Map Projections 3: The Tour de Topo
This post looks in detail at our own Topo50 series of 1:50,000 scale topographic maps, building upon previous posts about map projections
Finally. Some maps (he said in a huff).
Having previously covered the concepts and specifics of map projections, we’re now well placed to look in detail at how map projections come into play in our little part of the world. Land Information New Zealand (LINZ) is the government department charged with all things spatial, including producing our topographic maps. There are currently two series of topo maps that cover mainland NZ, one at a 1:50,000 scale (Topo50) and the other at a 1:250,000 scale (Topo250). We’ll look at the 1:50K series as they’re more commonly used and have the best detail. For the Topo50 series there are 448 maps sheets that cover the country, each coded with a sequence of letters (for rows) and numbers (for columns). Each is 36 km tall and 24 km wide (can’t do that with decimal degrees) and here’s how they cover the country:
Zooming in more locally we can start to see the logic of the numbering system:
Zooming in even more we can see the BX24 sheet which covers much of Christchurch:
From here on in we’ll look specifically at BX24 so if you’d like to play along at home, get yourself a copy from most reputable outfitters and some bookshops (or the library). Nothing like a paper map, eh?
When you come home with your map and unfold it, it will look something like this, with the map itself and some supporting material:
(By the way, I downloaded this from the LINZ website for free.) Lots to look at here but let’s focus on the map projection stuff. If we zoom in on the fine print at the bottom of the page we can see some details on the coordinate system:
From previous posts there should be some familiar looking stuff. The Horizontal Datum is NZGD2000. This is our spheroid and coordinate system. Note that NZGD2000 equates to WGS84 – a subtle but important point. The upshot of this is that GPS data will fit in very nicely with NZGD2000. this wasn’t always the case, as we’ll get into another time.
The Vertical Datum is Mean Sea Level – so all elevations on the map are tied to sea level. As we’ve seen previously, sea level differs depending on where you are in New Zealand. I would venture to guess that when this map series gets updated, it will probably use NZVD2016.
Projection: here we go. It’s New Zealand Transverse Mercator 2000 (NZTM2000). The Parameters relate to the spheroid and coordinate system. The datum uses GRS80 (Geodetic Reference System 1980) as its spheroid, which is geocentric (its centre coincides with the earth’s centre of mass). We know that map projections distort everything on a map, including linear distance; the Scale Factor is a measure of this. “It is the ratio of the “true” (undistorted) scale and the “nominal” (distorted) scale.” Values of 1 would mean no distortion (impossible) so this is an indication that the distortion is minimal (0.9996 is a common value for transverse Mercator projections).
Next we’ve got some coordinate values – here it gets a little tricky. First, the origin latitude and longitude are given as 0° South and 173° East respectively. Like any Cartesian space, this is our origin, our point of 0,0 on GRS80. Here’s where that point is, somewhere in the middle of the Pacific on the equator:
That point is given some false values in metres, namely a “False Northing” of 10,000,000 m N (the y-coordinate) and a “False Easting” of 1,600,000 m E (the x-coordinate). While those may seem like some crazy and arbitrary numbers, there is actually some logic behind them. Using those false values, it means that any northing on the NZ landmass will be between roughly 4,500,000 and 6,200,000 m while any easting will be between 1,000,000 and 2,100,000. With these ranges, it’s very difficult to confuse eastings and northings, especially once you get used to seeing these values for places your familiar with. These values then become the numbers you see in the lower right-hand corner of your map window in ArcMap:
And it’s those values that allow us to measure distance and area. In a sense, this could be the end point of our discussions of map projections but there’s still a lot more to talk about on the map, so let’s carry on. Here’s the lower left hand corner of BX24:
We see here a bit of mixing of our coordinate systems. In blue, the NZTM coordinates with our northings running vertically and the eastings laterally. The ten thousand and thousand places are shown in larger font and correspond to a grid of blue lines that cover the map. Each square is 1 km by 1 km which makes some distance measurements easier. As we’ll see later, we can also use the grid to find specific locations if we are given the coordinates (and vice-versa). We can also see a black outer scale with latitudes and longitudes in degrees and minutes (on GRS80). We could use raw GPS readings to find their locations on this map using these markings (again, that wasn’t the case with earlier map series).
Further to the left is our variation indicator:
This shows the difference between grid north (pointing to the geographic north pole) and magnetic north (pointing to the north magnetic pole). Note how it increases at a set rate – the magnetic north pole is on the move! (Note to self: blog post on this). Next the scale – an obligatory element that should be present on all maps:
Now we can translate measurements on the map to real world distances. This also tells us the vertical distance between elevation contours.
Next some locator maps to help the reader see where this particular map is. First a small scale (large area) locator and then a larger scale (smaller area) map:
Note how each map has a name as well as a code. Knowing the code is useful when adding one of the topo maps in ArcMap as it’s included as part of the file name (these can be found in J:\Data\Topoimages\South (or North) Island Geotiffs). Last thing we’ll cover is finding locations using grid references.
Sometimes you might be provided with a grid reference for a location that might look something like this:
As you might guess, the BX24 refers to the map sheet. The 771737 is actually an easting and a northing combined. With this reference you can roughly locate a point to within a 100 metres on this map. The 771 is the easting and the 737 is the northing.
Starting from the left hand side of the map and moving to the east, all the eastings start with 15. That gets dropped in a grid reference so the 771 is actually 77,100 metres minus the last two zeroes. Find the vertical line labelled 77 and then keeping moving to the right by, in this case, 1/10 of the distance to the next vertical line (which is 100 m).
Next, from the bottom of the map, all the northings start with 51, so again that gets dropped. Moving vertically from 771, find the horizontal line labelled 73 and then move up 7/10 of the distance to the next horizontal line (700 m in the real world). That should bring us here, Mt Cavendish:
Note the vertical grid lines labelled in blue at the bottom. On the map, the guide below helps reinforce how this works.
So those are the major elements of the Topo50 maps with respect to map projections. There’s still probably one more area we need to cover before we finish up with projections – the gory details of how the transformation from 2D to 3D works. But that’s probably enough for now.