Map Projections and Map Coordinate Systems

Transcription

1 Map Projections and Map Coordinate Systems Jamie Wolfe CITE Marshall University Huntington, WV Jawolfe@marshall.edu IS 645 Introduction to GIS Lecture 03, May 23, 2000

2 Today s class topics Map Projections Cylindrical Conical Azimuthal Map Distortions Map Coordinate Systems Latitude, Longitude, height Earth Centered, Earth Fixed XYZ Universal Transverse Mercator (UTM) Military Grid Reference System (MGRS) World Geographic Reference System (GEOREF) State Plane Coordinate Systems (SPCS) IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 2

3 Map Projections Map projections are attempts to portray the surface of the earth or a portion of the earth on a flat surface The map projection can be onto a flat surface or a surface that can be made flat by cutting, such as a cylinder or a cone If the globe, after scaling, cuts the surface, the projection is called secant. Lines where the cuts take place or where the surface touches the globe have no projection distortion IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 3

4 Map Projections Some distortions of conformality (shape), distance, direction, scale, and area always result from this process. Some projections minimize distortions in some of these properties at the expense of maximizing errors in others. Some projection are attempts to only moderately distort all of these properties IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 4

5 Map Projections Three common types of projections are: Cylindrical Conical Azimuthal (Planar) IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 5

6 Map projections: Cylindrical A cylindrical projection can be imagined in its simplest form as a cylinder that has been wrapped around a globe at the equator. If the graticule of latitude and longitude are projected onto the cylinder and the cylinder unwrapped, then a grid-like pattern of straight lines of latitude and longitude would result. The meridians of longitude would be equally spaced and the parallels of latitude would remain parallel but may not appear equally spaced anymore Cylindricals are true at the equator and distortion increases toward the poles IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 6

7 Map projections: Cylindrical Regular and Secant IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 7

8 Map projections: Cylindrical Oblique Tangent or secant to another point on the globe is called oblique IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 8

9 Map projections: Cylindrical Transverse Tangent or secant to a meridian is the transverse aspect IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 9

10 Map projections: Conical In the Conical Projection the graticule is projected onto a cone tangent, or secant, to the globe along any small circle (usually a mid-latitude parallel) In the normal aspect (which is oblique for conic projections), parallels are projected as concentric arcs of circles, and meridians are projected as straight lines radiating at uniform angular intervals from the apex of the flattened cone. Conic projections are not widely used in mapping because of their relatively small zone of reasonable accuracy IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 10

11 Map projections: Conical The secant case, which produces two standard parallels, is more frequently used with conics. Even then, the scale of the map rapidly becomes distorted as distance from the correctly represented standard parallel increases. Because of this problem, conic projections are best suited for maps of mid-latitude regions, especially those elongated in an east- west direction. The United States meets these qualifications and therefore is frequently mapped on conic projections IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 11

12 Map projections: Conical Conics are true along some parallel somewhere between the equator and a pole and distortion increases away from this standard IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 12

13 Map projections: Planar or Azimuthal Imagine a plane being placed against (tangent to) a globe. If a light source inside the globe projects the graticule onto the plane the result would be a planar, or azimuthal, map projection. If the imaginary light is inside the globe it is called Gnomonic If the light is antipodal (diametrically opposite) it is called Sterographic If light source is at infinity, it is called Orthographic Azimuthals are true only at their center point, but generally distortion is worst at the edge of the map IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 13

14 Map projections: Azimuthal or Planar IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 14

15 Map Projections: Distortions Projections can be based on axes parallel to the earth's rotation axis (equatorial), at 90 degrees to it (transverse), or at any other angle (oblique) A projection that preserves the shape of features across the map is called conformal A projection that preserves the area of a feature across the map is called equal area or equivalent No flat map can be both equivalent and conformal. Most fall between the two as compromises To compare or edge-match maps in a GIS, both maps MUST be in the same projection IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 15

16 Map Projections Distortions Conformality When the scale of a map at any point on the map is the same in any direction, the projection is conformal. Meridians (lines of longitude) and parallels (lines of latitude) intersect at right angles. Shape is preserved locally on conformal maps Area When a map portrays areas over the entire map so that all mapped areas have the same proportional relationship to the areas on the Earth that they represent, the map is an equal-area or equivalent map Distance A map is equidistant when it portrays distances from the center of the projection to any other place on the map IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 16

17 Map Projections Distortions Direction A map preserves direction when azimuths (angles from a point on a line to another point) are portrayed correctly in all directions Scale Scale is the relationship between a distance portrayed on a map and the same distance on the Earth Different map projections result in different spatial relationships between regions IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 17

18 Map projections: Cylindrical (Normal) Characteristics Lines of latitude and longitude are parallel intersecting at 90 degrees Meridians are equidistant Forms a rectangular map Scale along the equator or standard parallels is true Simple construction Can have the properites of equidistance, conformality or equal area IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 18

19 Map Projections: Cylindrical Equal Area: Behrmann Uses 30 degree North as parallel of no distortion IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 19

20 Map Projections: Cylindrical Conformal: Mercator Straight lines are lines of constant azimuth IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 20

21 Map Projections: Cylindrical Stereographic: Gall s Secant intersection at 45 N and 45 S Moderately distorts area, direction, distance, shape IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 21

22 Map projections: Conical Characteristics In the normal aspect (which is oblique for conic projections), parallels are projected as concentric arcs of circles, and meridians are projected as straight lines radiating at uniform angular intervals from the apex of the flattened cone. Conic projections are not widely used in mapping because of their relatively small zone of reasonable accuracy. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 22

23 Map projections: Conical Characteristics The secant case, which produces two standard parallels, is more frequently used with conics. Even then, the scale of the map rapidly becomes distorted as distance from the correctly represented standard parallel increases. Because of this problem, conic projections are best suited for maps of mid-latitude regions, especially those elongated in an east- west direction. The United States meets these qualifications and therefore is frequently mapped on conic projections. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 23

24 Map projections: Conical Equal Area Distorts scale and distance except along standard parallels Areas are proportional and directions are true in limited areas Used in the United States and other large countries with a larger east-west than north-south extent IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 24

25 Map projections: Conical Equal Area IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 25

26 Map projections: Conical Equidistant Equally Spaced Parallels. Equidistant Meridians converging at a common point Direction, area, and shape are distorted away from standard parallels. Used for portrayals of areas near to, but on one side of, the equator IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 26

27 Map projections: Conical Equidistant IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 27

28 Map projections: Conical Conformal Area, and shape are distorted away from standard parallels. Directions are true in limited areas. Used for maps of North America IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 28

29 Map projections: Conical Conformal. USA IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 29

30 Map projections: Azimuthal (Planar) Imagine a plane being placed against (tangent to) a globe. If a light source inside the globe projects the graticule onto the plane the result would be a planar, or azimuthal, map projection If the imaginary light is inside the globe it is called Gnomonic If the light is antipodal (diametrically opposite) it is called Sterographic If light source is at infinity, it is called Orthographic Azimuthals are true only at their center point, but generally distortion is worst at the edge of the map IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 30

31 Map projections: Azimuthal Lambert Equal Area Used to map large ocean areas. The central meridian is a straight line, others are curved. A straight line drawn through the center point is on a great circle. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 31

32 Map projections: Azimuthal Equidistant Used to show air-route distances. Distances measured from the center are true. Distortion of other properties increases away from the center point. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 32

33 Map projections: Azimuthal Conformal (Stereographic) Used for navigation in polar regions. Directions are true from the center point and scale increases away from the center point as does distortion in area and shape. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 33

34 Map projections: Azimuthal Orthographic Used for perspective views of hemispheres. Area and shape are distorted. Distances are true along the equator and other parallels. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 34

35 Map projections: Azimuthal Gnomonic Neither conformal nor equal area. Is used by navigators and aviators because greatcircle paths (shortest distances) are shown as straight lines. Less than one hemisphere can be viewed from a given origin. Scale is true only where the central parallel and meridian cross. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 35

36 Map projections: Azimuthal Gnomonic IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 36

37 Coordinates basics IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 37

38 Polar to cartesian coordinates IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 38

39 Coordinate Systems A coordinate system is a standardized method for assigning codes to locations so that locations can be found using the codes alone Standardized coordinate systems use absolute locations A map captured in the units of the paper sheet on which it is printed is based on relative locations or map millimeters IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 39

40 Coordinate Systems Some standard coordinate systems used are: geographic coordinates Lat-long, geodetic lat long, Earth Centered Earth Fixed XYZ Universal Transverse Mercator (UTM) system military grid state plane coordinate system To compare or edge-match maps in a GIS, both maps MUST be in the same coordinate system. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 40

41 Latitude, Longitude, Height The most commonly used coordinate system today is the latitude, longitude, and height system The Prime Meridian and the Equator are the reference planes used to define latitude and longitude Geographic coordinates are the earth's latitude and longitude system, ranging from 90 degrees south to 90 degrees north in latitude and 180 degrees west to 180 degrees east in longitude IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 41

42 Latitude, Longitude, Height A line with a constant latitude running east to west is called a parallel A line with constant longitude running from the north pole to the south pole is called a meridian The zero-longitude meridian is called the prime meridian and passes through Greenwich, England A grid of parallels and meridians shown as lines on a map is called a graticule IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 42

43 Latitude, Longitude, Height IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 43

44 Distances on the great circle IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 44

45 Geodetic Latitude, Longitude, Height The geodetic latitude of a point is the angle from the equatorial plane to the vertical direction of a line normal to the reference ellipsoid. The geodetic longitude of a point is the angle between a reference plane and a plane passing through the point, both planes being perpendicular to the equatorial plane. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 45

46 Geodetic Latitude, Longitude, Height The geodetic height at a point is the distance from the reference ellipsoid to the point in a direction normal to the ellipsoid. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 46

47 Earth Centered Earth Fixed XYZ Cartesian coordinates (XYZ) define three dimensional positions with respect to the center of mass of the reference ellipsoid. The Z-axis points toward the North Pole The X-axis is defined by the intersection of the plane defined by the prime meridian and the equatorial plane The Y-axis completes a right handed orthogonal system by a plane 90 degrees east of the X-axis and its intersection with the equator IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 47

48 Earth Centered Earth Fixed XYZ, Example IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 48

49 Eastings and Northings Traditionally, rectangular coordinates are used when reading maps, with the X value first and the Y value second When map is oriented with north on top, X value is called easting because it measures distances east of the origin and the Y value is called northing because it measures distances north of the origin Origin is placed so that all references are positive False origins may have to be placed at several places to ensure more accurate measurements. Easting and Northings from the false origin are called false eastings and false northings IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 49

50 Universal Transverse Mercator (UTM) Universal Transverse Mercator (UTM) is the most prevalent system used for mapping and other work UTM zone numbers designate 6 degree longitudinal strips (60 vertical zones) extending from 80 degrees South latitude to 84 degrees North latitude. Zone numbers start from the 180th meridian in an eastward direction IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 50

51 Universal Transverse Mercator Grid IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 51

52 Universal Transverse Mercator (UTM) For eastings, a false origin (easting value 500,000 meters) is established at the center of each zone UTM zone characters designate 8 degree zones extending north and south from the equator For northings, it has two primary ordinate points, one at the equator and the other at 80 degrees south For small scale maps, the last digit may be dropped to decrease resolution to 10 meters. Decimal may be used for more accuracy on large scale maps IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 52

53 UTM zones in the lower 48 IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 53

54 UTM Zone 14 Each zone has a central meridian. Zone 14, for example, has a central meridian of 99 degrees west longitude. The zone extends from 96 to 102 degrees west longitude. Eastings are measured from the central meridian (with a 500km false easting to insure positive coordinates). Northings are measured from the equator (with a 10,000km false northing for positions south of the equator). IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 54

55 Reading a UTM measurement IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 55

56 Military Grid Reference System (MGRS) MGRS is an extension of the UTM system. UTM zone number and zone character are used to identify an area 6 degrees in east-west extent and 8 degrees in north-south extent. UTM zone number and designator are followed by 100 km square easting and northing identifiers. The system uses a set of alphabetic characters for the 100 km grid squares. Starting at the 180 degree meridian the characters A to Z (omitting I and O) are used for 18 degrees before starting over. From the equator north the characters A to V (omitting I and O) are used for 100 km squares, repeating every 2,000 km. The reverse sequence IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 56 (from V to A) is used for southern hemisphere

57 Military Grid Reference System (MGRS) IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 57

58 Military Grid Reference System (MGRS) IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 58

59 Military Grid Reference System (MGRS) UTM zone number, UTM zone, and the two 100 km square characters are followed by an even number of numeric characters representing easting and northing values. If 10 numeric characters are used, a precision of 1 meter is assumed. 2 characters imply a precision of 10 km. From 2 to 10 numeric characters the precision changes from 10 km, 1 km, 100 m 10 m, to 1 m. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 59

60 Military Grid Reference System (MGRS) Example IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 60

61 World Geographic Reference System (GEOREF) The World Geographic Reference System is used for aircraft navigation. GEOREF is based on latitude and longitude. The globe is divided into twelve bands of latitude and twenty-four zones of longitude, each 15 degrees in extent. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 61

62 World Geographic Reference System (GEOREF) IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 62

63 World Geographic Reference System (GEOREF) These 15 degree areas are further divided into one degree units identified by 15 characters. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 63

64 State Plane Coordinate System (SPCS) In the United States, the State Plane System was developed in the 1930s and was based on the North American Datum 1927 (NAD27). NAD 27 coordinates are based on the foot While the NAD-27 State Plane System has been superseded by the NAD-83 System, maps in NAD- 27 coordinates (in feet) are still in use. The State Plane System 1983 is based on the North American Datum 1983 (NAD83) NAD 83 coordinates are based on the meter. State plane systems were developed in order to provide local reference systems that were tied to a national datum IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 64

65 State Plane Coordinate System (SPCS) Some smaller states use a single state plane zone. Larger states are divided into several zones. State plane zone boundaries often follow county boundaries. Lambert Conformal Conic projections are used for rectangular zones with a larger east-west than north- south extent. Transverse Mercator projections are used to define zones with a larger north-south extent. IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 65

66 State Plane Coordinate System (SPCS) IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 66

67 State Plane Coordinate System (SPCS) IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 67

68 State Plane Coordinate System (SPCS) IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 68

69 WV SPCS Has two zones: North and South Uses the Lambert Conformal Conic projection IS 645: Geographic Information Systems, Summer 2000, J. Wolfe 69