Question 14 of 15: Map projection

Question 14: Map projection (10 marks — a 6, b 4)

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.

(a) Definition. A map projection is a systematic mathematical transformation that maps positions on the curved surface of the Earth (a sphere or reference ellipsoid) onto a flat plane — a function that converts geographic coordinates (latitude φ, longitude λ) into planar Cartesian coordinates (x, y) = f(φ, λ), usually by first projecting onto a developable surface (a cylinder, cone or plane) that can be unrolled without stretching. Because a doubly-curved surface cannot be flattened without stretching or tearing, every projection necessarily distorts some combination of area, shape (angle), distance and direction; a given projection is designed to preserve one property — conformal (local shape/angles), equal-area, equidistant or azimuthal (true direction) — at the expense of the others, with distortion smallest near the projection's standard line or point and growing away from it.

(b) Why they are needed. Projections are required because GIS layers, computer screens and paper maps are two-dimensional and planar, whereas the Earth is three-dimensional and curved. A flat Cartesian (x, y) frame is needed so positions can be plotted and stored and — above all — so distances, areas and bearings can be computed with ordinary plane geometry instead of complex spherical/ellipsoidal formulae. Overlay and analysis further demand that all layers share one planar coordinate system with known, controlled distortion, otherwise features will not register. In short, a projection converts geodetic positions on the globe into the metric planar coordinates (for example UTM eastings/northings, or in Canada the 3° MTM zones) that mapping, measurement and spatial analysis depend on.