# Chapter 6 Part 3: Map Projection Distortions

In this chapter you will learn about distortions caused by map projections. You will also learn about the tissot indicatrix which can be used to understand those distortions better.

## 6.6: Determining Distortions

A common method used to determine deformation on a map projection is called Tissot’s indicatrix. As we know all flat maps distort shape, area, direction, or length when displaying features of a three-dimensional object such as the earth. Tissot’s indicatrix helps to quantify the distortion and projection properties shown on the map projection.

Tissot’s indicatrix is composed of infinitely small circles centered at points on the earth. The earth is then projected onto a map using a map projection and we consider the shape of the circles after projecting the map to determine the deformation and distribution of error throughout the map.

To gain a better understanding of of Tissot’s indicatrix look at the image with five map projections. Remember that the circles were all originally the same size and were perfect circles. We will interpret Tissot’s indicatrix by seeing what happens to the circles when the earth is projected.

Figure 42: Tissot's Indicatrix

Interpreting Tissot’s Indicatrix: As you learn how to interpret Tissot’s indicatrix consider how the circles change on an equal area and conformal map projection. On an equal area map projection the circles will be transformed into ellipses but the area of the ellipses will be the same as the area of the original circles. That means although they will change shape they will not change in size. On a conformal map projection the circles will continue to be perfect circles but their size will vary over the map.

Conformal Projection Property: Here are a few common map projections so you can see how Tissot’s indicatrix performs. Figure 40 is a Mercator map projection which is a conformal map projection the perfect circles would continue to remain circular however they will be larger or smaller than the original circle size. This is due to fact that the conformal map projection property keeps shape true but must distort size as shape and size are two major map projection properties, and are mutually exclusive.

Figure 43: Mercator Map Projection

Equal Area Projection Property: Consider the flat polar quartic projection and how Tissot’s indicatrix performs on it. As the flat polar quartic projection is an equal area projection circles will not have the same shape as a perfect circle; however, they will have the same area as the original circle. As we see on this map projection the circles roughly have the same size although we are still seeing some areal distortion towards the North and South Poles.

Figure 44: Flat Polar Quartic Projection

Equidistant Map Projection: On an equidistant map projection such as the equidistant cylindrical projection, where distances are true between two points, the circles are formed into different shapes throughout the map projection. This tells us that this map projection distorts both shape, and size, however, we are not able to immediately tell that this is an equidistant map projection based solely on the size and shape of the circles.

Figure 45: Equidistant Cylindrical Projection

Aphylactic Map Projection: On an aphylactic map projection, which is another way of referring to a compromise map projection, we do not have perfect circles anywhere on the map; however, the distortion is not severe anywhere on the map. Again this is because the aphylactic map projection distorts all four properties but no one property is distorted much greater than the other.

Figure 46: Robinson Projection

## 6.7: Map Projection Types

In order to choose the optimal map projection for your map there are many things that you must consider.

Projection Properties: You must consider the projection properties and whether those properties are compatible with the purpose of your map. For instance, if you want users to compare density across the globe you would want to choose an equal area map projection so that areas are kept true allowing users to do a true comparison with respect to density.

Deformational Patterns: You also need to consider the deformation patterns of the map projections. Some spatial phenomena are better represented using certain deformation patterns.

Projection Center: You will need to consider where you wish to center the projection. Typically you center the projection over the area that is of interest to the map user.

Familiarity: You should also consider how familiar your map users are to the map projection. Depending on your audience they may have a preconceived notion of what a map looks like. Then if you show them a complex mathematical map projection it may confuse the user too much and they may not be able to utilize your map to its fullest.

Software Support: Note: The rest of this lesson will discuss broad recommendations for what types of map projections you should use for particular situations. The discussion will not be exhaustive, nor must it be followed exactly. It is a good exercise and is often expected for a cartographer to render the map using many different map projections including changing properties to suit their needs before choosing the final map projection. You should consider software support if you are making a virtual map. Not all GIS and mapping software support the same map projections; therefore, if you are sharing your produced map you should make sure that the recipient has access to software that supports the same map projection of your map. If your map is part of a larger map series then the map projection you choose for your map should allow the map to have the same look and feel of the other maps for continuity and ease of use.

### Recommended Map Projections for World Projections

Equivalent: Equivalent map projections are recommended for world projections. Review the following types of equivalent map projections to become familiar with what their map projections entail.

Mollweide: The Mollweide map projections are useful for mapping world distributions as it does a good job of keeping relative sizes in check about the equator and mid-latitudes, however does distort greatly at the poles as can be seen with Tissot’s indicatrix.

Figure 47: Mollweide

Sinusoid: The sinusoid all map projection is also useful for mapping world distributions. Like the Mollweide map projection it too does a reasonable job of keeping relative size intact around the center of the map and between the middle latitudes. However, as you approach the extremities of the map, distortion becomes quite great.

Figure 48: Sinusoid

(Aitoff-) Hammer: The Aitoff-Hammer map projection is useful for mapping world distributions and excels at reducing the shape distortion at the extremities. The trade-off is that it does not hold the shape as well across the entire projection as the previous two map projections.

Figure 49: (Aitoff-) Hammer

Goode’s Homolosine: The Goode’s Homolosine map projection is interesting because it is an interrupted map projection that maximizes distortion over bodies of water and minimizes distortion over landmasses. One of the most negative aspects of using a good small sign map projection is that it is unfamiliar to many users and may confuse them.

Figure 50: Goode's Homolosine

Minimum Error: Minimum error map projections are also recommended for world projections. This section will focus on the various types of minimum error map projections.

Robinson: The Robinson map projection is a good choice for mapping the world. It does a reasonably good job of maintaining equal area and equal shape across the earth but it does distort once you reach the extremities especially at the North and South Pole.

Figure 51: Robinson

Winkel Tripel: The Winkle Tripel map projection is another good minimum error map projection choice when you want to map the entire world. Some consider this to be the best compromise map projection yet for world projections as it does a reasonably good job of minimizing distortion across the entire globe without any one area being distorted significantly more than other areas.

Figure 52: Winkel Tripel

Cylindrical: Cylindrical map projections are recognized as a recommended map projection type for world projections. Cylindrical maps are popular and commonly used although these map projections are not appropriate when mapping the entire Earth.

Mercator: The Mercator map projection was originally developed for navigation. It is an important and useful map projection because lines of constant bearing are shown as straight lines of a map projection. The downside is there is extreme distortion at high latitudes which unfortunately has imbued a massively distorted mental map of the world in many people’s minds. Figure 53 is an illustration of the Mercator map projection, perhaps the most famous map projection of all. Note the significant distortion as you move north or south away from the equator which causes significant distortion in the northern and southern latitudes.

Figure 53: Mercator

Gall-Peters: The Gall-Peters projection was made to counteract the Mercator projection. The problem is this projection distorts third world areas thus rendering the map not useful because the shape distortion is simply too great.

Figure 54: Gall-Peters

### Map Projections Recommended for Continental Areas

There are two types of map projections that are recommended for Continental areas: Lambert Azimuthal Equal Area, and Bonne.

Lambert Azimuthal: The Lambert Azimuthal equal area map projection is quite versatile as the standard point can be placed anywhere. If you are mapping a hemisphere then the equatorial aspect will be your best choice. For this map projection, distortion is radial. Figure 54 is the Lambert Azimuthal Equal area map projection centered over the North America region of the world.

Figure 55: Lambert Azimuthal Equal Area Map Projection

Bonne: The Bonne map projection is an equal area conical projection. Scale is true at the central Meridian and each parallel. However shape distortion is severe in the northeast and northwest corners of this map projection. This map projection is best for mapping compact areas on one side of the equator.

Figure 56: Bonne Map Projection

### Map Projections for Multiple Size Countries at Mid-Latitudes

There are three types of map projections recommended for multiple size countries at mid-latitudes: Lambert Azimuthal Equal Area, Albers Equal Area Conic, and Lambert Conformal Conic.

Both the Albers Equal Area Conic and Lambert Conformal Conic map projections are great for entities that have a pronounced east-west orientation. Remember that the conic family of map projections in the normal aspect minimizes distortion of the east-west directions. Since the continental United States has a pronounced east-west orientation, the Albers Equal Area Conic map projection is great for mapping the continental United States.

In general, conic projections are adequate for mapping large entities within east-west orientation. As areas being mapped become smaller, selection of a map projection becomes less critical as the curvature of the earth is less of a factor. Review Figures 56-58. Figure 56 is an example of the Lambert Azimuthal Equal Area map projection is great for groups a small countries as well as a single medium-size country. It excels at keeping distortion in check for countries of expand in all directions. Figure 57 is an illustration of the Albers equal area conic map projection. It is similar to the map scratch that is similar to the next map projection. Figure 58 is an example the Lambert Conformal Conic map projection is similar to the previous Albers equal area conic map projection and is also useful for mapping countries at mid-latitudes.

Figure 57: Lambert Azimuthal Equal Area Map Projection
Figure 58: Albers Equal Area Conic
Figure 59: Lambert Conformal Conic

### Map Projections Recommended for Low Latitudes

There are there types of map projections that are recommended for low latitudes.

The Lambert azimuthal equal area map projection is great for mapping low latitudes.

A Cylindrical projection with the standard parallel at the equator is also a good choice.

The Mercator map projection is great if the focus is on and around the equator.

## 6.8: Projected Coordinate Systems

For zones within the state plane coordinate system each zone uses either the Lambert conformal conic map projection or the transverse Mercator map projection. The only exception is that Alaska uses the oblique Mercator for one of its zones. Since one of the desires of creating the state plane coordinate system was that if you work within a single zone all coordinates are in the first quadrant and positive, the y-axis false origin is typically placed 600,000 m, or 2,000,000 feet, west of the zones central Meridian.

The placement of the x- axis false origin is more arbitrary and may change from zones. You may run into a situation where the area that you are mapping crosses multiple state plane coordinate system zones. There are two common solutions to this problem. One is to use the most central zone if the zones are prominently east-west in orientation. The second option is to choose one north-south zone and adjust the central Meridian to provide a balanced look.

Figure 59 is an illustration of all of the state plane coordinate system zones of the contiguous United States. The zones that have a predominant east-west orientation are mapped using the Lambert conformal conic map projections is that map projection minimizes distortion in the East and West directions. The zones that have a prominent north-south orientation are mapped using the transverse Mercator map projection as that map projection minimizes distortion in the North and South directions.

Figure 60: State Plane Coordinate System

Universal Transverse Mercator (UTM): The Universal Transverse Mercator (UTM) coordinate system is generally not as accurate as a state plane coordinate system. This coordinate system uses the transverse Mercator projection with the central Meridian running through the center of each zone. The y-axis false origin is placed 500,000 m west of the zone central Meridian. The x-axis false origin for each zone is at the equator for North Hemisphere and 10,000,000 m south of the equator for the southern hemisphere.

Figure 60 is an illustration showing the universal transverse Mercator map projection. There are 60 zones that start at zone one which is at -180° longitude and heads East 360° around the earth. The zones are split into North and South about the equator.

Figure 61: Universal Transverse Mercator Coordinate System

## Other Resources

Introduction to UTM (5 minute video)

Stateplane Coordinate System (2 minute video)

## Summary

In this chapter you learned about distortions caused by map projections. You also learned about the tissot indicatrix which can be used to understand those distortions better.

## 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.

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