Map Projections: Flatten the Sphere

Map Projections Example

What is a Map Projection?

The best way to represent the Earth is with a globe. We live on one big blue marble that’s the shape of a sphere (or close to it)

But globes are hard to carry in your backpack, you can only see one side of the globe, it’s hard to measure distances and they’re just not as convenient as paper maps.

This is why we project globes on two-dimensional planes with map projections. For the GIS analyst to make use of points, lines and polygons, it is necessary to locate them in two dimensions.

But Earth’s surface cannot be represented on a plane without distortion. This is because the Earth is curved. So this means that every map projection will distort the Earth in some way.

Different map projections have different strengths and weaknesses. Today, we explore everything you need to know about map projections in GIS.

Why use map projections?

Map Projection Georeference
Map Projection Georeference

How do you take this relatively large sphere that we live on and project it on a two-dimensional plane?

Use map projections!

A map projection is a method by which the cartographer translates a sphere or globe into a two-dimensional representation.

A map projection systematically renders a 3D ellipsoid (or spheroid) of Earth to a 2D map surface.

Because 3D surfaces cannot be displayed perfectly in a two-dimensional space, some distortions can occur. Examples of map distortions are of conformality, distance, direction, scale, and area. Map distortions always result from map projections.

There are multiple ways to represent a sphere on a two-dimensional surface… Like Jason Davies popular Map Projection Transition Visualizer.

Every projection has strengths and weaknesses. It is up to the cartographer to determine what projection is most favorable for its purpose.

Peel an Orange: Transform 3D to 2D

Orange Peel Map Projection: Goode Homolosine
Orange Peel Map Projection: Goode Homolosine

Imagine you have an orange (the fruit). You can pretend the orange is our three-dimensional Earth.

There’s no way to see all the sides of this orange any way you look at it.

When you peel the orange, flatten and stretch it on a table, you can begin to see all sides of the orange.

You’ve transformed a three-dimensional shape onto a two-dimensional plane.

Assumptions have to be made about the shape of the earth. The earth is an irregular shape though most map projections assume it to be a sphere or an ellipsoid.

Map Projections as Equations

Globe: New York to Tokyo
Globe: New York to Tokyo

If you stared at a globe and wanted to know the shortest distance between New York and Tokyo, how would you do this?

Typically, you’d draw a straight line from start to end point. But how do you draw a straight line on a sphere?

First, you need some background on spatial referencing systems. Lines of latitude run parallel to the equator and describe north and south positions on Earth. Lines of longitude (meridians) are east and west directions.

Latitude and longitude lines form geographic grids on Earth. Every place on Earth has a geographic grid coordinate. For example, the North pole is located at 90° N, 0° W

Map projections are actually equations that transform the earth’s angular geographic coordinates (latitudes and longitudes) to XY Cartesian coordinates on a flat projected surface.

But the problem is:

When we try to take this world and project it onto a two-dimensional plane, there’s going to be distortion.

Let’s examine the developable surfaces and distortions for map projections.

Developable Surface Map Projections

Developable Surface

Most map projections are based on developable surfaces. A developable surface is the geometric shape that a map projection can be built on.

Cylinders, cones, planes… Map projections take these types of developable surface shapes and flattens it in a two-dimensional plane. Each surface is mathematically flattened based on those geometric shapes.

The Earth can use various types of developable surfaces to create some pretty interesting map projections as shown below:




Conic Projections use a cone to develop its surface on a plane. Meridians converge at a single point, which may or may not be the South or North pole. Area is distorted, while scale is mostly preserved. Distance at the bottom of the image suffers the most distortion.

Cylindrical Projections use a cylinder to develop a plane surface on a map. A cylinder is wrapped around the globe and unraveled on a flat surface. Cylindrical map projections can be equidistant, conformal and equal-area. Countries near the equator has truest relative positions while the view of the poles are most distorted.

Azimuthal (plane) Projections use a plane to develop a surface on a map. They are often tangent to the ellipsoid at one point. The projection center must be specified. Only half of the globe will be viewed with more distortion occuring at the four edges.

Gnomonic Projections use the center of the spheroid as the projection center.

Gnomonic Projection

Stereographic Projections are diametrically opposite to the tangent point.

Stereographic Projection

Orthographic Projections has a projection center at infinity.

Orthographic

The most common map projections use developable surfaces. But some map projections do not. They are based on mathematical forms that aren’t cones, cylinders, planes or other three dimensional figures.

Some examples of map projections that don’t use developable surfaces are the Goode and Bonne projections.

Map Projection Examples




Through human history, maps have used a wide range of projections. Explorers used maps to accurately travel. The first known map originated in Greece and showed that the world was perceived to be cylindrical.

There are thousands of map projections that are in existence today!

Some map projections are useful for some things and other map projections are good for other things.

Two of the most common map projections used in North America are the Lambert conformal conic and the transverse Mercator.

Lambert Conformal Conic

Lambert Conformal Conic
North America: Lambert Conformal Conic

The Lambert Conformal Conic is derived from a cone intersecting the ellipsoid along two standard parallels. When you “unroll” the cone on a flat surface, this becomes the mathematically developed surface.

The most distortion occurs in the north-south directions. In general, distortion increases away from the standard parallels. For example, this map protection severely expands South America.

Universal Transverse Mercator

North America: Mercator
North America: Mercator

Universal Transverse Mercator (UTM) coordinate system is a standard set of map projections with a central meridian for each six-degree wide UTM zone. Even though Google maps used the Mercator projection because it preserves shape decently, and north is always up.

But Mercator map projections are really bad at preserving area. For most of us, the projection is common enough that it looks fine for us. In reality, Africa is huge on a globe. But Greenland appears to be as large as Africa, even though in reality it is only 1/14th the size. The Mercator puzzle game illustrates this point.

Conclusion

Spatial referencing systems (latitude and longitude) are used to locate a feature on the Earth’s spheroid surface. The location of any point on Earth can be defined using latitudes and longitudes. These points are expressed in angular units such as degrees, minutes and seconds.

Most maps in a GIS are in two-dimensional form. To make use of these maps, a referencing systems that uses a pair of coordinates measured along axes at right angles to one another is required. To obtain a pair of coordinates, a graticule is placed on the map. When you use map projections, the lines of latitude and longitude become the grid lines on a flat map. Graticules are obtained by projecting lines of latitude and longitude from a sphere to a flat surface using map projections.

However, when you transfer a spherical shape to a flat surface, you approximate the true shape of the Earth. Depending on the map projection you choose, some projections may cause distance between features on a map to be preserved while distortion is introduced to shape. In some cases, area may be preserved while direction is distorted.

Cartographers choose map projections that best represents the purpose, size and shape of the area of interest on the map.

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