGONES) CYLINDRICAL PROJECTION, AND THE SCALE INCREASES FROM THE EQUATOR TO THE POLES, WHERE IT BECOMES INFINITE Gall-Peters_projection The Gall-Peters projection is one specialization of a configurable equal-area map projection known as the equal-area cylindric or cylindric equal-area projection. The Gall-Peters achieved considerable notoriety in the late 20th century as the centerpiece of a controversy surrounding the political implications of map design. Maps based on the projection continue to see use in some circles and are readily available, though few major map publishers produce them. Mollweide_projection The Mollweide projection is a pseudocylindrical map projection generally used for global maps of the world (or sky). Also known as the Babinet projection, homolographic projection, or elliptical projection. As its more explicit name Mollweide equal area projection indicates, it sacrifices fidelity to angle and shape in favor of accurate depiction of area. It is used primarily where accurate representation of area takes precedence over shape, for instance small maps depicting global distributions. Sinusoidal projection The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson-Flamsteed or the Mercator equal-area projection Robinson projection - Neither Conformal or Equal-area The Robinson projection is a map projection of a world map, which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image. Map Projection Miller cylindrical projection The Miller cylindrical projection is a modified Mercator projection,Cylindrical, arbitrary compromise to Mercator Projections Classified by Geometry Plate carree Also known as Equirectangular, Equidistant Cylindrical, Simple Cylindrical, or Rectangular, this projection is very simple to construct because it forms a grid of equal rectangles. Because of its simple calculations, its usage was more common in the past. In this projection, the polar regions are less distorted in scale and area than they are in the Mercator projection. Plate carree GIS: Metadata for Map Projection Decisions Map Projections Exercise Properties, and then on Coordinate Systems tab and record the following information Central Meridian __ 0.000000__________ Standard Parallel _1: 45.000000___________ Where is Peters most accurate????? ON STANDART PARALLEL 45 Peters is an area accurate map, standard parallel 45 non-conformal equal area projection Standard parallel refers to the latitude on a globe where this projection is tangent or secant and therefore is most accurate. While we are still at this Properties screen, well try out a few other projections. Stay with current Projections of the World category for now, so you can look at different world projections. In the Properties dialog box, with the Coordinate Systems tab active, select Modifications, then Name, press the down arrow to see list of other types Select Mollweide and press OK and OK. Wait patiently! Is Mollweide an equal area projection? YES Right-click on Data Frame name, then select: Properties>Modifications>Name Select Sinusoidal and press OK and OK. Wait patiently. Is this an equal area projection? YES Summary question: Is the Peters Projection the only easily available equal area projection? Be ready to explain. NO, WE HAVE MANY OTHER EQUAL_AREA PROJECTIONS Extension: Try other world projections (e.g., Robinson, Miller, Plate Caree) and decide whether they are conformal, equal area, or neither. ROBINSON NEITHER MILLER NEITHER PLATE CAREER NEITHER (THE EQUIDISTANT CYLINDRICAL PROJECTION) Remember that conformal shows our pink shapes as circles but does not represent relative area properly, and equal area maintains proper relationships among areas of countries or regions. When you are done with this tutorial: File> Exit NO, dont save changes Tissots index pictures Tissots index how the circle will be deformed on map projection dependet from its latitude and longitude. Pink Schapes shows how the circle in 500 miles will be Projections Projections Datums Data Needs, Data Collection, and Verification Georeferenced Data Analysis Tutorial Georeferenced Data Geographic coordinate system Cartesian coordinate system Ordnance Survey National Grig System British national grid reference system British Grid Reference System Postcodes in the United Kingdom Postal code The UK Postcode System The Postcode Map of the United Kingdom Aerial Photographs and Satellite Images Aerial Photographs and Satellite Images USGS: Your source for science you can use Free Maps GENERAL COORDINATE SYSTEMS INTRODUCTION B. PLANE COORDINATE SYSTEMS - CARTESIAN COORDINATES Determining Coordinates Measuring Distance C. STORING COORDINATES Integers vs real numbers Computer Precision Precision of Cartesian Coordinates Propagation of coordinate errors D. PLANE COORDINATE SYSTEMS - POLAR COORDINATES E. GLOBAL COORDINATES - LATITUDE AND LONGITUDE Determining Coordinates Important Terms Measuring distance Question of Precision F. DETERMINING POSITION REFERENCES DISCUSSION AND EXAM QUESTIONS NOTES MAP PROJECTIONS INTRODUCTION Relevance to GIS B. DISTORTION PROPERTIES Tissot''s Indicatrix Conformal (Orthomorphic) Equal area (Equivalent) Equidistant C. FIGURE OF THE EARTH 1. Plane 2. Sphere 3. Spheroid or ellipsoid of rotation Accuracy of figures used D. GEOMETRIC ANALOGY Developable surfaces 1. Planar or azimuthal 2. Conic 3. Cylindrical 4. Non-Geometric (Mathematical) projections E. UNIVERSAL TRANSVERSE MERCATOR (UTM) Transverse Mercator Projection Zone System Distortion Coordinates Advantages Disadvantages F. STATE PLANE COORDINATES (SPC) Advantages Disadvantages Use in GIS REFERENCES DISCUSSION AND EXAM QUESTIONS NOTES This unit needs many overhead illustrations. Of course, the best figures are in commercially published books. To avoid copyright infringements, we have not included the masters for overheads needed here. Instead, you are directed to the References which lists several basic texts with good illustrations. We have included suggested overheads giving references and specific page numbers to help you locate suitable figures. UNIT 27 - MAP PROJECTIONS Compiled with assistance from Vicki Chmill, University of California, Santa Barbara A. INTRODUCTION a map projection is a system in which locations on the curved surface of the earth are displayed on a flat sheet or surface according to some set of rules mathematically, projection is a process of transforming global location (j,l) to a planar position (x,y) or (r,q) for example, the transformations for Mercator projection are: x = l y = loge tan(p/4 + j/2) Relevance to GIS maps are a common source of input data for a GIS often input maps will be in different projections, requiring transformation of one or all maps to make coordinates compatible thus, mathematical functions of projections are needed in a GIS often GIS are used for projects of global or regional scales so consideration of the effect of the earth''s curvature is necessary monitor screens are analogous to a flat sheet of paper thus, need to provide transformations from the curved surface to the plane for displaying data B. DISTORTION PROPERTIES angles, areas, directions, shapes and distances become distorted when transformed from a curved surface to a plane all these properties cannot be kept undistorted in a single projection usually the distortion in one property will be kept to a minimum while other properties become very distorted Tissot''s Indicatrix is a convenient way of showing distortion imagine a tiny circle drawn on the surface of the globe on the distorted map the circle will become an ellipse, squashed or stretched by the projection the size and shape of the Indicatrix will vary from one part of the map to another we use the Indicatrix to display the distorting effects of projections Conformal (Orthomorphic) Reference: Mercator projection (Strahler and Strahler 1987, p. 15) a projection is conformal if the angles in the original features are preserved over small areas the shapes of objects will be preserved preservation of shape does not hold with large regions (i.e. Greenland in Mercator projection) a line drawn with constant orientation (e.g. with respect to north) will be straight on a conformal projection, is termed a rhumb line or loxodrome parallels and meridians cross each other at right angles (note: not all projections with this appearance are conformal) the Tissot Indicatrix is a circle everywhere, but its size varies conformal projections cannot have equal area properties, so some areas are enlarged generally, areas near margins have a larger scale than areas near the center Equal area (Equivalent) Reference: Lambert Equal Area projection (Maling 1973, p. 72) the representation of areas is preserved so that all regions on the projection will be represented in correct relative size equal area maps cannot be conformal, so most earth angles are deformed and shapes are strongly distorted the Indicatrix has the same area everywhere, but is always elliptical, never a circle (except at the standard parallel) Equidistant Reference: Conic Equidistant projection (Maling 1973, p. 151) cannot make a single projection over which all distances are maintained thus, equidistant projections maintain relative distances from one or two points only i.e., in a conic projection all distances from the center are represented at the same scale C. FIGURE OF THE EARTH a figure of the earth is a geometrical model used to generate projections; a compromise between the desire for mathematical simplicity and the need for accurate approximation of the earth''s shape types in common use 1. Plane assume the earth is flat (use no projection) used for maps only intended to depict general relationships or for maps of small areas at scales larger than 1:10,000 planar representation has little effect on accuracy planar projections are usually assumed when working with air photos 2. Sphere assume the earth is perfectly spherical does not truly represent the earth''s shape 3. Spheroid or ellipsoid of rotation Reference: Ellipsoid of rotation (Maling 1973, p. 2) this is the figure created by rotating an ellipse about its minor axis the spheroid models the fact that the earth''s diameter at the equator is greater than the distance between poles, by about 0.3% at global scales, the difference between the sphere and spheroid are small, about equal to the topographic variation on the earth''s surface with a line width of 0.5 mm the earth would have to be drawn with a radius of 15 cm before the two models would deviate the difference is unlikely to affect mapping of the globe at scales smaller than 1:10,000,000 Accuracy of figures used the spheroid is still an approximation to the actual shape the earth is actually slightly pear shaped, slightly larger in the southern hemisphere, and has other smaller bulges therefore, different spheroids are used in different regions, each chosen to fit the observed datum of each region accurate conversion between latitude and longitude and projected coordinates requires knowledge of the specific figures of the earth that have been used the actual shape of the earth can now be determined quite accurately by observing satellite orbits satellite systems, such as GPS, can determine latitude and longitude at any point on the earth''s surface to accuracies of fractions of a second thus, it is now possible to observe otherwise unapparent errors introduced by the use of an approximate figure for map projections MAP PROJECTIONS D. GEOMETRIC ANALOGY Developable surfaces the most common methods of projection can be conceptually described by imagining the developable surface, which is a surface that can be made flat by cutting it along certain lines and unfolding or unrolling it Reference: Developable surfaces (Maling 1973, pp. 55-57) the points or lines where a developable surface touches the globe in projecting from the globe are called standard points and lines, or points and lines of zero distortion. At these points and lines, the scale is constant and equal to that of the globe, no linear distortion is present if the developable surface touches the globe, the projection is called tangent if the surface cuts into the globe, it is called secant where the surface and the globe intersect, there is no distortion where the surface is outside the globe, objects appear bigger than in reality - scales are greater than 1 where the surface is inside the globe, objects appear smaller than in reality and scales are less than 1 Reference: Projection equations (Maling 1973, pp. x-xi and 234-245) note: symbols used in the following: l - longitude j - latitude c - colatitude (90 - lat) h - distortion introduced along lines of longitude k - distortion introduced along lines of latitude (h and k are the lengths of the minor and major axes of the Indicatrix) commonly used developable surfaces are: 1. Planar or azimuthal a flat sheet is placed in contact with a globe, and points are projected from the globe to the sheet mathematically, the projection is easily expressed as mappings from latitude and longitude to polar coordinates with the origin located at the point of contact with the paper formulas for stereographic projection (conformal) are: r = 2 tan(c / 2) q = l h = k = sec2(c / 2) References: Azimuthal projections (Strahler and Strahler 1987, p. 13, Robinson et al 1984, p. 102) stereographic projection gnomic projection Lambert''s azimuthal equal-area projection orthographic projection 2. Conic the transformation is made to the surface of a cone tangent at a small circle (tangent case) or intersecting at two small circles (secant case) on a globe mathematically, this projection is also expressed as mappings from latitude and longitude to polar coordinates, but with the origin located at the apex of the cone formulas for equidistant conical projection with one standard parallel (j0 , colatitude c0) are: r = tan(c0) + tan(c - c0) q = n l n = cos(c0) h = 1.0 k = n r / sin(c) Examples References: Conic projections (Strahler and Strahler 1987 p. 14, Maling 1973, p. 164) Alber''s conical equal area projection with two standard parallels Lambert conformal conic projection with two standard parallels equidistant conic projection with one standard parallel 3. Cylindrical developed by transforming the spherical surface to a tangent or secant cylinder mathematically, a cylinder wrapped around the equator is expressed with x equal to longitude, and the y coordinates some function of latitude formulas for cylindrical equal area projection are: x = l y = sin(j) k = sec(j) h = cos(j) Examples References: Mercator and Lambert projections (Strahler and Strahler 1987, p. 15, Maling 1973, p. 72) note: Mercator Projection characteristics meridians and parallels intersect at right angles straight lines are lines of constant bearing - projection is useful for navigation great circles appear as curves 4. Non-Geometric (Mathematical) projections some projections cannot be expressed geometrically have only mathematical descriptions Examples Reference: Non-geometric projections (Robinson et al, 1984, p. 97) Molleweide Eckert E. UNIVERSAL TRANSVERSE MERCATOR (UTM) UTM is the first of two projection based coordinate systems to be examined in this unit UTM provides georeferencing at high levels of precision for the entire globe established in 1936 by the International Union of Geodesy and Geophysics adopted by the US Army in 1947 adopted by many national and international mapping agencies, including NATO is commonly used in topographic and thematic mapping, for referencing satellite imagery and as a basis for widely distributed spatial databases Transverse Mercator Projection results from wrapping the cylinder around the poles rather than around the equator the central meridian is the meridian where the cylinder touches the sphere theoretically, the central meridian is the line of zero distortion by rotating the cylinder around the poles the central meridian (and area of least distortion) can be moved around the earth for North American data, the projection uses a spheroid of approximate dimensions: 6378 km in the equatorial plane 6356 km in the polar plane Zone System Reference: UTM zones (Strahler and Strahler 1987, p. 18) in order to reduce distortion the globe is divided into 60 zones, 6 degrees of longitude wide zones are numbered eastward, 1 to 60, beginning at 180 degrees (W long) the system is only used from 84 degrees N to 80 degrees south as distortion at the poles is too great with this projection at the poles, a Universal Polar Stereographic projection (UPS) is used each zone is divided further into strips of 8 degrees latitude beginning at 80 degrees S, are assigned letters C through X, O and I are omitted Distortion to reduce the distortion across the area covered by each zone, scale along the central meridian is reduced to 0.9996 this produces two parallel lines of zero distortion approximately 180 km away from the central meridian scale at the zone boundary is approximately 1.0003 at US latitudes Coordinates coordinates are expressed in meters eastings (x) are displacements eastward northings (y) express displacement northward the central meridian is given an easting of 500,000 m the northing for the equator varies depending on hemisphere when calculating coordinates for locations in the northern hemisphere, the equator has a northing of 0 m in the southern hemisphere, the equator has a northing of 10,000,000 m Reference: UTM coordinates (Strahler and Strahler 1987, p. 19) Advantages UTM is frequently used consistent for the globe is a universal approach to accurate georeferencing Disadvantages full georeference requires the zone number, easting and northing (unless the area of the data base falls completely within a zone) rectangular grid superimposed on zones defined by meridians causes axes on adjacent zones to be skewed with respect to each other problems arise in working across zone boundaries no simple mathematical relationship exists between coordinates of one zone and an adjacent zone F. STATE PLANE COORDINATES (SPC) SPCs are individual coordinate systems adopted by U.S. state agencies each state''s shape determines which projection is chosen to represent that state e.g. a state extended N/S may use a Transverse Mercator projection while a state extended E/W may use a Lambert Conformal Conic projection (both of these are conformal) projections are chosen to minimize distortion over the state a state may have 2 or more overlapping zones, each with its own projection system and grid units are generally in feet Advantages SPC may give a better representation than the UTM system for a state''s area SPC coordinates may be simpler than those of UTM Disadvantages SPC are not universal from state to state problems may arise at the boundaries of projections Use in GIS many GIS have catalogues of SPC projections listed by state which can be used to choose the appropriate projection for a given state REFERENCES Maling, D.H., 1973. Coordinate Systems and Map Projections, George Phillip and Son Limited, London. Robinson, A.H., R.D. Sale, J.L. Morrison and P.C. Muehrcke, 1984, Elements of Cartography, 5th edition, John Wiley and Sons, New York. See pages 56-105. Snyder, J.P., 1987. Map Projections - A Working Manual, US Geological Survey Professional Paper 1395, US Government Printing Office, Washington. Strahler, A.N. and A.H. Strahler, 1987. Modern Physical Geography, 3rd edition, Wiley, New York. See pages 3-8 for a description of latitude and longitude and various appendices for information on coordinate systems. AFFINE AND CURVILINEAR TRANSFORMATIONS A. INTRODUCTION B. AFFINE TRANSFORMATION PRIMITIVES 1. Translation 2. Scaling 3. Rotation 4. Reflection C. COMPLEX AFFINE TRANSFORMATIONS D. AFFINE TRANSFORMATIONS IN GIS Simple Example City Fire Study Example E. CURVILINEAR TRANSFORMATIONS REFERENCES DISCUSSION AND EXAM QUESTIONS NOTES UNIT 28 - AFFINE AND CURVILINEAR TRANSFORMATIONS A. INTRODUCTION coordinate transformations are required when you need to register different sets of coordinates for objects in the same area that may have come from maps of different (and sometimes unknown) projections will need to transform one or more sets of coordinates so that they are represented in the same coordinate system as other sets are two ways to look at coordinate transformations: 1. move objects on a fixed coordinate system so that the coordinates change diagram 2. hold the objects fixed and move the coordinate system this is the more useful way to consider transformations for GIS purposes diagram (x,y) is the location of the object before transformation (in the old coordinate system) (u,v) is the location of the object after transformation (in the new coordinate system) are two major groups of transformations: affine transformations are those which keep parallel lines parallel as we will see they are a class of transformations which have 6 coefficients curvilinear transformations are higher order transformations that do not necessarily keep lines straight and parallel these transformations may require more than 6 coefficients B. AFFINE TRANSFORMATION PRIMITIVES affine transformations keep parallel lines parallel are four different types (primitives): handout - Affine transformation primitives 1. Translation origin is moved, axes do not rotate diagram u = x - a v = y - b origin is moved a units parallel to x and b units parallel to y 2. Scaling both origin and axes are fixed, scale changes diagram u = cx v = dy scaling of x and y may be different if the scaling is different, the shape of the object will change 3. Rotation origin fixed, axes move (rotate about origin) diagram u = x cos(a) + y sin(a) v = -x sin(a) + y cos(a) (note: a is measured counterclockwise) 4. Reflection coordinate system is reversed, objects appear in mirror image diagram to reverse y, but not x: u = x v = c - y this transformation is important for displaying images on video monitors as the default coordinate system has the origin in the upper left corner and coordinates which run across and down C. COMPLEX AFFINE TRANSFORMATIONS usually a combination of these transformations will be needed the combined equations are: u = a + bx + cy v = d + ex + fy often cannot actually separate the needed transformations into one or more of the primitives defined above as one transformation will cause changes that appear to be caused by another transformation, and order is important e.g. translation followed by scale change is not the same as scale change followed by translation, has different effect exception: reflection has occurred if bf < ce AFFINE TRANSFORMATIONS IN GIS . AFFINE TRANSFORMATIONS IN GIS frequently, when developing spatial databases for use in GIS, data will be provided on map sheets which use unknown or inaccurate projections in order to register two data sets, a set of control points or tics must be identified that can be located on both maps must have at least 3 control points since 3 points provide 6 values which can be used to solve for the 6 unknowns control points must not be on a straight line (not collinear) Simple Example control points are: x y u v 0 0 1 10 1 0 1 9 0 1 3 10 1 1 3 9 solution: hint: note that y and u are related, a change in y always produces the same change in u; x and v are similarly related therefore: v = 10 - x u = 1 + 2y complete equations are: u = 1 + 0x + 2y v = 10 - 1x + 0y note that bf = 0, ce = -2 therefore, bf > ce, there is no reflection involved City Fire Study Example handout - City fire study example (2 pages) Problem: given: two sets of spatial data: 1. Census tract boundaries in the city with coordinates given in UTM 2. Fire locations in the city plotted on a crude road map UTM is to be the destination system count the number of fires in each census tract and analyze the numbers of fires in relation to characteristics described by census data e.g. is number of fires related to number of houses constructed of wood? Solution: use major street intersections as control points for destination system, determine UTM coordinates of the intersections for the fire map system, use any arbitrary rectangular coordinate system (i.e. digitizer table) with fixed origin and axes using the two sets of coordinates for the control points and linear regression techniques, solve for the 6 coefficients in the two affine transformation equations examine the residuals to evaluate the accuracy of the analysis the spatial distribution of residuals may indicate weaknesses of the model may show that map has been distorted unevenly magnitude of the residuals gives an estimate of the accuracy of the transformation in this example, the magnitude of residuals indicates an accuracy of 150 m i.e. UTM coordinates of fire locations are +/- 150 m from their locations as indicated on the road map for this analysis, this is sufficient accuracy E. CURVILINEAR TRANSFORMATIONS simple linear affine transformation equations can be extended to higher powers: u = a + bx + cy + gxy or u = a + bx + cy + gx2 or u = a + bx + cy + gx2 + hy2 + ixy equations of this form create curved surfaces provides rubbersheeting in which points are not transformed evenly over the sheet, transformations are not affine (parallel lines become non-parallel, possibly curved) rubber-sheet transformations may also be piecewise map divided into regions, each with its own transformation equations equations must satisfy continuity conditions at the edges of regions curvilinear transformations usually give greater accuracy accuracy in the sense that when used to transform the control points or tics, the equations faithfully reproduce the known coordinates in the other system however if error in measurement is present, and it always is to some degree, then greater accuracy may not be desirable a curvilinear transformation may be more accurate for the control points, but less accurate on average REFERENCES Goodchild, M.F., 1984. "Geocoding and Geosampling," Spatial Statistics and Models, G.L. Gaile and C.J. Willmott, eds., Reidel Publishing Company, Dordrecht, Holland, pp. 33-53. DISCUSSION AND EXAM QUESTIONS 1. A map has been digitized using an origin in the lower left corner. The x axis is the lower edge of the map, and the y axis is the left edge. The study area ranges from 0 to 100 in x, and 0 to 50 in y. Calculate the transformations necessary to show this area on a screen containing 640 columns and 480 rows, with the columns and rows numbered from the top left corner, assuming that positions on the screen are referred to by row and column numbers. 2. Describe the transformations between x and y and UTM Northing and UTM Easting given in the text of the unit for the fire study example in terms of (a) translation of the origin, (b) scaling, (c) rotation and (d) reflection. 3. Discuss the factors which contribute to lack of fit between maps of the same area, and failure of affine transformations to register maps perfectly using control points. Discuss the relative importance of each of these factors. 4. Discuss the criteria you would use to select control points, including the number and distribution of such points, in setting up a coordinate transformation |
