4b.
Azimuthal Projections

Azimuthal projections are created by projecting onto a flat plane which is either tangent to the generating globe at a single point, or secant to the globe along a small circle. Distortion would be zero at the standard point or line, and would increase as you moved away. Azimuthal projections correctly represent directional relationships about the standard point or line.

Figure 1. The Tangent and Secant Cases of the Azimuthal Projection in the Normal Aspect Showing How Distortion Increases As You Move Away From the Standard Point or Line.

In the normal aspect, the plane is tangent at either the North or South Pole. When the projection is in the transverse aspect, the standard point would be located somewhere along the equator. In the oblique aspect, the point of tangency could be anywhere else. The normal aspect is most common as it places the pole in the centre of the map with the parallels of latitude forming concentric rings around the pole, and the meridians of longitude forming strait lines radiating out from the pole. These projections are frequently used for projections of polar regions, as they place the region of least distortion at or near the pole.

Let's look at some well-known examples of azimuthal projections.

Gnomonic Projection

This is an azimuthal projection in which the light source is located at the centre of the generating globe, as seen in Figure 2. This projection is limited to showing less than half the globe, as rays of light starting at the centre of the globe and passing through the equator would never reach the developable surface. Also, distortion becomes extreme as you move beyond 45 degrees away from the standard point (notice how the US is distorted in Figure 2).

This projection has one interesting and useful property. Great circles appear as strait lines on the final map. This would give the map some use as a navigational tool, since strait lines drawn between two points on the map would represent the shortest route between those points on the Earth.

This is thought to be one of the oldest map projections, having been developed by Thales of Miletus in the 6^th century BC.

Stereographic Projection

This is another very old projection, having been familiar to Greek mathematicians such as Ptolemy. This is an azimuthal projection in which the light source has been moved to a point on the generating globe opposite the point of tangency. As you can see in Figure 3, it is now possible to show more than one hemisphere on the map. Although as you move away from the point of tangency, distortion will increase. In practical terms, you would probably not show more than one hemisphere on a map using this projection. This projection is conformal, meaning that it preserves the shapes of features on the Earth (but not their area). Therefore, circles on the globe appear as circles on the map. This projection can be useful for showing radiating phenomena on the Earth, such as shock waves from an earthquake, or straight line distances from a point.

Orthographic Projection

Another projection that dates back to antiquity, the orthographic projection was used by the ancient Greeks and Romans for astronomical purposes and the construction of sundials. Like the Gnomonic and Stereographic projections, we are again projecting onto a flat plane. This time the light source is located an infinite distance away with the incoming rays of light parallel to one another. This is analogous to the light coming from the sun. With this projection, you are limited to showing one hemisphere of the Earth at a time. This projection essentially gives you a view of the Earth as it would appear from space (see Figure 4).

Equidistant Projection

Where the gnomonic, stereographic, and orthographic projections are conceptually constructed using the rays from a light source, we now turn to a projection that is constructed mathematically.

With the North Pole at the centre of the map, we adjust the spacing of the parallels of latitude so that they are proportional to their arc distance from the pole (see line L in Figure 5). Thus the parallels are equally spaced along the meridians. This gives a map that correctly shows distances measured from the standard point at the centre of the map. In this application, the map could be centered over any point of interest on the globe to depict distances from that point. This projection also correctly shows angular relationships around the standard point

Figure 6. The Emblem of the United Nations.

Lamber Azimuthal Equal-Area Projection

Figure 7. Lambert Equal-Area Projection.

Developed by the Swiss mathematician Johann Heinrich Lambert in 1772, this is another projection that is constructed mathematically. Therefore, this projection is also capable of depicting the entire surface of the Earth. The projection is constructed such that the spacing of the parallels is proportional to their chord distance from the pole (see line L in Figure 7). This projection preserves the area of features, although the shapes of those features will be distorted. This distortion can become quite dramatic as you move far away from the standard point. Because this is an azimuthal projection, the final map also correctly preserves angles from the standard point.

4a. Why We Need Maps and Projections

4c. Cylindrical Projections

4b.Azimuthal Projections