Have you looked at a world wall map and noticed how big Greenland is? It's huge. It's larger than China, the United States, and Australia, and is about as big as Africa. Too bad it's so cold, or we could fit a lot of people up there. Or could we?
Actually, Australia is about three and a half times as big as Greenland, China is almost four and a half times as big, and Africa is almost fourteen times as large!
What's going on? The Mercator projection is what's going on. It was developed by the Flemish cartographer Gerardus Mercator in 1569. Over time, it's become very popular, at least partially so because it fits nicely onto a rectangular page without wasting a lot of space around the edges, the way some projections do.
A map projection is a transformation of locations on a sphere or ellipsoid onto locations on a plane. You can think of it as a function that transforms latitudes and longitudes of the earth into the x and y coordinates on a sheet of paper. This allows us to take a point on a map and find it on the earth, take a point on the earth and find it on the map, or take a point on one map and find it on another.
Mercator is a common projection. It's created by wrapping a cylindrical sheet of paper around the globe, only touching along the equator. Then, the shapes on the globe are cast out onto the paper roll like beams of light spreading out. This was developed for navigation, and if you chart a course with a constant bearing, it plots on a Mercator map as a straight line. However, its major problem is that it distorts shapes around the edges, for example, Greenland or Antarctica.
There are a number of other common projections, such as the following:
- The Gall-Peters projection accurately shows the area but distorts the shape.
- The Eckert IV projection distorts the outer shape of the map onto an ovoid to minimize the area distortions of the Mercator projection, although it still distorts the shapes of things near the poles.
- The Goode homolosine projection attempts to accurately portray both the area and shape by cutting the skin off the globe into some awkward shapes. It's sometimes called the orange peel map because the outlines of the map look like you peeled an orange by hand and flattened it on the table top.
So how does this apply to our project?
On the one hand, we need some way to accurately measure the distances between points in the real world. For example, as we're working in the northern hemisphere, the points near the top of the map, to the north, will be closer together than the points near the bottom. We need to know the projection in order to measure these distances correctly and correctly calculate the interpolations.
To put it another way, the distance between two points that are a degree of longitude apart would be different, depending on their latitude. In Grand Forks, North Dakota, the distance between longitude -97 and -96 is approximately 46 miles (74.5 km). On the other hand, the distance between longitudes -97 and -96, just west of Houston, Texas, is almost 60 miles (96.52 km). Think of the way in which two lines that are parallel on the equator have to curve towards each other as they converge at the poles.
On the other hand, we also need to then be able to know which pixel a set of latitude and longitude correspond to. In order to actually plot the heat map on the screen, we have to be able to determine which pixel gets which color, depending on the interpolated points on the map.
Related to the projections, we also need to have a base layer to display the heat map on top of it. Without being able to see the context of the underlying geography, a heat map is more confusing than it is illuminating.
There are maps available that have their locations encoded in their metadata. GeoTIFF is one such format. GIS packages can layer the data and information on top of these base maps to provide more complex, interesting, and useful visualizations and analyses.