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Map Projections Part 2: The Allegory of the Cave

In this second post on map projections we cover the range of different types of projections.  Part 1 gave us a broad overview of different map projections

In thinking about map projections I’m reminded of an old story.  Now bear with me on this one: in this story, imagine some prisoners, chained in a cave all their lives, their heads unable to turn such that all they can see in front of them is the cave wall.   Behind them, a great fire burns, and in between prisoners and fire, puppetters march past with their puppets held high so that they cast shadows on the cave wall.  It looks something like this:

The idea here is that the prisoners, not knowing anything else, take the shadows to be reality.  The shadows have become how they understand the world.  In another part of the story, a prisoner breaks free and emerges from the cave, sees the sun for the first time and realises that things were not at all the way they seemed, and it’s all a bit frightening.  (Sorry, there’s really not a happy ending to this story.  Well, there is…sort of…but it’s not your typical Hollywood blockbuster ending.)

Map Projections

Okay, so I’ll admit, I’ve got some issues with this story too, but what does it have to do with map projections?   Like the prisoners, what we see on maps is a bit like the shadows thrown on the cave wall.  In fact, the reason they’re called projections is because map makers originally used shadows cast on a surface to translate the 3D earth onto a flat, 2D plane.  Imagine (if you will) a spherical globe constructed with a metal frame of meridians of longitude, parallels of latitude and the land masses.  Also imagine a light bulb at the center of this globe.  Take a large piece of paper and shape it into a cylinder.  Place that cylinder over the globe so it touches at the equator and turn the light on.  Now trace the outline of the shadows on the cylinder, et voila!  Unroll the paper and you’ve got a projected map.

And like the prisoners, it’s the shadows that give us our reality.   When the paper is unrolled,  the technique outlined above gives us the tried and (I was going to say true, but…) Mercator Projection.  (Do check out this link  where they actually show how this works.)  In a very analog way, this is how map projections work.  The main differences between different projections created this way are the kinds of surface the shadows are projection onto.  Above, it’s a cylinder.  But we can also use cones, or planes that touch the globe at a point.  This gives us the three main types of projections: azimuthal (planar), conic and cylindrical:

When unfolded, each lays flat and the traced shadows give us what’s where.  Now we’re really only just beginning to look at the range of possible map projections (there are thousands built from these three simple approaches) especially when we realise (like our prisoners) that we can also work with mathematical projections that use equations to go from 3D to 2D.  Here’ a small list of projections (69 in total, a mere smattering).  As noted earlier, with any projection, it’s impossible to transfer the features on the surface of a sphere onto a flat plane without creating some sort of distortion.  As soon as we turn the light on (metaphorically speaking) we sacrifice something: area, shape, bearing, or distance.

Another way we can classify projections is what aspect of the surface they preserve.  Here are the main categories:

  • Direction Preserving (azimuthal or zenithal)
  • Shape Preserving (conformal or orthomorphic)
  • Area Preserving (equal-area or equiareal or equivalent or authalic)
  • Distance Preserving (equidistant), or
  • Shortest Route Preserving (gnomonic)

Often, the choice of which one to use depends on what the maps will be used for.  Mercator first drew his map to be used by mariners, so it was designed to preserve bearing (direction) at the cost of shape and area (hence, an overly large Greenland).  Sometimes, a projection can be chosen so that it minimises several distortions at one time, a sort of compromise that has been used on many New Zealand maps (we’ll look at this more closely later on).

Geodetic Datums

When talking about map projections we also need to talk abut geodetic datums.  We’ve covered datums a bit already; they are mathematical shapes (spheroids, to be exact) that attempt to represent the shape of the earth.  WGS84 is one example but there are many others.  Each datum has its own geometric parameters and uses latitude and longitude to locate points on the surface, though the same point, say Cathedral Square, may have two different sets of coordinates between two different spheroids.  So in a sense, you could be in two (or more) places at one time…  (I’ll show you how another time.)

So lat/long is used to locate a point on the spheroid surface – that’s referred to as the horizontal datum.  We also need a vertical datum to measure heights.  There’s a lot I’m going to gloss over about heights because they’re often surprisingly not as simple as we think they are, but for now, the vertical datum is a reference point that allows us to say, “Here be zero” against which we can determine if other points are above or below it.  The most common one we’re used to is sea level (and we know how complex that can be…).  We’ve already seen how different parts of New Zealand had, until very recently, different vertical datums so it’s far from simple to get your head around this.

Every map projection is based on a geodetic datum and you can usually find information about that by looking on the map.  the geodetic datum is sort of the like the metal globe framework we talked about earlier.  The map projection then is how (mathematically) the lat/long/height is converted to a flat, Cartesian x, y, z set of coordinates that are shown on the map.  I guess in this case, the map projection is like the rays of light in that same example.

My head hurts…

In the next installment of our look at map projections we’ll focus on projections (and datums) used on New Zealand maps, but before we go, let’s put some thought into one of the fundamental reasons why we need projections in the first place.  Aside from that being how maps are made, there is a more basic reason.

The 3-dimensional nature of latitude and longitude is the best way to specify location on the surface of a spher(oid).  Degrees of latitude and longitude can be broken down into divisions of 60 minutes per degree and further into 60 seconds of arc per minute (and there’s your space-time continuum, maybe).  But the big problem with degrees is they not all created equal, particularly when we think about longitude.  At the equator, a degree of longitude roughly equals 111 km.  But as we move north, the meridians converge, such that a degree of longitude reduces to zero at the north and south poles.  This makes proper distance measurements a challenge (and why any good navigator worth their salt uses the latitude scales on either side of a chart to make distance measurements rather than the scales across the top and bottom).  One nice thing about meters (and we might as well include feet, and leagues, and cubits) is that a meter is a meter is a meter, no matter where you are.  Projections allow us to use maps to make reliable, standard length and area measurements which are crucial is most GIS analyses, not to mention putting things in the correct place.  With map projections I don’t need to tell someone that I live 0.399 degrees away from Lincoln at 43.53 degrees south latitude.

I think I’ll crawl back into my cave now, where life is much simpler.



• May 25, 2017

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  1. Dorje McKinnon May 29, 2017 - 9:04 pm Reply

    Crile – you write wonderfully well about a very complex subject.
    In reading this post I though of the movie Mars, and just how complex space travel must be given how complex mapping one world is.

    • doscherc May 30, 2017 - 1:48 am Reply

      Thanks Dorje – very kind of you. I’m often reminded of people like Captain Cook, sailing into (literally) uncharted territory before there were any maps/charts (thought they certainly weren’t the first ones there). These days we’re seldom in the position of not knowing what’s around the corner, let alone what’s just outside the cave.

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