# Chapter 6 Part 2: Map Projections

In this chapter you will learn about map projections. You will learn about the determining factors to consider when creating map projections, deformation and distribution of the projection, and how to determine which map projection to use for a project. You will also learn about the light source location.

## 6.3: Map Projections

Defined, a map projection is a systematic rendering of locations from the curved earth surface onto a flat map surface. This allows us to flatten the curved surface onto a flat surface such as a piece of paper, or computer screen. The reason why we employ map projections is because globes are not very portable, or practical to use in some cases. Therefore we use map projections to flatten the earth into a map.

Figure 9 is a basic illustration displays the concept of a map projection. The map projection is broadly composed of three parts, the ellipsoid, which models the shape of the earth, a light source which is used to project features on the earth surface, and a developable surface commonly a flat piece of paper onto which the Earth’s features are projected, and flattened to be used as a map.

Figure 9: Basic Example of Map Projection

Developable Surfaces: A developable surface is a geometric surface on which the curved surface of the earth is projected; the end result being what we know as a map. Geometric forms that are commonly used as developable surfaces are planes, cylinders, cones, and mathematical surfaces.

No matter which developable surface is used to create a map the basic idea remains the same. The features of the curved earth are projected onto one of the four geometric forms to produce a flat map.

Figure 10: (left to right) Examples of Plane, Cylinder, Cone, and Mathematical

The way in which an ellipsoid and developable surface interacts with each other is to place the ellipsoid in different locations and different rotations is to gain a desired view at map properties. In the Figure 11 illustration the blue circle is an ellipsoid representing the earth. The triangle represents a cone although we are representing it in 2-D. The idea is that the developable surface can be placed on top of the ellipsoid as a hat, or pulled down through the surface of the earth, and even rotated side to side, or forwards and backwards to create the desired view that we are looking for in a flat map.

Figure 11: Developable Surface Examples

Interaction: These developable surfaces will interact in a few different ways with the ellipsoid. Generally, the developable surfaces will touch the ellipsoid in either two places which creates two secant lines or at a single location and creates a single tangent line. In the illustration shown the tangent line is touching the ellipsoid of the South Pole which should give us a polar view of the earth.

The secant intersection is intersecting the earth at two locations which provides us with an abnormal view of the northern portion of the ellipsoid. Both types of interactions are correct as they depend on the purpose of the map and the way in which we want to portray the earth on a flat map.

Figure 12: Two Secant Lines, One Point, or Tangent Line

To further illustrate the idea of a tangent and secant intersection, two more illustrations are provided. For the tangent interaction there is a line that can be drawn around the earth at its widest point that is perpendicular to the point tangents. This is known as the tangent line. For secant interactions two lines are drawn following the earth’s curvature between the secant intersection points.

Figure 13: Tangent Interaction
Figure 14: Secant Interaction

Map Projection Parameters: There are five map projection parameters that are going to be discussed in this section. They are: standard points and lines, projection aspect, central Meridian, latitude of origin, and light source location. Together these five map projection properties allow an individual to selectively display and distort the earth to create a map that is suitable based on need. The lesson will start with standard points and lines and continue on with details covering each of the additional four map projection parameters.

Standard Points and Lines: Defined, a standard point and line is a point or line of intersection between the developable surface and the spheroid or ellipsoid. In the case of a secant intersection, there will be two standard lines that would define where the developable surface intersects with the spheroid.

If the developable surface happens to intersect the spheroid along a line of latitude it is known as a standard parallel. Additionally if a standard line falls on a line of longitude exactly it is known as a standard Meridian. When defining a map projection you must define the standard points and lines. It is not uncommon to have a map projection follow a standard parallel or standard Meridian.

In this illustration the cone is in a normal aspect and with the secant intersection, the secant intersecting lines follow parallels. Therefore they are both known as standard parallels.

Figure 15: Standard Parallels

Importance of Standard Points and Lines: The reasons why standard points and lines are important are because the corresponding places act or along the standard point to lines will have no scale distortion. That means that where the developable surface intersects with the spheroid there is little to no distortion on our flat map. The further away from the standard point or lines the greater the distortion or deformation that will occur on the map. Secant intersection between the developable surface and the spheroid can help minimize distortion over a large area by providing more control than a tangent intersection at a single point or line. Therefore placement of standard points and lines is one of the most important parameters to consider when defining a map projection.

Projection Aspect: The next projection parameter to discuss is the projection aspect. A projection aspect is the position of the projected graticule relative to ordinary position of the geographic grid on earth. What that means is that if the developable surface’s vertical axis coincides with the vertical axis of the earth this defines a normal aspect. Should the vertical axis of the developable surface differ from the vertical axis of the earth this would be an abnormal aspect. Figure 16 displays the normal axis of the globe, which runs from the North Pole through the center of the Earth to the South Pole.

Figure 16: Globe Normal Axis

Figure 17 shows a cone developable surface at a secant intersection with the earth. The vertical axis of the cone, displayed as a blue line, coincides with the vertical axis of the earth displayed as a red dotted line which is considered a normal aspect.

Figure 17: Normal Aspect

Figure 18 is an illustration that shows a non-aspect. Note that the vertical axis of the cone does not coincide with the vertical axis of the earth thus creating a non-aspect.

Figure 18: Non-Normal Aspect

Central Meridian: The next projection parameter to discuss is the Central Meridian. The Central Meridian defines the center point of the projection. That means that this is the Meridian, or longitude line, that displays in the center of the map. Essentially this allows you to rotate the earth about the vertical axis to determine what portion of the earth you want to have in the center of the map.

Latitude of Origin: The three primary positions of light sources are gnomonic, stereographic, and orthographic.

Related to the Central Meridian is latitude of origin. Latitude of origin is the latitude that defines the center of the projection. That means that this is the latitude that will be in the center of the map projection. Changing the latitude of origin moves the projection about the horizontal axis to determine which portion of the earth will be shown in the middle the map.

### Light Source Location

There are three possible light source locations.

Gnomonic: In the gnomonic light source position the light source is placed at the center, or core, of the earth. The light is then projected through the earth’s surface and projects the landmasses onto the developable surface. In the illustration, the earth is the bottom circle, the lines are the light source, and the solid white line is the developable surface, in this case a plane. The top circle and the dotted lines represent where the earth will be compressed, or stretched, based on the position of the light source. In the case of the gnomonic light source, looking at a polar projection, locations at the center of the earth are held closer to true. However the locations towards the extremities are elongated as they are stretched out to meet the developable surface.

The light source location map projection parameter is the location of the hypothetical light source in reference to the globe being projected. Remember that there are three parts of a map projection: ellipsoid, the light source, and the developable surface. The light source is what projects the surface of the earth onto the developable surface and there are three primary positions in which we can place the light source.

Figure 19: Gnomonic Light Source
Figure 20: Stereographic

Stereographic: In the stereographic projection the light sources are placed at the opposite side of the earth from where the developable surface has its secant or tangent intersection. In this case we see a less severe differential between where the earth is compressed and elongated, but no location is clearly free from distortion.

Orthographic: In the orthographic position the light is placed at a theoretical infinite distance from Earth opposite from the point of intersection or tangency. This allows formidable distortion in the center the projection however there is significant compression at the extremities of the map.

Figure 21: Orthographic

Great Circle: A great circle is a circle that results when a plane intersects the earth going through the center effectively dividing the earth into two equal parts. Based on this definition, all meridians and the equator are considered to be great circles as they all divide the earth into two equal halves. An arc segment of a great circle is the shortest path between two points on a spherical surface. Therefore, to determine the shortest distance between two points on earth you would first construct a great circle and then calculate the distance along that segment.

Figure 22: Examples of Great Circles

Small Circle: A small circle is a circle the results when a plane intersects the earth but does not go through the center effectively dividing the earth into two equal parts. Based on this definition of a small circle, all parallels, except the equator, and other lines that do not pass through the center of the earth are small circles. A small circle is not the shortest distance between two points on the spherical surface even though it may appear to be.

Figure 23: Small Circle

Rhumb Line: A Rhumb Line is a curve that crosses each Meridian at the same angle. This is also known as a loxodrome. Rhumb lines are useful because they are easier for navigation because a rhumb line follows a constant bearing or azimuth. While the Rhumb line is easy to use for navigation it does require that you follow a longer distance in the shortest possible distance which would be a great circle.

Figure 24: Rhumb Line

Importance of Great and Small Circles: There are three important reasons that you should know about circles. First, on an azimuthal projection all lines drawn through or to the center of the projection are great circles. Second, on an azimuthal projection all lines not drawn to the center are considered small circles. And third on a gnomonic azimuthal projection all lines drawn are great circles.

These three points illustrate that even though the map user may draw a straight line on a map projection, it may not represent the shortest distance between two points. Therefore it is important that you consider how different map projections portray small and big circles and what implications that will have for your map user.

## 6.4: Map Projection Families

The azimuthal map projection family, also known as the Planar or Zenithal map projection family is when a spheroid is projected onto a flat plane. Based on this interaction between the spheroid and a developable surface we see deformation outward a series of concentric bands from the center. The azimuthal family of map projections is commonly used to display larger scale maps and Polar Regions of the earth. The azimuthal map projection family has three aspects: polar, which is considered the normal aspect, oblique, and equatorial. Figure 22 is an illustration of an azimuthal map projection with a polar aspect. The plane’s pointed tangency is the North Pole, and the earth is then unfolded onto the plane which causes deformation in a radial pattern way from the North Pole.

Figure 25: Polar Aspect

Figure 26 is an illustration of an azimuthal map projection with an oblique aspect along the mid-latitudes of earth centered over Europe. Here, the developable surface’s pointed tangency is centered over Europe, and the earth is folded out onto the plane.

Figure 26: Oblique Aspect (Mid-Latitudes)

Figure 27 is an illustration of an azimuthal map projection within equatorial aspect. The developable surface’s pointed tangency is on the equator.

Figure 27: Equatorial

Cylindrical Family: The second map projection family we will discuss is the cylindrical family. The cylindrical family projects the spheroid onto a cylinder. The spheroid is deformed and increasing bands of exaggeration towards the outer edges of the map plane. The cylindrical map projection family is commonly used to display the entire world, or medium and large scale mapping. One unique characteristic of the cylindrical family is that all parallels and meridians intersect at 90° angles. Figure 28 is an illustration of the Miller cylindrical projection. In this projection, the center of the cylinder touches the equator.

Figure 28: Miller Cylindrical Projection

Cylindrical Aspects: There are three cylindrical aspects: equatorial, just consider this the normal aspect, transverse, which is also known as the polar aspect, and the oblique aspect. Figure 29 is an illustration of the equatorial aspect for the cylindrical family of map projections. For this aspect, the line of tangency follows the equator.

Figure 29: Equatorial (normal)

In Figure 30 you can see the transverse, or polar, aspect for the cylindrical projection. In the case of the transverse aspect, the cylinder can have a single line of tangency following a Meridian from North to South Pole, or a secant intersection for the cylinder is still centered over the pole.

Figure 30: Transverse (polar)

In the oblique aspect, shown in Figure 31, the line or lines of intersection between the developable surface and the spheroid touch the Earth between the equator and the poles.

Figure 31: Oblique

Conic Family: The third map projection family is the conic family. In the conic family the developable surface is a cone onto which the spheroid is projected. We see deformation and concentric bands parallel to the standard parallels of the map projection. The conic family is commonly used to display areas of earth having a greater east-west extent. Figure 32 is an illustration that shows how the cone developable surface interacts with the spheroid. In this case, the cone is pushed through the surface of the earth so that it has two secant lines of intersection. The lines of intersection follow parallels which reduce distortion in the east-west direction.

Figure 32: Conic Projection

With respect to aspects, the conic map projection family is typically presented in the normal aspect where the axis of the cone is in line with the axis of the spheroid as shown in Figure 33.

Figure 33: Conic Aspects

Mathematical Family: The final map projection family we will discuss is the mathematical family. To create a map projection of the mathematical family the spheroid is projected onto a mathematical surface that is not a cone, plane, or cylinder. Deformation can vary quite widely depending on the shape of the developable surface.

Additionally, since the developable surfaces in this mathematical family can vary so greatly, there is no common map purpose that is used with the mathematical family of map projections. In some cases developable surfaces in the mathematical family of map projections may resemble cylinders, cones, or planes that have been slightly deformed or warped for a specific need. In these cases the developable surfaces may be referred to as pseudo-cylindrical, pseudo-conic, or pseudo-azimuthal.

Figure 34: Sinusoidal
Figure 35: Hammer-Aitoff

## 6.5: Map Projection Properties

A map projection property is an alteration of area, shape, distance, and direction on a map projection. These map projection properties exist because the conversion from a three dimensional object, such as the earth, to a two-dimensional representation, such as a flat paper map, requires the deformation of the three-dimensional object to fit onto a flat map. The three-dimensional spherical surface is torn, sheared, or compressed to flatten it onto a flat developable surface.

Four Map Projection Properties: These map projection properties are area, shape, distance, and direction. These four map projection properties described for facets of a map projection that can either be held true, or be distorted.

Area and Shape: Area and shape are considered major properties and are mutually exclusive. That means, that if area is held to its true form on a map, shape must be distorted, and vice versa.

Distance and Direction: Distance and direction, on the other hand, are minor properties and can coexist with any of the other projection properties. However, distance and direction cannot be true everywhere on a map.

Equal Area Map Projection: The equal area map projection, also known as the equivalent map projection, aims to preserve the area relationships of all parts of the globe. You can easily identify most equal area map projections by noting that the meridians and parallels are not at right angles to each other. Additionally, distance distortion is often present on equal area map projections and the shape is often skewed.

Even with the distortion of distance and shape, equal area map projection is useful for general quantitative thematic maps when it is desirable to retain area properties. This is especially useful for choropleth maps, when the attribute is normalized by area. Holding area properties to be true allows for an apple to apple comparison of density between different enumeration units such as counties. Two types of equal area map projections include: Cylindrical Equal Area and Hammer-Aitoff.

Cylindrical Equal Area: The cylindrical equal area map projection is an example of an equal area, or equivalent map projection, which aims to keep the areal relationships of all parts of the globe correct.

Figure 36: Cylindrical Equal Area

Hammer-Aitoff: A second example of an equal area projection is the Hammer-Aitoff map projection. Again, like the cylindrical equal area projection, this map projection aims to hold areas true. Also note that on this map projection, the parallels and meridians do not intersect at 90° angles, which is a hint that lets us know that this may be an equal area projection.

Figure 37: Hammer-Aitoff

Conformal Map Projections: Conformal map projections, also known as orthomorphic map projection, preserves angles around points and shapes of small areas. Additionally it allows for the same scale in all directions to or from a single point on the map. Conformal map projections can usually be identified by the fact that meridians intersect parallels at right angles, areas are distorted significantly, its’ small scales, and the shapes of large regions may be severely distorted. Even with the potential for large shape distortion conformal map projections are useful for large scale mapping and phenomena with circular radial patterns such as radio broadcasts for average wind directions.

The Mercator projection, perhaps the most famous of all map projections, is a conformal map projection that preserves shape. However notice the massive amount of distortion in the lower latitudes towards the South Pole and the northern latitudes near the North Pole. Also note that the parallels and meridians intersect at 90° angles.

Figure 38: Conformal Map Projections

Equidistant Map Projection: The third map projection family is the equidistant map projection which aims to preserve great circle distances. That means a distance can be held true from one point to all other points or from a few select points to others but not from all points to all other points. It is also important to note the scale is uniform along these lines a true distance from the select points on the map. Identifying marks of the equidistant map projection are that they are neither conformal nor equal area, and look less distorted. Equidistant map projections are useful for general purpose maps and Atlas maps.

An example of an equidistant map projection is this equidistant cylindrical map projection. Notice that compared to the conformal map projection there is less distortion at the North and South Poles, and the shapes do not look overly distorted.

Figure 39: Equidistant Cylindrical

Azimuthal Map Projection: The azimuthal map projection, also known as the true direction map projection, preserves direction from one point to all other points in the map. It is important to note that direction is not true between non-central points. Direction is only true when measured to or from the specific points chosen to have true direction. Azimuthal map projection is most useful for preserving direction two or one from point, often used for navigation.

The azimuthal equidistant map projection is an example of a true direction map projection that also holds distance to be true. While not all azimuthal map projections look like this, this particular map projection allows you to measure across the poles and around the world to determine true distance and direction from a single point.

Figure 40: Azimuthal Equidistant

Combined Map Projections: As seen on a few example map projections previously we can combine map projection properties onto a single projection. For example an equal area map projection can also combine parts of it azimuthal map projection. Conformal can combine with azimuthal, equidistant can combine with azimuthal, and azimuthal can combine with equal area, conformal, and/or equidistant.

Note: Yes denotes map projections can be combined. No denotes that map projections cannot be combined.

Compromise Map Projection: There is another map projection family that does not try to hold a single map projection property true. This map projection family is known as the minimum error, or compromise map projection. The goal of the compromise map projection is to simultaneously minimize all for map projection properties, but may not hold any of the four map projection properties as true. The compromise map projection is useful for general geographic cartography.

Robinson Map Projection: An example of a compromise map projection is the Robinson map projection. It does not greatly distort any of the four map projection properties nor does it hold any of the four properties true. However what is nice about the Robinson map projection is that it does a reasonable job of showing the true shape, distance, direction, and size of the features of the earth.

Figure 41: Robinson Map

## Other Resources

### Map Projection Reference Websites

Here are three recommended map projection reference websites.

1. USGS Map Projections Poster: The USGS map projections poster provides illustrations and information about many common map projections, and useful matrices to show when the use of a particular map projection is appropriate.
2. Radical Cartography Projection Reference: The radical cartography projection reference also shows helpful illustrations, and information about when to use certain map projections.
3. Flex Projector: Flex projector is a free piece of software that allows you to create your own map projections, and export them into projection files for use elsewhere.

## Summary

This chapter covered map projections. This part of the chapter identified the determining factors one should consider when creating map projections, deformation and distribution of the projection, and how to determine which map projection to use for a project.

## Credits

This work by the National Information Security and Geospatial Technologies Consortium (NISGTC), and except where otherwise noted, is licensed under the Creative Commons Attribution 3.0 Unported License.

Authoring Organization: Del Mar College

Written by: Richard Smith

Copyright: © National Information Security, Geospatial Technologies Consortium (NISGTC)

Development was funded by the Department of Labor (DOL) Trade Adjustment Assistance Community College and Career Training (TAACCCT) Grant No. TC-22525-11-60-A-48; The National Information Security, Geospatial Technologies Consortium (NISGTC) is an entity of Collin College of Texas, Bellevue College of Washington, Bunker Hill Community College of Massachusetts, Del Mar College of Texas, Moraine Valley Community College of Illinois, Rio Salado College of Arizona, and Salt Lake Community College of Utah.

This workforce solution was funded by a grant awarded by the U.S. Department of Labor's Employment and Training Administration. The solution was created by the grantee and does not necessarily reflect the official position of the U.S. Department of Labor. The Department of Labor makes no guarantees, warranties or assurances of any kind, express or implied, with respect to such information, including any information on linked sites, and including, but not limited to accuracy of the information or its completeness, timeliness, usefulness, adequacy, continued availability or ownership.