Projections - CGRSC

Projections form a category of coordinate systems. They are based on mathematical formulas designed to project coordinates to a different but similar surface to achieve a particular purpose, such as preserving the scale, form or surface area of objects. Each type of projection has its own advantages and disadvantages, and the selection of one or another depends mainly on the needs of the user. Since the projection cannot be done without distortion, one must decide about the characteristic, which is intended to be shown accurately at the expense of others, or a compromise of several characteristics. In surveying and engineering, where many projects are conducted over small areas, it is generally simpler to work with planar coordinates, as systematic errors caused by the Earth’s curvature are negligible. For these reasons, Universal Transverse Mercator (UTM) and Modified Transverse Mercator (MTM) coordinates are commonly used in different fields of activities. The next two sections describe briefly these two projections.

Figure 6 shows the concept of the Universal Transverse Mercator (UTM) coordinate system involving an imaginary secant cylinder intersecting a spherical Earth to define projection zones. In this system, the Earth between latitudes of N84° and S80° is divided into 60 zones of 6° in longitude (3° on each side of a central meridian). The first zone (Zone 1) covers longitude 180° to 174° west of the Greenwich meridian; zone numbering increases eastward. In the northern hemisphere, the intersection of the equator and the central meridian of the zone makes the origin where the pair (x, y) is (500,000 m, 0 m). In the southern hemisphere, the origin is the same intersection, but the pair (x, y) is now (500,000 m, 10,000,000 m) to avoid negative y coordinates. In both cases, the planar coordinates increase eastward and northward. The lines of true scale are approximately 180 km east and west of the central meridian. The scale factor is less than 1 between them, reaching a minimum of 0.9996 at the central meridian, and larger than 1 outside them. The distortion due to the scale factor is less than 0.1%. UTM is a conformal projection system because the scale factor is applied in all directions. Canada is covered by sixteen UTM zones, numbered from 7 to 22.

Figure 6. Concept of the Universal Transverse Mercator (UTM) projection.

The Modified Transverse Mercator (MTM) coordinate system is similar to UTM, but the zones are reduced to 3° of longitude (1.5° on each side of a central meridian). The narrower zones can fit better to the survey area, and therefore, the scale factor along the central meridian becomes 0.9999, resulting in less distortion when compared to UTM. The first MTM zone (Zone 1) has its central meridian at W53°, which crosses near St-John’s, Newfoundland. The zone numbering increases westward. There are 32 zones for Canada. However, some zones (between 1 and 13) are modified by half a degree to avoid having a zone boundary to go through a city or town. The MTM projection, which is also called a 3-degree Transverse Mercator (3TM) projection, may be considered as a densification of the UTM projection.

In order to avoid negative coordinates, the true origin of the grid coordinates is shifted by introducing the false northing and false easting. As Canada is completely located in the northern hemisphere, there is only one false northing (0 m at the equator) and one false easting (304,800 m at the central meridian). Figure 7 shows the concept of the MTM projection for Zone 10, which includes the city of Toronto. The province of Ontario is covered by Zones 8–17 (10 zones). As the figure shows, the scale factor changes from 0.9999 at the central meridian to 1.00008 at the zone boundary.

Figure 7. The concept of the Modified Transverse Mercator (MTM) projection