Mercator Charts - Properties — Page 297, Lesson 264

I want to walk you through the properties of the Mercator chart, starting with the projection itself. This is the normal or direct Mercator projection. The projection surface—the cylinder—touches the Reduced Earth at the Equator. That means the cylinder is wrapped around the globe so it makes contact all the way around the Equator line. The geographic poles cannot be projected because they lie on the axis of the cylinder; geometrically, there is no way to map them onto the cylinder's surface. Now let's talk about scale. On a Mercator chart, the scale expands as the secant of the latitude. That's a key relationship, and it comes directly from the departure formula. You'll remember that departure equals change of longitude in minutes multiplied by the cosine of the latitude. Departure is the east-west distance on the Earth's surface between two meridians at a given latitude. Mercator realized that by using a cylindrical projection, the meridians are drawn as parallel straight lines. That means a given change of longitude on the chart is always represented by the same distance on the chart—so many centimetres—no matter what the latitude. So we are holding the chart length constant for a change of longitude, and we need to see how the Earth distance—the departure—changes with latitude. That tells us how the scale changes with latitude. We rearrange the departure formula. Instead of departure equals change of longitude times cosine latitude, we write: change of longitude equals departure divided by cosine latitude. In mathematics, there is a trigonometric function called the secant. Its definition is that secant of an angle theta equals 1 divided by the cosine of theta. So we can rewrite the equation as: change of longitude equals departure multiplied by secant of the latitude. Because the east-west scale has to change as a function of the secant of the latitude—since the meridians are parallel at all latitudes—Mercator realized that the north-south scale has to change by the same amount. The chart must be conformal, preserving angles locally, so the scale must be the same in all directions at any given point. This gives us the equation: scale at any latitude lambda equals scale at the Equator multiplied by secant of lambda. Let's work through a practical example. Suppose the scale at the Equator is 1 over 1,000,000. We want to calculate the scale at 8 degrees north or south latitude. So scale at 8 degrees north equals scale at the Equator times secant of 8 degrees. That is 1 over 1,000,000 multiplied by 1 over cosine of 8 degrees. There's no point evaluating secant as a numerator because when multiplying representative fractions, we want 1 on the top line. It's better to write secant as 1 over cosine, because then we are multiplying denominators. Cosine of 8 degrees is approximately 0.990268. So we have 1 over 1,000,000 multiplied by 1 over 0.990268, which gives 1 over 990,268. This shows that the scale at 8 degrees north or south latitude is 99 percent of the scale at the Equator, or within 1 percent of the correct scale. That 1 percent figure is important in navigation. Up to a scale error of 1 percent, we can regard a chart as having constant scale. That means you can measure distances directly using a ruler. Once the scale error increases beyond 1 percent, you must find distances either by calculation or by using the local latitude scale and measuring small distances at a time with a pair of dividers. The most important figure to remember is that Mercator scale is within 1 percent up to 8 degrees from the Equator.