What’s This UTM All About? by Norman Thyer
We use topographical maps a lot in our activities, and, I hope, carry one with us whenever we venture off well-marked trails. Many of those maps have a grid of squares marked on them. What do these squares represent, and how are their positions determined?
Most maps that we are familiar with depict on a flat surface a portion of the earths curved surface. The process used to transfer points from one surface to another is called a mapping or projection, and in a transfer from a curved to a flat surface, some distortion is inevitable. The scale of such a mapping generally changes from one point to another, and at a given point, the scale may change with direction too. So a distance of 1 km in a northerly direction and a distance of 1 km in an easterly direction may not measure the same on a map.
However, there are some projections in which scale at a given point is independent of direction, and these are called conformal projections. Conformal projections also have the property that the intersection angle between two lines on the map is equal to the corresponding angle on the ground.
The earth is not a perfect sphere. Because of flattening at the poles, any cross-section through the poles and along a meridian is approximately an ellipse. In the 19′ century, Carl Gauss devised a conformal projection for a strip of the earth’s surface centred on a meridian, such that the scale would be the same everywhere on that meridian. It is known as the Gaussian projection. However, while the scale is the same everywhere on the central meridian, it changes gradually as one moves away from that meridian. So for instance, while the scale might be exactly 1:100000 on that meridian, it would be different elsewhere, and significantly different far from the central meridian. To minimize this distortion of scale, the Gaussian projection has been modified. It is normally applied to a zone of width six degrees, extending three degrees to each side of the central meridian. Also the scale on the central meridian is modified by a factor of 0.9996, to minimize deviations of scale from the nominal value across the 6-degree-wide strip.
So if the scale of your map is given as 1:50000, it will have exactly this value only along a line about two or three degrees from the central meridian. The resulting projection is called the Universal Transverse Mercator (UTM) projection, and it appears to be the most common projection used on Canadian topographic maps. One can draw a grid of squares on a map. When this is done on a map with UTM projection, with one grid line along the central meridian, we have the UTM grid. 1 km squares on the grid generally represent 1 km distances on the ground to within 0.04% accuracy. However, distortion in its various forms is always lurking, and although the grid line corresponds to true north on the central meridian, it deviates from it elsewhere. So a correction, known as the convergence of meridians, is needed to relate grid north to true north.
In our area, the UTM zone stretches between longitudes 114 and 120 degrees west, and is centred on the 117 degree meridian, which passes through Sunshine Bay. There, grid and true north coincide, but by the time one gets to Cranbrook, they differ by over one degree, and on the edge of the zone in Calgary, at 114 degrees, the discrepancy is over two degrees.
Once we have the UTM grid marked on the map, we can use it to refer to locations of various points, as an alternative to latitude and longitude. In the usual way of analytic geometry, we can use X to refer to distance east and Y to distance north from a suitable origin. It is customary to give X the value 500km, or 500,000m, on the central meridian, to avoid negative values within a zone, while Y is the distance from the equator.
The numbers marked on the grid lines and in the margins of Canadian 1:50000 and 1:100000 topographic maps give the tens and units digits in kilometres. For the hundreds of digits, you must look in the margins near the corners of the map. On 1:250000 maps, only the tens digit is given for each of the grid lines, which are at 10km spacing.
One advantage of UTM coordinates over latitude and longitude is that the computation of distance and direction between two points is greatly simplified. On a basic scientific calculator, it amounts to using the difference in X-values and the difference in Y-values as inputs to the rectangular-to-polar-conversion function. However, one must interchange the X and Y inputs from what the calculator instructions tell you, because mathematicians measure angles anticlockwise from the X-axis, while navigators measure them clockwise from the Y-axis. To the resulting direction, you must then add the correction for meridian convergence at the observing station. The “direct line” between two points is actually slightly curved on the map, but the correction for this is so small that it is needed only in work of very high accuracy.
UTM coordinates can be related to latitude and longitude. Unfortunately, the formulas for doing so are complicated, but for limited accuracy over a limited area, simpler formulas may be adequate. For our area, between latitudes 49deg and 51deg, and between longitudes 115deg and 119deg, the following formulas are accurate to about 100 metres. LA = 45.15856 + .008992 • Y – (X – 500)A 2 * (7.954•1 0A – 7 + 9• 1 0A – 1 1 • Y) LO = 117-(X – 500)*(.01255 + 2.6•10A – 6 • Y) X = 500 + (LO – 117)* (LA *1.49 – 146.153) -5022.017 +111.20857 • LA +(LO – 117)A2 • (1.195- .0143 * LA)
For compass work, the “convergence of meridians” is given adequately by: (X 500) • (0.0107 + 0.0000034 • (Y 538.65)) This amount should be added to the grid bearing to give the true bearing. More approximately, it changes by about 0.75 degrees for each longitude degree away from 117deg, being positive east of the 117 meridian and negative west of it.
Also, the magnetic declination (east) for 1 January, 1999 can be given by the formula: 18.435 – 0.00392 • X + 0.004286 • Y In all these formulas, LA and LO refer to latitude and longitude in degrees. X and Y refer to UTM coordinates in kilometres. Throughout out region, the value of Y lies between 5000 and 6000km, so for V. the initial digit “5” representing 5000 can be omitted without ambiguity, and it should be omitted for use in these formulas. In other words, in values of X and V. there should be 3 digits before the decimal point. As examples, for Old Glory X=433.6 and Y=444.3, while for Mt. Loki X=517.9 and Y=520.7 Hence Mt. Loki is in the direction 47.1deg from Old Glory, at a distance 114km. In the formulas, 42 means “to the power of 2”. These formulas can be used in this form in a computer program using the BASIC language.
Finally, a word of warning is justified. It was mentioned that the earth’s cross-section along a meridian is approximately an ellipse. For UTM purposes, a perfect ellipse is used, and it is chosen to coincide with the earth’s real surface as closely as possible. However, different ellipses have come into use, depending on whether one wants the closeness of fit to be the best for the whole earth or for a certain region, such as North America. Most of the topographic maps that we use are based on an ellipse referred to as the North American 1927 datum, and the Where Are We? formulas given above apply to it. Some of the new 1:20000 maps use another ellipse known as NAD83. Consequently the grid on one map may differ in location from the grid on another map by as much as 200 metres in our area.
Also, especially with maps that are photocopied, one must beware of distortions, probably arising in the photocopying process, in which the scale in the W-E direction differs from that in the S-N direction.
For the Kootenays, a practical rule for converting UTM coordinates from the NAD27 to the NAD83 (NGS84) system is: For the easting (x-coordinate), subtract 79 metres. For the northing (y-coordinate), add 212 metres.