Map Projections

The process of systematically transforming positions on the Earth's spherical surface to a flat map while maintaining spatial relationships, is called map projection. This projection process is accomplished by the use of geometry and, more commonly, by mathematical formulas. In geometric terms, the Earth as a spheroid (that is, a slightly flattened sphere), is considered an undevelopable shape, because, no matter how the Earth is divided up, it cannot be unrolled or unfolded to lie flat. Some of the simplest projections are made onto geometric shapes that can be flattened without stretching their surfaces. These shapes or forms are considered to be developable. Examples of shapes that reflect these properties are cones, cylinders, and planes.

The CONE, CYLINDER and PLANE are developable geometric shapes. The curved surface of the Earth can be projected on to these forms which can be unrolled to make a flat map. (The PLANE is already a flat surface!)

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Figure 1. Diagram of Developable Surfaces

These geometric shapes can either be tangent or secant to the spheroid. In the tangent case the cone, cylinder or plane just touches the Earth along a single line or at a point. In the secant case, the cone, or cylinder intersects or cuts through the Earth as two circles. (The secant case for the plane intersects as one circle.) Whether tangent or secant, the location of this contact is important because it defines the line or point of least distortion on the map projection. This line of true scale is called the standard parallel or standard line.

With conical and cylindrical projections, the axis of these shapes usually corresponds to the axis of the spheroid (Earth); the exception is the oblique case. When a cone or cylinder is cut along any meridian to produce the final projection, the meridian opposite the cut line is called the central meridian. Planar projections may be oriented in different ways: polar, equatorial and oblique.

Projections may be oriented in different ways with respect to the Earth's axis. For cylindrical projections this is achieved by changing the position of the lines used for tangency or secancy. For planar projections, the point of contact with the Earth can be altered. This point determines the aspect used and functions as the focus of the projections.

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Figure 2. Drawing of Projection Orientation

Most projections are derived from mathematical formulas, but some are easier to visualize as projected on to a developable surface. Therefore, projections are commonly classified according to the geometric surface from which they are derived: conical, cylindrical, and planar (azimuthal or zenithal). The many projections that cannot be easily related to these three surfaces are described as pseudo, modified or individual (or unique).

Conical Projection

In the conical case, we can visualize the Earth projected onto a tangent or secant cone, which is then cut lengthwise and laid flat. The parallels (lines of latitude), are represented by concentric circular arcs, and the meridians (lines of longitude), by straight, equally spaced, radiating lines.

This type of projection is used for mapping mid-latitude regions, such as Canada and the United States. The result is less overall shape distortion of land and water areas. The Lambert Conformal Conic projection is a commonly used version of the conic type.

The polyconic projection (from the Greek, "poly" meaning many), envelopes the globe with an infinite number of cones, each with its own standard parallel. The parallels are non-concentric, while the central meridian is straight. Other meridians are complex curves. Scale is true along each parallel and along the central meridian.

Cylindrical Projection

In the cylindrical case, the Earth is projected on to a tangent or secant cylinder which is also cut lengthwise and laid flat. The result is an evenly spaced network of straight, horizontal parallels and straight, vertical meridians. A straight line between any two points on this projection follows a single direction or bearing, called a rhumb line. This feature makes the cylindrical projection useful in the construction of navigation charts.

When the cylinder is used as a surface to project the entire World on to a single map, significant distortion occurs at the higher latitudes, where the parallels become further apart, and the poles cannot be shown. The famous Mercator projection, is the best known example of this class and one of the earliest of all projections (circa 1569).

Planar or Azimuthal Projection

With the planar projection, a portion of the Earth's surface is transformed from a perspective point to a flat surface. In the polar case, the parallels are represented by a system of concentric circles sharing a common point of origin from which radiate the meridians, spaced at true angles. This projection shows true direction only between the centre point and other locations on the map.

Although these projections are most often used to map polar regions, they may be centred anywhere on the Earth's surface. The gnomonic is one type of planar projection on which any great circle appears as a straight line. A great circle is the circle created when a plane cuts the Earth through its centre. This term is most often used in the expression, "great circle route", which is the shortest path between two points on the Earth. This information is most useful in air navigation, because aircraft usually travel along great circle routes. This explains why aircraft flying from Toronto or Montréal to Japan fly near the North Pole!

The AZIMUTHAL family of projections, also called ZENITHAL or PLANAR, is produced by transforming the Earth's surface onto a plane. Members of the family are distinguished from each other by the different perspective points used to construct them. For the GNOMONIC projection, the perspective point (like a source of light rays), is the centre of the Earth. For the STEREOGRAPHIC this point is the opposite pole to the point of tangency, and for the ORTHOGRAPHIC the perspective point is an infinite point in space on the opposite side of the Earth.

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Figure3. Drawing of The Azimuthal Family of Projections

Other Projections

The pseudoconic and pseudocylindrical projections are both constructed in the same manner as their unprefixed counterparts, except, they both have curved meridians instead of straight ones.

Modified projections are versions of a projection to which changes have been made to reduce or modify the pattern of distortion, or to add more standard parallels.

Many other projections, some of which are in common use, cannot be easily related to one of the three developable geometric forms. These can be classed as individual or unique projections. Examples of this group are the Bacon Globular, Peirce Quincuncial, Armadillo, Adams World in a Square I, and Van der Grinten I, II, III, or IV.

The Challenge: The Earth is a spheroid, and the best way to represent it is with a globe. This scale model retains all of the desired properties necessary to produce the perfect map: area, distance, direction, and shape are all accurately represented. However, when this spheroid is projected on to a flat map, all these properties cannot be retained simultaneously. In fact, each projection is a compromise, showing some properties accurately, while at the same time, allowing others to be distorted. The extent to which these properties are preserved, provides another method of classifying projections.

Despite the problems related to distortion, all projections do retain one important feature, that of positional accuracy. By transforming the graticule (a gridded reference network of latitude and longitude lines, encompassing the globe) to a map, the spatial relationship between points on both surfaces is maintained.

The Factor of Scale: Our interest in the significant properties of map projections begins with map scale. A small-scale map portrays a large area and a large-scale map portrays a small area of the Earth. If the area to be mapped is small (only a few square kilometres, as for example, a county, township or city), then the occurrence of error that results from projecting the curved surface of the Earth, to the flat surface of the map, is negligible. In relation to the surface of the entire Earth, a small area is conceptually as flat as the sheet of paper on which we wish to represent it. Only when larger areas of the Earth are to be mapped, such as provinces, countries or continents, do the following properties play a more important role in the selection of projections.

A map projection is said to be equal-area or equivalent if it portrays areas over the entire map so that they retain the same proportional relationship to the areas on the Earth they represent. The creation of this projection results in shapes and angles being greatly distorted. This distortion increases with distance away from the point of origin.

A projection which is equidistant maintains constant scale (i.e., true distance), only from the centre of the projection or along great circles (meridians), passing through this point. For example, a planar equidistant projection centred on Montréal, would show the correct distance to any other location on the map, from Montréal only. This property is accomplished at the expense of distorting area and direction.

A projection is azimuthal or zenithal when angles or compass directions from one central point are shown correctly to all other points on the map. However, to achieve this property, shapes, distances and areas are badly distorted.

A map projection is conformal, (also known as orthomorphic or equiangular) when all angles at any point are preserved. Or, the scale at any point is the same in every direction. Lines of latitude and longitude intersect at right angles, and shapes are maintained for small areas. However, in the process of projection, the size of large areas is distorted.

The following table shows which pairs of properties can be combined in one projection:

Table 1. Which pairs of properties can be combined in one projection?

There are many map projections in use that do not possess any of the desired properties mentioned above. However, they are still suitable for certain applications and, indeed, may be very useful if a compromise is reached and a number of properties are reasonably preserved.

Those projections that succeed in showing the entire World on one map, often encounter serious problems of distortion. World projections, by their nature, usually distort regions shown at the extremes of the projection. To improve the depiction of these distorted areas, "interrupted" forms, splitting the projection into gores, have been developed. Following this approach, many landmasses (or oceans), can have their own central meridian, resulting in true shapes or conformality in each region of the projected map.

Examples of World Map Projections:

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Figure 4. Goode's Homolosine Equal-area Projection. (with the oceans interrupted to show the continents)

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Figure 5. Miller Cylindrical World Map Projection

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Figure 6. Eckert IV Equal-area World Map Projectio

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Figure 7. Sinusoidal Equal-area World Map Projection

A situation worth noting in regard to map projections, and their properties, is the edge matching of adjacent regions. This problem is frequently encountered by cartographers and map readers, particularly when dealing with maps in a series. In order to fit two or more separate maps exactly along their edges, a number of parameters must be maintained:

1. the maps must be constructed with the same projection;
2. they must be at the same scale;
3. they must have the same standard parallels; and
4. they should be based on the same ellipsoid reference datum
(in other words, longitude and latitude are calculated from an ellipsoid, such as the reference surface NAD83 - North American Datum)

The Transverse Mercator projection, which lends itself to edge-matching operations, is commonly used for map series, such as the 1:50 000 and 1:250 000 scale National Topographic System (NTS), produced by Geomatics Canada. Other factors influencing the accuracy of edge matching are: the instability of map paper when exposed to changes in temperature and humidity and errors of cartographic drafting or surveying.

The selection of the best map projection depends upon the purpose for which the map is to be used. For navigation, correct directions are important; on road maps, accurate distances are of the major concern and on thematic maps (depicting area-related data), the correct size and shape of regions is important. Other considerations in choosing the best projection are the extent and location of the area to be mapped. With reference to map extent, the larger the area being mapped the more significant is the curved surface of the Earth and, therefore, the greater the distortion of the desirable properties. For map location, the following conventions can be applied: for low-latitude regions, use cylindrical projections; for middle-latitude areas, use conical; and for polar regions, use planar.

The following table provides a summary of more common map projections, their properties and use:

Table 2. Summary of More Common Map Projections, Their Properties and Use

* Limitations apply
** United States Geological Survey - the supplier of base and thematic maps covering the United States of America.

The following maps show Canada projected in different ways:

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Figure 8. Gnomonic Azimuthal Projection

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Figure 9. Lambert Conformal Conic Projection

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Figure 10. Transverse Mercator Projection

Dana, Peter H. 1995. Map Projections. The Geographer's Craft Project. Austin. Dept. of Geography, University of Texas at Austin.

ESRI (Environmental Systems Research Institute, Inc.). 1994. Map Projections, Georeferencing spatial data. Redlands, California: ESRI.

Gersmehl, Philip J. 1991. The Language of Maps. Pathways in Geography Series, title no. 1. Indiana, Pennsylvania: Indiana University of Pennsylvania.

Greenhood, David. 1964. Mapping. Chicago: The University of Chicago Press.

Pearson II, Frederick. 1990. Map Projection: Theory and Applications. Boca Raton, Florida: CRC Press, Inc.

Raisz, Erwin. 1962. Principles of Cartography. New York: McGraw-Hill Book Company.

Robinson, Arthur H., and Randall D. Sale. 1969. Elements of Cartography. Third edition. New York: John Wiley & Sons.

Snyder, John P., and Philip M. Voxland. 1989. An Album of Map Projections. U.S. Geological Survey Professional Paper 1453. Denver: United States Government Printing Office.