Question 4 of 7: Map Projections — Grid Factor, Convergence, UTM/MTM

Question 4: Map Projections — Grid Factor, Convergence, UTM/MTM (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Terrain distance and azimuth observations that must be reduced to the ellipsoid and then to a transverse-Mercator grid (UTM, and the Canadian MTM/3TM).

Find. (a) definition and use of the grid (combined) factor; (b) meaning and magnitude of meridian convergence at the central meridian and the zone edge, and whether it must be applied under UTM; (c) what MTM/3TM is and two conceptual differences from UTM.

(a) Grid (combined) factor. The grid factor, or combined scale factor, is the single multiplier that converts a horizontal ground distance directly to its grid (map) distance. It is the product of two reductions: the elevation (sea-level) factor, which brings the ground distance down to the ellipsoid, and the scale (projection) factor, which carries the ellipsoidal distance up to the grid. Writing \(EF=\dfrac{R}{R+H}\) for the elevation factor (with \(R\) the mean Earth radius and \(H\) the height above the ellipsoid) and \(k\) for the point scale factor,

$$\text{Grid factor } CF = EF \times k = \frac{R}{R+H}\,k.$$

As a worked example, for \(H = 1000\ \text{m}\), \(R = 6371\ \text{km}\), \(EF = 6371000/6372000 = 0.9998431\); with a point scale factor \(k = 0.99970\), the combined factor is \(CF = 0.9998431 \times 0.99970 = \boxed{0.9995431}\). A 1000 m ground line therefore plots as \(1000\times0.9995431 = 999.543\ \text{m}\) on the grid. In use, every field distance is multiplied by \(CF\) before it is entered into grid (plane) coordinate computations; the inverse divides grid distances back to ground.

(b) Meridian convergence. Meridian convergence \(\gamma\) is the angle between grid north (the direction of the grid’s central meridian, parallel to the vertical grid lines) and true (geodetic) north at a point. It arises because the meridians converge toward the pole while the grid lines stay parallel. To first order,

$$\gamma \approx \Delta\lambda\,\sin\varphi,$$

where \(\Delta\lambda\) is the longitude difference from the central meridian and \(\varphi\) the latitude. At the UTM central meridian \(\Delta\lambda = 0\), so \(\gamma = \boxed{0}\): grid and true north coincide. At the UTM zone boundary a 6°-wide zone gives \(\Delta\lambda = 3^\circ\); at Canadian latitudes this yields, for example, \(\gamma \approx 3^\circ\sin 49^\circ = \boxed{2.26^\circ}\) (\(\approx 2^\circ16'\)) at \(\varphi = 49^\circ\), rising to \(\approx 3^\circ\sin 60^\circ = 2.60^\circ\) at \(\varphi = 60^\circ\). Yes, the correction must be applied: to convert an observed geodetic azimuth to a grid azimuth (grid bearing) one uses \(\text{grid azimuth} = \text{geodetic azimuth} - \gamma\;(+\,\delta,\text{ the arc-to-chord})\). Since \(\gamma\) reaches 2–2.6° — thousands of times a survey’s angular tolerance — neglecting it corrupts every grid bearing except on the central meridian.

(c) MTM (3TM). MTM — the Modified (or 3-degree) Transverse Mercator — is the Canadian provincial grid built on the same transverse-Mercator projection as UTM but with 3°-wide zones (hence “3TM”). Two conceptual differences from UTM: (1) Zone width and scale distortion — MTM zones are half as wide (3° vs 6°), so the maximum scale distortion is much smaller, which is why MTM is preferred for high-accuracy provincial cadastral and engineering work. (2) Central-meridian scale factor — MTM uses \(k_0 = 0.9999\) on its central meridian, whereas UTM uses \(k_0 = 0.9996\); MTM also uses different central meridians and false-easting/zone-numbering conventions defined provincially rather than the global UTM scheme.