A critical distance is a threshold of distance beyond which the requisite travel costs or efforts are too great to make the journey viable or worthwhile for an individual or group

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

The cardinal points north, east, south, and west correspond to absolute direction. Their directions are absolute because they are based on physical astrological and geographical phenomena, such as the movement of the sun and the stars, which do not vary from place to place or culture to culture

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

Cylindrical projections represent the globe as if it were enclosed within a cylinder, much like the thought of placing a tennis ball inside an empty soup can, then peeling off the soup can label to derive a two-dimensional image of the globe inside. In this example, the cylindrical surface is closest to the globe at its midregion, the earth’s equator, where the globe is the greatest in circumference. Another way to state this relationship would be to say that the cylindrical projection is usually tangent to the earth’s equator. Because cylindrical projections are typically tangent to the earth’s equator, the middle latitudes are typically less distorted than the higher and lower latitudes in cylindrical projections

Area Distortion 827,000 square miles 6.8 million square miles Mercator Projection 827,000 square miles 6.8 million square miles

General Map Projection: Cylindrical 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

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.

Map Projection-distortion Map projections that preserve and accurately represent the shape of the geographical areas and features are said to be conformal 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

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.

Map Projection-distortion Direction:Projection can distort true directions (angle or azimuth) between locations; azimuthal projections maintain true direction with respect to the center point. Some azimuthal map projections maintain direction between any two points, so that the angle of a line drawn between any two locations on the projection gives the correct direction with respect to true north. © 2005, Austin Troy

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 © 2005, Austin Troy

Distance distortion 4,300 km: Robinson 5,400 km: Mercator © 2005, Austin Troy

Shape distortion Mercator (left) World Cylindrical Equal Area (above) The distortion in shape above is necessary to get Greenland to have the correct area; The Mercator map looks good but Greenland is many times too big © 2005, Austin Troy

Some Examples of distortion Robinson Mercator—goes on forever sinusoidal © 2005, Austin Troy

Some examples of distortion 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 © 2005, Austin Troy

Map Projection-Distortion Tissot’s indicatrix, made up of ellipses, is a method for measuring distortion of a map © 2005, Austin Troy

Map Projection-Distortion Tissot’s indicatrix; here is the Robinson projection © 2005, Austin Troy

Map Projection-Distortion Here is Sinusoidal Area of these ellipses should be same as those at equator, but shape is different © 2005, Austin Troy

Cylindrical Map Types Tangent to great circle: 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 © 2005, Austin Troy Source: http://nationalatlas.gov/articles/mapping/a_projections.html

Cylindrical Map Types Secant projections are more accurate because projection is more accurate the closer the projection surface is to the globe and when the projection surface touches twice, that means it is on average closer to the globe The distance from map surface to projection surface is described by a scale factor, which is 1 where they touch Earth surface .9996 Projection surface Standard meridians Central meridian © 2005, Austin Troy

Cylindrical Map Types 3. Transverse cyclindrical projections: in this type the cylinder is turned on its side so it touches a line of longitude; these can also be tangent © 2005, Austin Troy

Cylindrical map distortion 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 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 distance from equator (latitude) changes Requires alternating Scale Bar based on latitude © 2005, Austin Troy

Cylindrical map distortion If such a map has a scale bar, 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 © 2005, Austin Troy

Cylindrical Map Distortion 50 ◦ latitude 25 ◦ latitude 0 ◦ atitude X miles © 2005, Austin Troy

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 © 2005, Austin Troy

General Projection Types: Conic 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 © 2005, Austin Troy

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) © 2005, Austin Troy

Map Projection-General Types Can be tangent or secant Secant are more accurate for reasons given earlier © 2005, Austin Troy

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 © 2005, Austin Troy Source: ESRI

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 © 2005, Austin Troy