Mercator Charts - Properties — Page 297, Lesson 262

Let’s start with the core idea: a Mercator chart is a conformal — or orthomorphic — chart, and that word means something very precise. At any point on the chart, the scale must be the same in all directions, or if it changes, it must change at the same rate in all directions. That’s the second requirement of orthomorphism, and it’s the whole reason the Mercator projection exists. To see why, look at Figure 18.1, the simple cylindrical projection. The shapes there are stretched in a north–south direction. Mercator realised the cause: on that simple projection, the north–south scale was changing at a different rate from the east–west scale. That failure to meet the conformality requirement is exactly what distorts the shapes. Here’s the mathematics of the problem. On the simple cylindrical projection, the east–west scale at any latitude was proportional to the secant of the latitude. Secant is simply 1 divided by the cosine — so secant of latitude equals 1 over cosine of latitude. But the north–south scale was changing at a different rate: at any latitude it was proportional to the tangent of the latitude. That mismatch — E–W proportional to secant, N–S proportional to tangent — is what produced the north–south stretching you see in Figure 18.2. The latitude spacing there is a function of tan(latitude). Mercator’s solution was elegant. He adjusted the positions of the parallels of latitude. Originally, the parallels had been projected as parallel lines, with the separation between them increasing in proportion to the tangent of the latitude. Mercator changed that: he adjusted the parallels so their separation increased only at a rate proportional to the secant of latitude — matching the east–west scale change. Once the N–S spacing matched the E–W spacing at every latitude, the chart became orthomorphic, or conformal. Two important consequences follow. First, because the chart is mathematically produced — the parallels were repositioned by calculation, not by simple geometric projection — it is a non-perspective chart. There is no single viewpoint from which the Earth is projected onto the paper; it’s a mathematical construction. Second, the date: Mercator solved this in 1569. His solution was so simple, elegant, and correct that we still use his projection today, over 400 years later. A modern Mercator chart looks very different from his own, simply because we have since discovered and explored so many more countries. But the basic principles of the graticule — the grid of meridians and parallels — have not changed. He got it right. Figure 18.3 shows an example of the adjusted Mercator projection, where you can see the parallels spaced according to the secant rule, giving you the conformal chart we still navigate with today.