Projections Supporting tagline
Mapping involves determining the geographic locations of features on the earth, transforming these locations into positions on a flat map through use of map projection.
The first step is to choose a model of reference for the shape of the earth.
1. Modeling the shape of the earth
Until the late 1600s, the earth was thought to be perfectly spherical in shape. The change came in 1670 with Isaac Newton who proposed that a consequence of its theory of gravity would be a slight bulging of the earth at the equator due the greater centrifugal force generated by the earth’s rotation.
From 1800 to the present, at least 20 determinations of the earth’s flattening and radii have been made: Everest (1830), …, Bessel (1841), Clarke (1866), WGS72 (1972), GRS80 (1980), WGS84 (1984).
Insert table 4.2 from E.C
WGS84 and WGS72 determined from satellite orbital data, are considered more accurate than the earlier ground measurements determinations.
The shape of the earth is approximatively an ellipsoid with the following dimensions:
- semi-major axis: 6378 km
- semi-minor axis: 6356 km
The earth is a very smooth geometrical figure. Most of the earth’s surface appears rugged and rough to us, but even the highest peaks and deepest ocean trenches are barely noticeable irregularities. Imagine for instance the earth reduced to 1m in diameter, Mt Everest would be but a 70 mm.
1.1 Geoid
Due to irregularities of earth’s gravity field, earth’s shape is not perfectly an ellipsoid as shown below (note that variations are exaggerated in this illustration for educational purpose). This shape is called ‘Geoid’.
One way to define a simple mathematical model of such complex shape is to define local ellipsoids best fitting the local shape of the earth as shown below. As a result many local ellipsoids exists according to the location of interest.
Hopefully, the WGS84 global ellipsoid though not giving the best fit for a particular part of earth is today considered as a standard. In our case WGS84 will be systematically used.
To summarize, the surface of the earth can be modelled by:
- authalic spheres (spheres whose surface is equal to the surface of the ellipsoid modelling the earth shape)
- local ellipsoids
- global ellipsoids
Once a model of earth’s shape has been defined, the second step is to define a coordinate reference system.
2. Geographic coordinates
The geographical coordinate system employing latitude and longitude can be traced to the 2nd century B.C. The geographical coordinate system is the primary locational reference system for the earth. It was devised to make possible a unique statement of location for each earth feature.
The north and south poles, where the axis of rotation intersects the earth’s surface, are the starting points on which to base the system. Specyfing a locationon the earth requires determining latitude, the north-south angular distance from the equator, and longitude, the east-west angular distance from a prime meridian. All points on the earth having the same latitude form a line called a parallel; all points of the same longitude form a meridian line.
The equator, the line on the earth formed by points halfway between the two poles, is the natural starting place for latitude. Latitude ranges pole to pole from 90°N to 90°S or +90° to -90° (when using a digital databases).
Longitude, our weast-west position on the earth, is associated with an infinite set of meridians, arranged perpandicularly to the parallels. Unlike the equator in the latitude system, no meridian has a natural basis for being the starting line from which to measure east-west positions. During the 19th century, many nations accepted the meridian of the Royal Observatory at Greenwich near London, England, as 0° (it was officialy agreed in 1884). Longitude ranges from 180°W to 180°E of the prime meridian (from -180° to +180° for digital databases).
2.1 Geographic coordinates units and conversions
To specify longitude and latitude coordinates, three numeral systems can be used:
- sexagesimal (base 60) also known as Degree, Minutes, seconds
- decimal (base 10)
- a mix of decimal and sexagesimal also known as Degree, Decimal Minutes
For instance Irapuato (lon/lat) geographical coordinates can be specified as:
- lon: E 101°21’16.1856” / lat: N 20°40’43.194”
- lon: -101.354496 / lat: +20.678665
- lon: E 101°21.26976’ / lat: N 20°40.7199’
The conversion process is elementary as explained below. Let’s consider you want to convert coordinates from system 1 to system 2:
- considering that 1° = 60’ = 3600”
- decimal longitude = 101 + (21/60) + (16.1856/3600) = 101.354496 (then -101.354496 to take into account that Irapuato is East from Greenwich)
- and decimal latidue = 20 +(40/60) + (43.194/3600) = 20.678665 (the +20.678665 as North pole).
Thus, it is easy to create your own converter using Excel for instance (look at the example provided into course DropBox account). Additionnaly many website proposes such tools online. For instance boulter.com, …
3. Map projections
One simple way of mapping the earth without distortion is to map it on a globe. When we do so all we change is the size (scale). On the other hand, globes have many practical disadvantages:
- expensive to make
- difficult to reproduce
- cumbersome to handle
- awkward to store
- difficult to measure and draw on
- less than half of the globe is visible at any one time
Projecting earth surface on a flat surface eliminates these drawbacks. Nevertheless developping a sphere onto a plane introduces distortions (modification of angles or/and area -see below-).
3.1 Projections on a flat surface
It is not possible to flatten a sphere onto a plane without stretching it (think about the situation where you want to flatten an orange skin onto a flat surface. The orange skin will be necessary torn). Transforming a spherical surface to plane surface always create distortion. However, different projection techniques allows to choose the type and degree of distortion introduced.
3.1.1 The three families of map projections
The process of creating map projections can be visualised by positioning a light source inside a transparent globe on which opaque earth features are placed. Then project the feature outlines onto a two-dimensional flat piece of paper.
Different ways of projecting can be produced by surrounding the globe in a cylindrical fashion, as a cone, or even as a flat surface. Each of these methods produces what is called a map projection family.
Therefore, there is a family of planar projections, a family of cylindrical projections, and another called conical projections (see Illustration above)
a. Cylindrical b. Conical c. Planar
3.1.2 Type of distortions introduced
Map projections are never absolutely accurate representations of the spherical earth. As a result of the map projection process, every map shows distortions of angular conformity, distance and area. A map projection may combine several of these characteristics, or may be a compromise that distorts all the properties of area, distance and angular conformity, within some acceptable limit.
When a map projection preserves:
- angle: it is said conformal or orthomorphic
- area: it is said equal-area or equivalent
- distance: it is said equidistant
It is worth noting that map projection finding a compromise between angle and area distortions can be designed (see examples below).
3.2 Examples of famous map projections
3.2.1 Mercator projection
Characteristics: Cylindrical, conformal
Look at the relative size of Greenland and Africa or South America. Area of South America is in reality more than 8 times bigger. However, its conformal property is interesting for navigation purpose.
3.2.2 “Plate carrée” or Equidistant Cylindrical projection
Characteristics: Cylindrical, equidistant
Look at the arrangement of the graticule (longitude and latitude lines) at equal distance and perpendicular.
Often, in GIS, when no cartographic projection is specified, this default one is used.
3.2.3 Mollweide
Characteristics: Cylindrical, equivalent (equal-area)
3.2.4 Robinson
Characteristics: Pseudo-cylindrical, neither conformal nor equivalent but a good compromise
See this link for further information on pseudo-cylindrical projections. It’s basically a cylindrical projection but using several cylinders of different diameters.
3.2.5 Universal Transverse Mercator (UTM)
Characteristics: Cylindrical, conformal
We saw earlier that the Mercator projection while offering very interesting properties in preserving angles, it introduced huge area distortions. Actually, the distortion increases as we move away from the line of tangency between the sphere and the cylinder.
You will see below, the Mercator projection with its ‘Tissot’s indicatrix’. Tissot’s indicatrix allows to visualize the degree of distortions of areas (relative size of circles) and angles (when we see circles there is no angles distortion. When circles change to ellipse, there is angular distorion, the semi-major and semi-minor values of the ellipsoid indicating the degree of distortion).
Thus, the idea of the UTM projection is to propose a set of Mercator projection whose cylinder is centered on areas of interest. First the cylinder is positionned with axis perpendicular to the poles axis as shown below in transverse position:
Then, to avoid too much distortion in points far from the tangency line of the cylinder and sphere/ellipsoid, the world is divided into 60 equal zones that are all 6 degrees wide in longitude from East to West. The UTM zones are numbered 1 to 60, starting at the international date line (zone 1 at 180 degrees West longitude) and progressing East back to the international date line (zone 60 at 180 degrees East longitude) as shown below:
Several online website allows to find your are of interest quickly. For instance http://www.apsalin.com/utm-zone-finder.aspx
Irapuato Longitude, Latitude is: (-101.35, 20.67), try to identify the corresponding UTM zone.
IMPORTANT: On the globe (when no map projection is used) the coordinate sytem use pairs of longitude and latitude (spherical coordinates). As soon as we use a map projection distances and positions are shown in meters.
4. Spatial Reference System (SRS) codes
Geographic coordinate systems (the choice of the ellipsoid (semi-major, semi-minor axis and origin) and map projections can be collectively called Spatial Reference System.
There are hundreds of Geographic coordinate systems and cartographic/map projections, the EPSG (European Petroleum Survey Group) today absorbed by the International Association of Oil & Gas Producers associated ‘almost’ unique idendifiers to any projections in order to ease their use and identification.
See Wikipedia page for further information on EPSG
The following website http://spatialreference.org/ allows to searc reference ID of any projections.
Hereafter, a minimal list of indispensable SRS (Spatial Reference System) ID:
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EPSG:4326: WGS84 (World Geodesic System 84)
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EPSG:900913 or EPSG:3785 or EPSG:3857: Spherical Mercator (Mercator projection based on a sphere instead of an ellipsoid). Some of these codes are deprecated but still widely used. So it is important to know them. The following page describes this projection.