Modified transverse Mercator

The modified transverse Mercator (MTM) coordinate system is a metric grid-based method of specifying geographic locations, similar to the Universal Transverse Mercator coordinate system (UTM). However, MTM uses a transverse Mercator projection with zones spaced 3° of longitude apart, one half of the spacing of UTM zones.

Properties

Like any Mercator projection, MTM is a conformal map projection, so angles at any point, and the shape of small areas, are true. However, scale varies slightly by longitude.

Due to its narrower zones, MTM exhibits a scale factor of 0.9999 (distortion of 1:10,000) in the midpoint of a zone, versus 0.9996 (1:2,500) for UTM. This makes it more suitable for parametrizing and displaying land surveys, since grid distances (calculated from MTM coordinates) differ less from ground measurements. For moderate-scale surveys, there is easier conformance (based on calculation, without correction) with land surveyors' target maximum distance (closure) error of 1:5000.^[1]

Like UTM, the direction of true north is not perfect grid north away from each zone's central meridian, i.e. non-central meridians are slightly curved. The deviation between grid and true north is called the angle of convergence, and is roughly proportional to the east-west (longitudinal) distance away from the central meridian, and the sine of the latitude. This deviation can be on the order of 1° in temperate latitudes and so needs to be taken into account.

Usage

The MTM coordinate system is used by various government institutions in Canada in particular.^[2][3][4] Canada is covered by 32 zones, generally following the canonical 3° longitude spacing, with some adjustments to avoid zone boundaries passing through densely populated areas. Just like UTM, MTM eastings (X coordinates) do not align across zone boundaries, so a given map needs to be displayed wholly in one MTM zone. However, the distortion from using a given MTM zone's projection slighly outside its intended 3°-wide area is minimal, so slight mapping excursions outside the zone area are tolerated. Nevertheless, for large-scale maps, such as of a whole Province or all of Canada, different map projections need to be used.

MTM is conceptually similar to the Gauss-Kruger coordinate system historically used in parts of continental Europe, even though the parameters (and scale factor) are different.

Mathematical formulae

Coming from the same family of projections, conversion between latitude and longitude and MTM coordinates uses the same mathematical formulae as those for UTM. However, the parameters are adapted for the different zone configuration.

• The central meridians ${\displaystyle \lambda _{0}}$ reflect the 3° spacing.
• The factor ${\displaystyle k_{0}=0.9999}$.
• The false easting ${\displaystyle E_{0}=304800}$m by convention, rather than ${\displaystyle E_{0}=500000}$m for UTM.
• The false northing ${\displaystyle N_{0}}$ remains 0 as for UTM. However, a given geographical point's northing (Y coordinate) is in general slightly different in UTM and MTM since ${\displaystyle \lambda _{0}}$ and ${\displaystyle k_{0}}$ are not the same.

Note MTM (and UTM) coordinates depend meaningfully on the horizontal datum being used, so for instance (in Canadian applications) it is important to clarify whether the datum is NAD 27, NAD 83, or NAD 83 (CSRS). Technically, the choice of datum affects geodetic coordinates, not their projection in MTM (or UTM). However, the datums underlying concrete MTM and UTM realizations in common use often vary, and therefore appropriate conversion between those datums plays a material role in coordinate conversion. Nevertheless, over a small area being represented, relative easting and northing coordinate differences between MTM and UTM differ only by a scaling given by the ratio of the (local) scale factors ${\displaystyle k}$, and by a rotation given by the difference of the respective angles of convergence. So differences in grid bearing and grid distance measured on two small-scale UTM and/or MTM maps are wholly determined by their longitude deviation from their respective central meridians.