Question 4 of 8: Map Projections

Question 4: Map Projections (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. The reduction of terrain measurements to the mapping plane through the Universal Transverse Mercator (UTM) projection, with central scale factor \(k_0 = 0.9996\), zone width 6° of longitude, and mean Earth radius \(R\approx 6371\ \text{km}\).

Find. (a) the intermediate reference surface and its purpose; (b) the grid/combined scale factor and its use; (c) central vs standard meridian and their count in a zone; (d) the scale factors across a zone; (e) meridian convergence and whether it must be applied to azimuths; (f) MTM/3TM and two differences from UTM.

(a) Intermediate reference surface. The terrain observations are first reduced to the reference ellipsoid (in Canada, GRS80/NAD83(CSRS)). We need this intermediate surface because a map projection is a rigorous mathematical mapping from the ellipsoid to the plane — it is defined for points on the ellipsoid, not for points floating at arbitrary terrain heights. Reducing to the ellipsoid removes the effect of station elevation (the height/sea-level reduction) and the deflection of the vertical, giving clean ellipsoidal distances and geodetic azimuths that the projection formulae can accept.

(b) Grid (combined) scale factor. The grid factor \(K\) is the ratio of a grid distance (on the map) to the corresponding ground/terrain distance, and it is the product of two effects: the elevation factor \(F_e=R/(R+H)\), which shrinks the terrain distance down to the ellipsoid, and the projection (point) scale factor \(k\), which stretches the ellipsoidal distance onto the grid: \(K = F_e\cdot k = \dfrac{R}{R+H}\cdot k\). It is used by multiplying a measured ground distance by \(K\) to obtain the grid distance for coordinate computation (or dividing a grid distance by \(K\) to recover ground distance for setting-out). Because \(k\) varies across the zone and \(H\) varies with terrain, \(K\) is computed for the mean location and mean elevation of each line.

(c) Central and standard meridians. The central meridian is the meridian of longitude at the centre of a UTM zone, along which the projection cylinder’s axis is oriented; there is exactly one per zone, and the scale there is deliberately reduced to \(k_0=0.9996\). The standard (secant) meridians are the two lines where the secant cylinder cuts the ellipsoid and the scale is true, \(k=1\); there are exactly two per zone, located about 180 km either side of the central meridian.

(d) Scale factors across the zone. The point scale factor is \(k=k_0\left(1+\dfrac{E'^2}{2R^2}\right)\), where \(E'\) is the distance from the central meridian. It equals \(k_0=\mathbf{0.9996}\) on the central meridian, rises to \(k=\mathbf{1}\) at the two standard meridians (\(E'\approx\) 180 km), and reaches its maximum at the zone edges. At the equator (where the 3° half-zone is widest, \(E'\approx 334\ \text{km}\)) the edge value is

$$k_{\text{edge}}=0.9996\left(1+\frac{(334\ \text{km})^2}{2(6371\ \text{km})^2}\right)=1.00097.$$

So across a zone the scale ranges from \(0.9996\) (centre) to \(\approx 1.00097\) (equatorial edge) — the two-sided (secant) design keeps the distortion below about 1 part in 1000 everywhere, with the secant design splitting the distortion between an under-scale of 1:2500 at the centre and an over-scale of up to about 1:1000 at the equatorial edge. (At mid-latitudes the edge value shrinks with \(\cos\varphi\); e.g. \(k_{\text{edge}}\approx 1.0003\) at \(\varphi=45^\circ\).)

(e) Meridian convergence. Meridian convergence \(\gamma\) is the angle at a point between grid north (parallel to the central meridian) and true/geodetic north (the meridian through the point); to first order \(\gamma=\Delta\lambda\,\sin\varphi\), where \(\Delta\lambda\) is the longitude difference from the central meridian. It is zero on the central meridian and grows toward the zone edges — e.g. at \(\varphi=45^\circ,\ \Delta\lambda=3^\circ\), \(\gamma=3^\circ\sin45^\circ=2.12^\circ\) (larger at higher latitude, \(\approx2.6^\circ\) at \(\varphi=60^\circ\)). Yes, it must be applied: a geodetic (true) azimuth and a grid azimuth differ by exactly \(\gamma\) (\(\text{grid azimuth}=\text{geodetic azimuth}-\gamma\), plus the small arc-to-chord (\(t-T\)) term). Two degrees is far larger than survey angular tolerances, so ignoring convergence would rotate the whole survey relative to the grid.

(f) MTM / 3TM. MTM (Modified Transverse Mercator, also “3TM”) is a transverse-Mercator projection used in Canadian provinces for large-scale/cadastral work. Two conceptual differences from UTM: (1) Zone width — MTM zones are only 3° wide (versus UTM’s 6°), which halves the maximum distance distortion and keeps the grid factor much closer to unity for precise engineering surveys; (2) Central scale factor — MTM uses \(k_0=\mathbf{0.9999}\) (versus UTM’s 0.9996), because the narrower zone needs less scale reduction to balance the distortion. (Provinces also assign MTM their own false eastings/zone numbering rather than the global UTM scheme.)