# The World is not Flat

by | Sep 20, 2016 | Web Mapping | 0 comments

## Introduction to Datums and Projections

I wrote this post because there seems to be a lot of confusion among casual users of GIS and other mapping applications around projections. The three take-away lessons I would like to share are:

1. Projections are a necessary requirement for representing three dimensional shapes on two dimensional surfaces.
2. Projections are built up from other model elements – including a spheroid, datum and coordinate system.
3. Different projections are optimized for different purposes.  There is no single “best” projection.

## Drawing Three Dimensional Objects

All maps (or GIS layers) are representations of three dimensional surfaces on a two dimensional plane. As such they can never be completely accurate – a curved surface cannot be faithfully represented on a flat plane without introducing distortions in shape, distance, area and direction. If you doubt this, try wrapping a piece of paper around an orange in such a way that all of the paper surface is in contact with the orange surface. You can’t. On the other hand, try the same thing using plastic wrap – you can; but only because the plastic wrap is elastic, and can stretch as necessary to remain in contact with the orange surface everywhere.

The best you can do is define a rule so that points on the 3D surface can be predictably associated with points on the 2D surface. In a computerized environment this rule necessarily assumes the form of a mathematical formula.

## Modelling the Earth

### Spheroids

Before you can translate a 3D coordinate to a 2D equivalent, you first have know what the original coordinate is. This requires two things:

1. A mathematical model describing the shape and size of the Earth.
2. A way of describing the location of a specific point on the surface of this shape.

A simple way to model the Earth is as a sphere. This is good enough for many practical applications, although more exact models have described the Earth as ellipsoid or even vaguely pear shaped. If we assume a sphere, this raises the question of what radius to use. It is insufficient to state “at sea level”, since there is no consistent sea level. The radius from the centre of the Earth to a random point in the Pacific will not be the same as the radius terminating somewhere in the Atlantic. Moreover, sea level rises and falls with tides; therefore either an averaged value or value at some specific point in the cycle is mandated.

Most modern models of the earth use a spheroid. A spheroid is based on an ellipse which has been rotated around its shorter axis (semi-minor axis) to produce a three-dimensional, flattened sphere. In this model the half-length of the longer axis (semi-major axis) replaces the radius as the key dimension.

### Datums

Once a shape and its parameters are defined, an origin point for the coordinate reference system must be established. Generally, this is the geometric centre of the Earth (in other cases the gravitational centre of mass is used). The shape plus the relative placement of the origin point constitute the datum.

### Geographic Coordinates

Angles measured from the origin are used to identify a specific point on the surface. These angles are referred to as latitude (North-South direction) and longitude (East-West direction), with units most often in decimal degrees or degrees-minutes-seconds.

Displayed on a globe latitudes run parallel to the equator (which itself is defined as the latitude of zero degrees); all latitudes north of the equator assume positive values up to +90⁰ and latitudes south of the equator assume negative values up to -90⁰ (respectively defining the two poles).

Longitudes all pass through the both poles to form orange-like segments; unlike latitudes they are all the same circumference. One longitude is chosen as the zero degree reference and is known as the prime meridian (meridian is an alternate term for longitude). Longitudes travelling east from the prime meridian assume positive values up to +180⁰ while those travelling west assume negative values up to -180⁰.

The combination of spheroid (or other shape modelling the Earth), datum and geographic coordinates define a Geographic Coordinate System (GCS).

## Projections – Transforming Coordinates

A projection is a mathematical formula for translating the angle-based 3D geographic coordinates to 2D grid-based coordinates. It is important to note that all map representations on a flat surface – whether on a screen or a paper map – are projected even if the author is unaware they are implementing a translation. Even if you define the grid as using latitude values for y-coordinates and longitude values for x-coordinates, this is still a projection (in fact this particular projection is variously known as Simple Cylindrical, Equirectangular or Plate Carrée). The important thing to note is that projection always introduces distortion in shape, area, distance or direction.

### Linear Units

Although you can use degrees as the units in your grid, they are inconvenient. Most projections other then a Plate Carrée will result in a grid with curved vertical lines because the distance between longitudes decreases as you move towards the poles. Also for most practical map applications linear units – which measure distances from point to point in a straight line – are more intuitive and correspond to the units more typically used for expressing distance and area. Using linear units, distances between grid lines remain constant.

### Cartesian Coordinates

Often the grid used in a projection uses a Cartesian coordinate system which involves two perpendicular lines that intersect at an origin point to define (0,0). One line represents the horizontal axis (x-axis) and the other the vertical axis (y-axis). In the standard Cartesian grid values to the right of the origin on the x-axis are positive and those to the left are negative; similarly, for the y-axis, values above the origin are positive and values below the origin are negative.

## Choosing a Projection

### Distortion

As noted earlier, all projections distort the true representation of the Earth’s surface to some extent, but they do so in various ways. This means different projections are more suitable to different mapping purposes. In addition, most projections cannot display the entire Earth’s surface on one map; at most a single hemisphere can be depicted. Some projections are only suitable for mapping a limited region – for example the Universal Transverse Mercator (UTM) divides the Earth into sixty zones; if the desired map extends across several zones, another projection needs to be used. Additionally, the UTM projection’s distortions become extreme as you approach the poles and therefore the zones are limited to +80⁰ to -80⁰ latitude.

The four types of distortion that can occur from a projection are:

1. Shape
2. Area
3. Distance (from point to point)
4. Direction (from point to point)

A map projection can only faithfully preserve at most one of these characteristics. Most map projections attempt to compromise to display acceptable levels of distortion across all four factors. There are four general classes of projections that are particularly good at minimizing each type of distortion.

1. Conformal – Shapes are well preserved.
2. Equal Area – Areas are well preserved.
3. Equidistant – Distances between points are well preserved.
4. Azimuthal – Angle directions between points are well preserved.

### Factors in Choosing a Projection

If all you want to do is to display data in relation to its geographic context, the choice of projection is not critical. Anything that covers the mapping extent and where distortion does not confuse the message is acceptable. However, if any geographic statistics are to be derived from the map – for instance comparison of areas or identifying distances between points – then the choice of projection becomes impactful. A non-exhaustive list of factors in choosing a projection are:

1. Geographic extent that must be covered.
2. Geographic statistics that will be obtained.
3. What projections are supported by software tools.
4. Projection that source data is supplied in.
5. Standards for the industry or geographic region.