Lecture 9: Introduction to Projections and Coordinate Systems By Austin Troy, University of Vermont,

1. ------Using GIS-- Introduction to GIS Lecture 9: Introduction to Projections and Coordinate Systems By Austin Troy, University of Vermont, with sections adapted from ESRI’s online course on projections

2. Introduction to GIS The Earth’s Shape and Size • It is only comparatively recently that we’ve been able to say what both are • Estimates of shape by the ancients have ranged from a flat disk, to a cube to a cylinder to an oyster. • Pythagoras was the first to postulate it was a sphere • By the fifth century BCE, this was firmly established. • But how big was it?

3. Introduction to GIS The Earth’s Size • It was Posidonius who used the stars to determine the earth's circumference. “He observed that a given star could be seen just on the horizon at Rhodes. He then measured the star's elevation at Alexandria, Egypt, and calculated the angle of difference to be 7.5 degrees or 1/48th of a circle. Multiplying 48 by what he believed to be the correct distance from Rhodes to Alexandria (805 kilometers or 500 miles), Posidonius calculated the earth's circumference to be 38,647 kilometers (24,000 miles)--an error of only three percent.” • More info -source: ESRI

4. Introduction to GIS So, what shape IS the earth? • Earth is not a sphere, but an ellipsoid, because the centrifugal force of the earth’s rotation “flattens it out”. Source: ESRI • This was finally proven by the French in 1753 • The earth rotates about its shortest axis, or minor axis, and is therefore described as an oblate ellipsoid

5. Introduction to GIS And it’s also a…. • Because it’s so close to a sphere, the earth is often referred to as a spheroid: that is a type of ellipsoid that is really, really close to being a sphere Source: ESRI • These are two common spheroids used today: the difference between its major axis and its minor axis is less than 0.34%.

6. Introduction to GIS Spheroids • We have several different estimates of spheroids because of irregularities in the earth: there are slight deviations and irregularities in different regions • We must use a different spheroid for different regions to account for irregularities, or we get positional errors • The International 1924 and the Bessel 1841 spheroids are used in Europe while in North America the GRS80, and decreasingly, the Clarke 1866 Spheroid, are used • In Russia and China the Krasovsky spheroid is used and in India the Everest spheroid

7. Introduction to GIS Spheroids • Note how two different spheroids given slightly different major and minor axis lengths Source: ESRI One more thing about spheroids: If your mapping scales are smaller than 1:5,000,000 (small scale maps), you can use an authalic sphere to define the earth's shape to make things more simple

8. Introduction to GIS The Geographic Graticule/Grid • This is a location reference system for the earth’s surface, consisting of: • Meridians: lines of longitude and • Parallels: lines of latitude • Prime meridian is at Greenwich, England (that is 0º longitude) • Equator is at 0º latitude Source: ESRI

9. Introduction to GIS The Geographic Graticule/Grid • This is like a planar coordinate system, with an origin at the point where the equator meets the prime meridian • The difference is that it is not a Grid because grid lines must meet at right angles; this is why it’s called a graticule instead • Each degree of latitude represents about 110 km, although, that varies slightly because the earth is not a perfect sphere

10. Introduction to GIS The Geographic Grid/Graticule • Latitude and longitude can be measured either in degrees, minutes, seconds (e.g. 56° 34’ 30”); minutes and seconds are base-60, like on a clock • Can also use decimal degrees (more common in GIS), where minutes and seconds are converted to a decimal • Example: 45° 52’ 30” = 45.875 °

11. Introduction to GIS The Geographic Grid/Graticule • Latitude lines form parallel circles of different sizes, while longitude lines are half-circles that meet at the poles • Latitude goes from 0 to 90º N or S and longitude to 180 º E or W of meridian; the 180 º line is the date line Source: ESRI

12. Introduction to GIS Map Projection • This is the method by which we transform the earth’s spheroid (real world) to a flat surface (abstraction), either on paper or digitally • Because we can’t take our globe everywhere with us! • Remember: most GIS layers are 2-D 2D 3D Think about projecting a see-through globe onto a wall Source: ESRI

13. Introduction to GIS Map Projection • The earliest and simplest map projection is the plane chart, or plate carrée, invented around the first century; it treated the graticule as a grid of equal squares, forcing meridians and parallels to meet at right angles • If applied to the world as mapped now, it would look like:

14. Introduction to GIS Map Projection-distortion • The problem with map projection is that it distorts one or several of these properties of a surface: • Shape • Area • Distance • Direction • Some projections specialize in preserving one or several of these features, but none preserve all

15. Introduction to GIS Map Projection-distortion • Shape: projection can distort the shape of a feature. Conformal maps preserve the shape of smaller, local geographic features, while general shapes of larger features are distorted. That is, they preserve local angles; angle on map will be same as angle on globe. Conformal maps also preserve constant scale locally

16. Introduction to GIS Map Projection-distortion • Area:projection can distort the property of equal area (or equivalent), meaning that features have the correct area relative to one another. Map projections that maintain this property are often called equal area map projections. • For instance, if S America is 8x larger than Greenland on the globe will be 8x larger on map • No map projection can have conformality and equal area; sacrifice shape to preserve area and vice versa.

17. Introduction to GIS Map Projection-distortion • Distance: Projection can distort measures of true distance. Accurate distance is maintained for only certain parallels or meridians unless the map is very localized. Maps are said to be equidistant if distance from the map projection's center to all points is accurate. We’ll go into this more later.

18. Introduction to GIS Map Projection-distortion • Direction:Projection can distort true directions between geographic locations; that is, it can mess up the angle, or azimuth between two features; projections of this kind maintain true directions with respect to the map projection's center. Some azimuthal map projections maintain the correct azimuth between any two points. In a map of this kind, the angle of a line drawn between any two locations on the projection gives the correct direction with respect to true north.

19. Introduction to GIS Map Projection-distortion • Hence, when choosing a projection, one must take into account what it is that matters in your analysis and what properties you need to preserve • Conformal and equal area properties are mutually exclusive but some map projections can have more than one preserved property. For instance a map can be conformal and azimuthal • Conformal and equal area properties are global (apply to whole map) while equidistant and azimuthal properties are local and may be true only from or to the center of map

20. Introduction to GIS Map Projection-distortion • Some examples: • Mercator maintains shape and direction, but sacrifices area accuracy • The Sinusoidal and Equal-Area Cylindrical projections both maintain area, but look quite different from each other. The latter distorts shape • The Robinson projection does not enforce any specific properties but is widely used because it makes the earth’s surface and its features look somewhat accurate

21. Introduction to GIS Map Projection-distortion robinson Mercator—goes on forever sinusoidal

22. Introduction to GIS Map Projection-Distortion • Tissot’s indicatrix, made up of ellipses, is a method for measuring distortion of a map; here is Robinson

23. Introduction to GIS Map Projection-Distortion • Here is Sinusoidal Area of these ellipses should be same as those at equator, but shape is different

24. Introduction to GIS Map Projection-General Types • Cylindrical projection: created by wrapping a cylinder around a globe and, in theory, projecting light out of that globe; the meridians in cylindrical projections are equally spaced, while the spacing between parallel lines of latitude increases toward the poles; meridians never converge so poles can’t be shown Source: ESRI

25. Introduction to GIS Map Projection-General Types • In the simplest case, the cylinder is North-South, so it is tangent (touching) at the equator; this is called the standard parallel and represents where the projection is most accurate • If the cylinder is smaller than the circumference of the earth, then it intersects as a secant in two places • A north-south cylindrical Projections cause major distortions in higher latitudes because those points on the cylinder are further away from from the corresponding point on the globe

26. Introduction to GIS Cylindrical map distortion • Think of the problem with this cylindrical equatorial projection • Scale is constant in north-south direction and in east west direction along the equator for an equatorial projection but non constant in east-west direction as move up in latitude

27. Introduction to GIS Cylindrical map distortion • Why is this? Because meridians are all the same length, but parallels are not. • This sort of projection forces parallels to be same length so it distorts them • As move to higher latitudes, east-west scale increases (2 x equatorial scale at 60° N or S latitude) until reaches infinity at the poles; N-S scale is constant

28. Introduction to GIS Cylindrical map distortion • If such a map has a scale bar (see map in 104 Aiken), know that it is only good for those places and directions in which scale is constant—the equator and the meridians • Hence, the measured distance between Nairobi and the mouth of the Amazon might be correct, but the measured distance between Toronto and Vancouver would be off; the measured distance between Alaska and Iceland would be even further off

29. Introduction to GIS Map Projection-General Types • Conic Projections: projects a globe onto a cone • In simplest case, globe touches cone along a single latitude line, or tangent, called standard parallel • Other latitude lines are projected onto cone • To flatten the cone, it must be cut along a line of longitude (see image) • The opposite line of longitude is called the central meridian Source: ESRI

30. Introduction to GIS Map Projection-General Types • Conic Projections: • Projection is most accurate where globe and cone meet—at the standard parallel • Distortion generally increases north or south of it, so poles are often not included • Conic projections are typically used for mid-latitude zones with east-to-west orientation. They are normally applied only to portions of a hemisphere (e.g. North America)

31. Introduction to GIS Map Projection-General Types • Planar or Azimuthal Projections: simply project a globe onto a flat plane • The simplest form is only tangent at one point • Any point of contact may be used but the poles are most commonly used • When another location is used, it is generally to make a small map of a specific area • When the poles are used, longitude lines look like hub and spokes Source: ESRI

32. Introduction to GIS Map Projection-General Types • Planar or Azimuthal Projections: • Because the area of distortion is circular around the point of contact, they are best for mapping roughly circular regions, and hence the poles

33. Introduction to GIS Map Projection-Specific Types • Mercator: This is specific type of cylindrical projection • Invented by Gerardus Mercator during the 16th Century • It was invented for navigation because it preserves azimuthal accuracy—that is, if you draw a straight line between two points on a map created with Mercator projection, the angle of that line represents the actual bearing you need to sail to travel between the two points Source: ESRI

34. Introduction to GIS Map Projection-Specific Types • Mercator: Of course the Mercator projection is not so good for preserving area. Take a look at how it enlarges high latitude features like Greenland Antarctica and shrinks mid latitude features

35. Introduction to GIS Map Projection-Specific Types • Transverse Mercator: Invented by Johann Lambert in 1772, this projection is cylindrical, but the axis of the cylinder is rotated 90°, so the tangent line is longitudinal, rather than the equator • In this case, only the central longitudinal meridian and the equator are straight lines All other lines are represented by complex curves: that is they can’t be represented by single section of a circle Source: ESRI

36. Introduction to GIS Map Projection-Specific Types • Transverse Mercator: • Transverse Mercator projection is not used on a global scale but is applied to regions that have a general north-south orientation, while Mercator tends to be used more for geographic features with east-west axis. • It is used in commonly in the US with the State Plane Coordinate system, with north-south features

37. Introduction to GIS Map Projection-Specific Types • Lambert Conformal Conic:invented in 1772, this is a form of a conic projection • Latitude lines are unequally spaced arcs that are portions of concentric circles. Longitude lines are actually radii of the same circles that define the latitude lines. Source: ESRI

38. Introduction to GIS Map Projection-Specific Types • The Lambert Conformal Conic projection is very good for middle latitudes with east-west orientation. • It portrays the pole as a point • It portrays shape more accurately than area and is commonly used for North America. • The State Plane coordinate system uses it for east-west oriented features

39. Introduction to GIS Map Projection-Specific Types • The Lambert Conformal Conic projection is a slightly more complex form of conic projection because it intersects the globe along two lines, called secants, rather than along one, which would be called a tangent • There is no distortion along those two lines • Distortion increases as move away from secants Source: ESRI

40. Introduction to GIS Map Projection-Specific Types • Albers Equal Area Conic projection: Again, this is a conic projection, using secants as standard parallels but while Lambert preserves shape Albers preserves area • It also differs in that poles are not represented as points, but as arcs, meaning that meridians don’t converge • Latitude lines are unequally spaced concentric circles, whose spacing decreases toward the poles. • Developed by Heinrich Christian Albers in the early nineteenth century for European maps

41. Introduction to GIS Map Projection-Specific Types • Albers Equal Area Conic: It preserves area by making the scale factor of a meridian at any given point the reciprocal of that along the parallel. • Scale factor is the ratio of local scale a point on the projection to the reference scale of the globe; 1 means the two are touching and greater than 1 means the projection surface is at a distance

42. Introduction to GIS Other Selected Projections • MoreCylindrical equal area: (have straight meridians and parallels, the meridians are equally spaced, the parallels unequally spaced) • Behrmann cyclindrical equal-area: single standard parallel at 30 ° north • Gall’s stereographic: secant intersecting at 45° north and 45 ° south • Peter’s: de-emphasizes area exaggerations in high latitudes; standard parallels at 45 or 47 ° Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder for links

43. Introduction to GIS Other Selected Projections • Azimuthal projections: • Azimuthal equidistant: preserves distance property; used to show air route distances • Lambert Azimuthal equal area: Often used for polar regions; central meridian is straight, others are curved • Oblique Aspect Orthographic • North Polar Stereographic Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder for links

44. Introduction to GIS Other Selected Projections • More conic projections • Equidistant Conic: used for showing areas near to, but on one side of the equator, preserves only distance property • Polyconic: used for most of the early USGS quads; based on on an infinite number of cones tangent to an infinite number of parallels; central meridian straight but other lines are complex curves Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder for links

45. Introduction to GIS Other Selected Projections • Pseudo-cylindrical projections: resemble cylindrical projections, with straight, parallel parallels and equally spaced meridians, but all meridians but the reference meridian are curves • Mollweide: used for world maps; is equal-area; 90th meridians are semi-circles • Robinson:based on tables of coordinates, not mathematical formulas; distorts shape, area, scale, and distance in an attempt to make a balanced map Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder for links