# UK British National Grid Map projection

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### #1 mikerouge Posted 28 May 2010 - 04:38 PM

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I use a lot of UK data projected in British National Grid. The data are already in that format, and they are elevation data at 50m postings. So they fit real nice with the GB National Grid which of course is measured in metres.

I do a lot of grid operations like line of sight. Sometimes I'm dealing with a line of sight of 100miles. Say from one mountaintop to the other.

Now, if we ignore earth curvature and refraction for a second (and their horizontal component shouldn't matter too much anyway as they ought to apply only in the vertical dimension...) I think I'm right in saying that light travels in straight lines. So let's imagine a nice straight line from mountain A to mountain B. 100 miles apart. Sorry, let's say 200km apart, keep it all metric. A large distance.

Now, if you were to plot that straight line on the OS BNG map, would it also be a straight line? Or would it be slightly curved? This is what I can't work out. Because this has implications for line of sight algorithms.

How far does the straight line on the map grid deviate from the horizontal position of the real line of sight line on the real world? How much are these two lines different?

I need to know, but I can't figure it out. If anyone can explain to what extent the British National Grid distorts directions, I'd find it really helpful. I know that azimuthal equidistant map projections preserve direction (great circles etc) but before using that I need to confirm what kind of horizontal deviation I'd expect over a 200 km line of sight on UK data. Any ideas? Thanks a lot

Edited by mikerouge, 29 May 2010 - 03:28 AM.

### #2 Melita Kennedy Posted 01 June 2010 - 09:01 PM

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Hi,

BNG uses tranverse Mercator ™ which is a conformal projection. It maintains shapes, and relative directions (for instance a 90deg angle on the earth is 90deg in the map. However, a rhumb line, a line of equal azimuth from the north pole, is not straight in TM line it is in Mercator.

The amount that a rhumb line is going to curve will depend upon its location in the coordinate system (distance and "angle" from the central meridian).

One way to test this is to create a dense line that represents your LOS in lat/lon coordinates. Project it to BNG.

Melita

### #3 pghardy Posted 07 June 2010 - 04:53 AM

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...
Now, if you were to plot that straight line on the OS BNG map, would it also be a straight line? Or would it be slightly curved? This is what I can't work out. Because this has implications for line of sight algorithms.

As a resident Brit who has had to understand this sort of problem before, I'll give some background to Melita's reply, although I stress that I'm not a geodecist, nor a projections expert.

Firstly, British National Grid is a Transverse Mercator projection. This is similar to the familiar Mercator projection of the world, except that a meridian (line of longitude) is taken as the defining line rather than the Equator (line of latitude). In the case of BNG, that meridian is 2 degrees West.

My way of thinking about it (not strictly accurate, but not far off) is that I take a big sheet of paper, and wrap it round the world, forming a cylinder that touches the world along the 2W meridian (and 178E), and goes through both poles. I put a bright light in the centre of the world, and project through the world onto the paper. I cut the cylinder along the 178E line, flatten it out, and shrink it down to manageable size.

The resulting projection is conformal (preserves direction locally), but does not preserve scale - it is true along the 2W meridian and gets worse as you get away from that.

A straight line between two points on a Mercator project is a Loxodrome or Rhumb Line - a line of constant compass bearing. This is excellent for navigation (which is why Mercator was used so much), but it is not quite the great circle, which is the shortest path. The line of sight is not a great circle either, but I suspect that the horizontal differences are negligible for you. For Transverse Mercator, a straight line is not a rhumb line, but similarly will deviate from the great circle.

For BNG, there is then a refinement in that there is a smale scale correction applied (0.9996), which makes the scale accurate 180 Km either side of the 2W line rather than along it. Physically, think of moving the cylinder of paper slightly inwards, so that rather than touching along 2W it cuts through the earth's surface along two meridians close to 2W. I don't think this affects the discussion though.

In summary, the straight line on the BNG map is not the same as the line of sight, but because of the conformal nature of Mercator, is closer to it than most other projections would be.

How far does the straight line on the map grid deviate from the horizontal position of the real line of sight line on the real world? How much are these two lines different?
...

The answer will vary according to which points, and where on the map.
--
Paul Hardy
ESRI Europe (phardy@esri.com)

### #4 mikerouge Posted 12 July 2010 - 04:09 PM

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Thanks Paul for explaining. That was very kind of you.

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