People have been drawing maps for a long time, and one of the biggest problems is figuring out how to squeeze a curved surface (the Earth) onto a flat piece of paper. There will always be some kind of distortion, and no single type of projection is perfect – the only question is which kind of distortion you want to minimize.
My girlfriend and I recently visited the Mathematisch-Physikalischer Salon, a collection of historical scientific instruments at the beautiful Zwinger building in Dresden. There was an exhibition on globes, and coincidentally I had been reading on the train ride there a very informative book on map projections, Map Projections: A Working Manual by John Snyder, published by the US Geological Survey (freely available as a pdf).
One exhibit was a toy globe which didn’t have the usual spherical shape – instead it looked somewhat like a Chinese lantern. Unfortunately I didn’t take a picture of the original, but below is a picture of my version. It has a cylindrical middle part, conical upper and lower parts, and is flat at the poles. An ugly shape, but these are incidentally representing the three major families of map projections: Cylindrical, Conical, and Azimuthal (the flat part).
It’s a beautiful learning tool to illustrate the three types of projections and their relative strengths. Cylindrical projections are best suited for plotting areas near the equator. The Mercator projection (think Google Maps) is an example, and if you recall, is very distorted near the poles. Conical projections are suitable for intermediate latitudes. Many maps of the United States, for example, are cylindrically projected. Azimuthal projections, such as most maps of Antarctica, are suitable for the poles.
I have previously blogged about an open source cartography software called GMT (Generic Mapping Tools). This tool is great because it lets you plot maps in any projection you can imagine, add annotations, custom scales, etc. In this blog post I shall show how I made my own Three-Projection Globe (I don’t know the actual name), and provide a template that you can print out and build for yourself!
Making the maps
The maps themselves are relatively easy to plot, once you are familiar with GMT. There are many different variants within the three major families of projections, and in principle you could choose many combinations of them. The main consideration for building a globe out of these maps is that they have to join up at the seams properly, i.e. you want to choose the scales and parameters properly such that at the lines where they join up, they have the same scale and are distortion-free.
The way this globe is designed has the seams at lines of latitude. In principle you can orient it arbitrarily, so that the central axis doesn’t pass through the poles but instead some other axis, but that’s for another time.
The following list summarizes the parameters that I chose:
- Cylindrical part: between 30 N and 30 S, centered at 0 E, cylindrical equal-area projection, with standard latitude at 30 N
- Conical parts: between 30 and 60 degrees (N or S respectively), centered at 0 E and 45 degrees (N or S respectively), and standard latitudes at 30 and 60 degrees (N or S respectively), Albers equal-area projection
- Azimuthal parts: Above 60 degrees (N or S), centered at the respective pole, Lambert azimuthal projection
What are standard latitudes? I chose the standard latitudes above correspond to the seams where the projections join up. Most map projections strive to minimize some kind of distortion, but cannot eliminate them entirely. Depending on how they are formulated, many have specific points or lines which are “true”, where there is zero distortion of size or scale. As we move away from those specific points or lines, the distortion increases. In the projections that I chose for my globe, the standard latitudes are such lines that are true to scale, i.e. free of size distortion, so that when I try to stick the different parts together, they will match up properly.
It can be helpful to imagine the Three-Projection Globe being inscribed by a sphere representing the ideal globe. Then the standard latitudes, and thus the seams where the parts join, are the lines where the surface of the Three-Projection Globe touch the inside surface of the sphere. In cartographic jargon – these are secant projections. Okay maybe for most people this paragraph wasn’t so helpful.
Here’s the GMT commands required to make these plots, using the vector coastline data bundled with the GMT software package.
# Center piece representing the equatorial belt gmt pscoast -Jy0/30/0.03i -R-180/180/-30/30 -Gchocolate -Ba > center.ps # Conical part from Northern hemisphere gmt pscoast -Jb0/45/30/60/0.03i -R-180/180/30/60 -Gchocolate -Ba > conical_N.ps # The Arctic Circle! gmt pscoast -JA0/90/1.72i -R-180/180/60/90 -Gchocolate -Ba > azimuthal_N.ps # Conical part from Southern Hemisphere gmt pscoast -Jb0/-45/-30/-60/0.03i -R-180/180/-60/-30 -Gchocolate -Ba > conical_S.ps # Antarctica! gmt pscoast -JA0/-90/1.72i -R-180/180/-90/-60 -Gchocolate -Ba > azimuthal_S.ps
I then used ps2pdf to convert the Postscript files to PDF format, and arranged the pieces to fit in a single page using Inkscape. Download the pdf here.
A point about the scaling parameter in GMT: The units for cylindrical and conical projections are in (units) per degree. In this case, 0.03 inches per degree. However for the azimuthal part, I had trouble figuring out the syntax of the scaling parameter, so I just provided it as a width (0.03 in per degree so 180 degrees = 5.4 inches and diameter of a circle with that circumference is 1.72 inches).
Putting the pieces together
Once I had printed the pieces out with the correct scales, I cut out the maps (without the black and white borders!) and tried to stick them together. My first thought was to try leaving little tabs on the margins so that I could attach the parts. Unfortunately the edges of attachment are all curved, so the tabs were a nightmare and could not follow the curve and still fold properly at the same time.
What I did in the end was to cut clean edges and use really thin slivers of sticky film to paste them together. There was no way to avoid having tape on the outside because you have to close it up somehow. Any suggestions for more elegant methods of pasting them together are welcome!
It’s also important to keep the parts aligned, and it’s a bit of effort to do that while trying not to deform the paper and trying to put a tiny piece of sticky film in the right place. Some patience is necessary.
Step 1 – Make the cylinder and cones
Cut out the cylinder, leaving a little tab along one of the short edges so that you can join it up when you roll it into a cylimder. Do the same for the cones. Make the cylinder and partial cones. The seam is at 180 degrees longitude.
Step 2 – Stick the azimuthal part to the conical part
Cut out the azimuthal parts corresponding to the poles, but before you do, make a marking on the reverse side of the paper at 180 deg longitude. This lets you align the azimuthal part to the conical part, when you stick them from the inside.
Put the azimuthal part face down.
Position the cone (check that it’s the correct hemisphere!) face down on top of the azimuthal part, and align the seam with the marking you made earlier.
Carefully stick together with strips of sticky tape.
Step 3 – Stick the conical part to the cylindrical part
The conical and cylindrical parts can be aligned with the seam at 180 deg longitude. Stick them together, as before, with thin slivers of sticky tape. Unfortunately there’s no way to avoid having to apply tape to the outside surface – aesthetically this is not ideal but there doesn’t seem to be an alternative….
The final product
And voila! You have a globe built from three different map projections that you can use to confuse children and bore your family and friends!
Fun ideas for going further
- If you are the creative sort, make it into a lamp with LED lights inside (please don’t use candles/incandescents and set your house on fire)
- Rotate the axis of the globe so that the azimuthal parts are centered on somewhere else other than the poles
- Plot landmasses or countries in different colors, or color by topography or bathymetry, or add annotations
- Make a medieval-style map with dragons and sea-monsters and vast swathes of Terra Incognita
- Make a globe from an extraterrestrial body such as the Moon or Mars