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Copyright, © Qiming Zhou GEOG1150. Cartography Map Scale and Projection

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Map Scale and Projections2 Map scale and transformations Distortions resulting from map transformations Analysis and visualisation of distortion Choosing a map projection Commonly used map projections For details about the contents of this topic, please refer to Peter H. Dana, 1999: Map Projection Overview, (http://www.colorado.edu/geography/gcraft/notes/mappr oj/mapproj_f.html).Map Projection Overview

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Map Scale and Projections3 Map scale and transformations Map, to be useful, are necessarily smaller than the areas mapped. Map scale - the ratio between measurements on the map to those on the earth. Transformation from globe to map means that the map’s scale will vary from place to place.

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Map Scale and Projections4 Globe and a flat map Comparison between globe and flat map of the earth. From Robinson, et al., 1995

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Map Scale and Projections5 Statements of scale Representative fraction (RF): a simple ratio (e.g. 1:20,000) Verbal statement Graphical or bar scale (scale bar) Map Scale 1:1,000, km Represents 1 km 2

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Map Scale and Projections6 Scale factor The scale factor (SF) at a point is computed by: On the reference globe, SF = 1. As the earth is essentially spherical, while the view represented by a map is orthogonal, the SF will be various from place to place. The sphere and the plane are not applicable.

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Map Scale and Projections7 Scale factor of distance a b' c’ d’ e’ f’ g’ i’ h’ b c d e f g h i 90° Orthographic projection of an arc to a tangent straight line. After Robinson, et al., 1995

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Map Scale and Projections8 Scale factor of direction b’a dc’ b c ad b’c’ bc Projection of rectangle abcd to rectangle ab’c’d with side ad held constant. Left: the perspective view shows the geometric relations of the two rectangles. Up: the relation of the two rectangles when they are each viewed orthogonally. After Robinson, et al., 1995

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Map Scale and Projections9 Whenever the spherical surface is transformed to a plane, it is certain that all of the geometrical relationships on the sphere cannot be entirely duplicated. The major alternations have to do with angles, areas, distances and directions. Distortions resulting from map transformations

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Map Scale and Projections10 Tissot’s indicatrix 0 A A’ B B’ B” a b M M’ P P’ U’ U OA = OB = 1.0 For the ellipse OA’ = a = 1.25 OB’ = b = 0.80 For the dashed OA’ = a = 1.25 OB” = b = 1.25 = U - U’

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Map Scale and Projections11 Transformation The great circle (solid line) and the rhumb (dashed line) as they appear on Mercator’s projection. From Robinson, et al., 1995

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Map Scale and Projections12 Parallels are parallel. Parallels are spaced equally on meridians. Meridians and great circles on a globe appear as straight lines. Meridians converge toward the poles and diverge toward the equator. Meridians are equally spaced on the parallels, but their spacing decreases to the pole. Evaluation of visual characteristics of the earth’s coordinate systems Analysis and visualisation of distortion

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Map Scale and Projections13 Meridians and parallels are equally spaced at or near the equator. Meridians at 60° latitude are half as far apart as parallels. Parallels and meridians always intersect at right angles. The surface area bounded by any two parallels and two meridians is the same anywhere between the same parallels. Evaluation of visual characteristics of the earth’s coordinate systems Analysis and visualisation of distortion (cont.)

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Map Scale and Projections14 Visual analyses Different centerings of the sinusoidal projection produce different appearing graticules. Nevertheless, the arrangement or pattern of the deformation is the same on all, since the same system of transformation is employed. From Robinson, et al., 1995 EquatorialOblique Polar

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Map Scale and Projections15 Graphical portrayal of distortions A head drawn on the Mollweide projection (top) has been transferred to Mercator’s projection (centre) and to the cylindrical equal-area projection with standard parallels at 30° (bottom). From Robinson, et al., 1995

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Map Scale and Projections16 Distortions of directions Selected great circle arcs on an equatorial case of the sinusoidal projection showing their departures from straight lines. Each uninterrupted and interrupted arc is 150° long. Courtesy W.R. Tobler, cited in Robinson, et al., 1995

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Map Scale and Projections17 Choosing a map projection Cartographers need to be thoroughly familiar with map projections. Cartographers frequently transfer data from one projection to another.

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Map Scale and Projections18 Guidelines Projection’s major property: conformality, equivalence, azimuthality, reasonable appearance, etc. Amount and arrangement of distortion: Mean distortion (angular or area). Map series have special projection requirements. To show the same pattern of distortion for large areas as for small areas. The overall shape of the area.

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Map Scale and Projections19 Common distortion patterns on map projection Azimuthal patterns of deformation. (a) The pattern when the plane is tangent to the sphere at a point, and (b) the pattern when the plane intersects the sphere. Conical patterns of deformation. (a) The pattern when the cone is tangent to one small circle, and (b) the pattern when the cone intersects the sphere along to small circles.

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Map Scale and Projections20 Cylindrical patterns of deformation. (a) The pattern when the cylinder is tangent to a great circle, and (b) the pattern when the cylinder is secant. Common distortion patterns on map projection (cont.)

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Map Scale and Projections21 Example of equatorial world map projections A few of the many equivalent world map projections. (a) cylindrical equal-area with standard parallels at 30°N and S latitude; (b) sinusoidal projection; and (c) Mollweide’s projection. From Robinson, et al., 1995

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Map Scale and Projections22 Minimising distortion Modified-stereographic conformal projection of Alaska, with lines of constant scale superimposed. From Robinson, et al., 1995

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Map Scale and Projections23 Minimising distortion (cont.) Modified- stereographic conformal projection of 48 United States, bounded by a near rectangular area of constant scale.. From Robinson, et al., 1995

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Map Scale and Projections24 Commonly used map projections Conformal projections Mercator Transverse Mercator Lambert’s conformal conic (with two standard parallels) Equal-area projections Alber’s equal-area Lambert’s equal-area

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Map Scale and Projections25 Mercator’s projection

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Map Scale and Projections26 Mercator and transverse mercator projections Right: The conceptual cylinder for the normal form of Mercator’s projection is arranged parallel to the axis of the sphere. Up: To develop the transverse Mercator projection, the cylinder is turned.

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Map Scale and Projections27 Lambert’s conformal conic projection

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Map Scale and Projections28 World equal-area projections Cylindrical equal-area Sinusoidal Mollweide’s projection Goode’s homolosine projection

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Map Scale and Projections29 Albers’ equal-area conic projection

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Map Scale and Projections30 Goode’s homolosine projection Goode’s homolosine projection is an interrupted union of the sinusoidal projection equator-ward of approximate 40° latitude and the pole-ward zones of Mollweide’s projection. From Robinson, et al., 1995

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Map Scale and Projections31 Condensing (a) An interrupted flat polar quartic equal- area projection of the entire earth. By deleting unwanted areas, i.e. condensing as in (b), additional scale is obtained within a limiting width. From Robinson, et al., 1995

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Map Scale and Projections32 Azimuthal projections The stereographic: conformal. Lambert equal-area: equal-area. Azimuthal equidistant: the linear scale is uniform along the radiating straight lines through the centre. The orthographic: perspective view. The gnomonic: all great-circle arcs are represented as straight lines anywhere on the projection.

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Map Scale and Projections33 Class of azimuthal projections The hypothetical positions of the points of projection for the class of azimuthal projections: (1) gnomonic, (2) stereographic, (3) equidistant, (4) equivalent, and (5) orthographic. From Robinson, et al., 1995

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Map Scale and Projections34 Comparison of the classes “The important thing to note is that the only variation among the projections is in the spacing of the parallels. That is, the only difference among them is the radial scale from the centre”. Robinson, et al., 1995

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Map Scale and Projections35 The five well-known azimuthal projections (a)Stereographic; (b)Lambert’s equal-area; (c)azimuthal equidistant; (d)orthographic; and (e)gnomonic.

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Map Scale and Projections36 Azimuthal equidistant

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Map Scale and Projections37 Other map projections Plane chart (equidistant cylindrical) Simple conic Polyconic projection Robinson’s projection Space oblique Mercator projection

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Map Scale and Projections38 Plane chart

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Map Scale and Projections39 Simple conic projection

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Map Scale and Projections40 Polyconic projection

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Map Scale and Projections41 Polyconic projection (cont.) The distribution of scale factors on a polyconic projection in the vicinity of 40° latitude.

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Map Scale and Projections42 Robinson’s projection “The Robinson projection is neither conformal nor equal- area but a compromise between the two”. Robinson, et al., 1995

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Map Scale and Projections43 Space oblique Mercator The central line of the space oblique Mercator projection is slightly curved and oblique to the equator. It crosses the polar area at about 81°N and S latitude. Along this line, which represents the Landsat ground-track, the SF is essentially 1.0. The conceptual basis for the projection is similar to that of the transverse Mercator projection, but the central line is not a great circle. From Robinson, et al., 1995

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