NAME
Geo::Coordinates::OSGB::Background - Backround and extended description
VERSION
V2.13
DESCRIPTION
These notes are part of Geo::Coordinates::OSGB, a perl implementation of latitude and longitude co-ordinate conversion for England, Wales, and Scotland based on formulae and data published by the Ordnance Survey of Great Britain.
These modules will convert accurately between an OSGB national grid reference and coordinates given in latitude and longitude using the WGS84 model. This means that you can take latitude and longitude readings from your GPS receiver, (or read them from Wikipedia, or Google Earth, or your car's sat nav), and use this module to convert them to an accurate British National grid reference for use with one of the Ordnance Survey's paper maps. And vice versa, of course.
These notes explain some of the background and implementation details that might help you get the most out of them.
The algorithms and theory for these conversion routines are all from A Guide to Coordinate Systems in Great Britain published by the OSGB, April 1999 (Revised Dec 2010) and available at http://www.ordnancesurvey.co.uk/.
You may also like to read some of the other introductory material there. Should you be hoping to adapt this code to your own custom Mercator projection, you will find the paper called Surveying with the National GPS Network, especially useful.
Coordinates and ellipsoid models
This section explains the fundamental problems of mapping a spherical earth onto a flat piece of paper (or computer screen). A basic understanding of this material will help you use these routines more effectively. It will also provide you with a good store of ammunition if you ever get into an argument with someone from the Flat Earth Society.
It is a direct consequence of Newton's law of universal gravitation (and in particular the bit that states that the gravitational attraction between two objects varies inversely as the square of the distance between them) that all planets are roughly spherical. (If they were any other shape gravity would tend to pull them into a sphere). On the other hand, most useful surfaces for displaying large scale maps (such as pieces of paper or screens) are flat. There is therefore a fundamental problem in making any maps of the earth that its curved surface being mapped must be distorted at least slightly in order to get it to fit onto the flat map.
This module sets out to solve the corresponding problem of converting latitude and longitude coordinates (designed for a spherical surface) to and from a rectangular grid (for a flat surface). A spherical projection is a fairly simple but tedious bit of trigonometry, but the problem is complicated by the fact that the earth is not quite a sphere. Because our planet spins about a vertical axis, it tends to bulge out slightly in the middle, so it is more of an oblate spheroid (or ellipsoid) than a sphere. This makes the arithmetic even more tedious, but the real problem is that the earth is not a regular ellipsoid either, but an irregular lump that closely resembles an ellipsoid and which is constantly (if slowly) being rearranged by plate tectonics. So the best we can do is to pick an imaginary regular ellipsoid that provides a good fit for the region of the earth that we are interested in mapping.
An ellipsoid model is defined by a series of numbers: the major and minor semiaxes of the solid, and a ratio between them called the flattening. There are three ellipsoid models that are relevant to the UK:
- OSGB36
-
The OSGB's own ellipsoid is a revision of work begun by George Airy the Astronomer Royal in 1830, when the OS first undertook to make a series of maps that covered the entire country. It provides a good fit for most of the British Isles.
- EDM50
-
The European standard ellipsoid is a compromise to get a good fit for most of Western Europe.
- WGS84
-
As part of the development of the GPS network by the American military in the 1980s a new world-wide ellipsoid was defined. This fits most populated regions of the world reasonably well.
The latitude and longitude marked on OS maps are in the OSGB36 model. The latitude and longitude you read from your GPS device, or from Wikipedia, or Google Earth are in the WGS84 model. So the point with latitude 51.4778 and longitude 0 in the OSGB36 model is on the prime meridian line in the courtyard of the Royal Observatory in Greenwich, but the point with the same coordinates in the WGS84 model is about 100 yards away to the east, in the park.
There is no intrinsic merit to using one model or another, but there's an obvious need to be consistent about which one you choose, and with the growing ubiquity of GPS systems, it makes sense to standardize of WGS84. It is perhaps possible that future editions of OS maps will show WGS84 latitude and longitude along the edges rather than OSGB36.
In these modules the shape used for the projection of latitude and longitude onto the grid is WGS84 unless you specifically set it to use OSGB36.
The British National Grid and OSTN02
A Mercator grid projection like the British National Grid is defined by the five parameters defined as constants at the top of the module.
- True point of origin Latitude and Longitude = 49N 2W
- False origin easting and northing = 400000 -100000
- Convergence factor = 0.9996012717
One consequence of the True Point of Origin of the British Grid being set to +4900-00200/ is that all the vertical grid lines are parallel to the 2W meridian; you can see this on the appropriate OS maps (for example Landranger sheet 184), or on the plotmaps.pdf picture supplied with this package. The effect of moving the False Point of Origin to the far south west is to make all grid references positive.
Strictly grid references are given as whole numbers of metres from this point, with the easting always given before the northing. For everyday use however, the OSGB suggest that grid references need only to be given within the local 100km square as this makes the numbers smaller. For this purpose they divide Britain into a series of 100km squares identified in pair of letters: TQ, SU, ND, etc. The grid of the big squares actually used is something like this:
HP
HU
HY
NA NB NC ND
NF NG NH NJ NK
NL NM NN NO NP
NR NS NT NU
NW NX NY NZ OV
SC SD SE TA
SH SJ SK TF TG
SM SN SO SP TL TM
SR SS ST SU TQ TR
SV SW SX SY SZ TVSW covers most of Cornwall, TQ London, HU the Shetlands, and there's one tiny corner of a beach in Yorkshire that is in OV. Note that it has the neat feature that N and S are directly above each other, so that most Sx squares are in the south and most Nx squares are in the north. The system logically extends far out in all directions; so square XA lies south of SV and ME to the west of NA and so on. But it becomes less useful the further you go from the central meridian of 2W.
Within each of the large squares, we only need five digit coordinates --- from (0,0) to (99999,99999) --- to refer to a given square metre. For daily use however we don't generally need such precision, so the normal recommended usage is to use units of 100m (hectometres) so that we only need three digits for each easting and northing --- 000,000 to 999,999. If we combine the easting and northing we get the familiar traditional six figure grid reference. Each of these grid references is repeated in each of the large 100km squares but for local use with a particular map, this does not usually matter. Where it does matter, the OS suggest that the six figure reference is prefixed with the identifier of the large grid square to give a `full national grid reference', such as TQ330800. This system is described in the notes in the corner of every Landranger 1:50,000 scale map.
This system was originally devised for use on top of the OSGB36 model of latitude and longitude, so the prime meridian used and the coordinates of the true point of origin are all defined in that system. However as part of standardizing on an international GPS system, the OS have redefined the grid as a rubber sheet transformation from WGS84. This transformation of called OSTN02 and consists of a large dataset that defines a three dimensional shift for each square kilometre of the country. To get from WGS84 latitude and longitude to the grid, you project from the WGS84 ellipsoid to a pseudo-grid and then look up the relevant shifts from OSTN02 and adjust the easting and northing accordingly to get coordinates in the OSGB grid. Going the other way is slightly more complicated as you have to use an iterative approach to find the latitude and longitude that would give you your grid coordinates.
It is also possible to use a three-dimensional shift and rotation called a Helmert transformation to get an approximate conversion. This approach is used automatically by these modules for locations that are undefined in OSTN02, and, if you want to, you can explicitly use it anywhere in the UK by using the grid_to_ll_helmert and ll_to_grid_helmert routines.
Modern GPS receivers can all display coordinates in the OS grid system. You just need to set the display units to be `British National Grid' or whatever similar name is used on your unit. Most units display the coordinates as two groups of five digits and a grid square identifier. The units are metres within the grid square. However you should note that your consumer GPS unit will not have a copy of the whole of OSTN02 in it. It will be using either a Helmert transformation, or an even more approximate Molodenksy transformation to translate from the WGS84 coordinates it is getting from the satellite to find the OSGB grid figures.
Note that the OSGB (and therefore this module) does not cover the whole of the British Isles, nor even the whole of the UK, in particular it covers neither the Channel Islands nor Northern Ireland. The coverage that is included is essentially the same as the coverage provided by the OSGB "Landranger" 1:50000 series maps. The coverage of the OSTN02 data set is slightly smaller, as the OS do not define the model for any points more than about 2km off shore.
Implementation of OSTN02
The OSTN02 is the definitive transformation from WGS84 coordinates to the British National Grid. It is published as a large text file giving a set of corrections for each square kilometre of the country. The OS also publish an algorithm to use it which is described on their website. Essentially you take WGS84 latitude and longitude coordinates and project them into an (easting, northing) pair of coordinates for the flat surface of your grid. You then look up the corrections for the four corners of the relevant kilometre square and interpolate the exact corrections needed for your spot in the square. Adding these exact corrections gives you an (easting, northing) pair in the British grid.
The distributed data also includes a vertical height correction as part of the OSGM02 geoid module, but since this is not used in this module it is omitted from the table of data in order to save space. The table of data contains 876951 rows with entries for each km intersection between (0,0) and (700000, 1250000). However nearly 2/3 of these entries (actually 567472 of them) refer to places that are more than 10 km away from the British mainland (either in the sea or in Eire), and these are set to zero indicating that OSTN02 is not defined at these places. In order to save more space, these are omitted from the beginning and end of each row in the data.
The OSTN02 data is included in the OSGB.pm module after the __DATA__ line, and is read using the magic <DATA> file handle. In my tests this proved to be the fastest way to load all that data, by a long way.
There are 1229 rows of data, and each row contains upto 701 pairs of shift data encoded as pairs of hexadecimal integers representing the shift in mm. Leading and trailing zeros are omitted, and the number of leading zeros omitted is stored in the first three characters of each row.
Earlier versions of the module had the data in a hash, but this was much too slow to load. Later versions includes a `memo-izing' hash to cache the results of the data look ups, but this made very little improvement to performance so it was removed.
Accuracy, uncertainty, and speed
This section explores the limits of accuracy and precision you can expect from this software. Almost certainly, it provides more accuracy than you need.
In particular if you are converting readings taken from your own handheld GPS device, the readings themselves will not be very accurate. To convince yourself of this, try taking your GPS on the same walk on different days and comparing the track: you will see that the tracks do not coincide. If you have two units take them both and compare the tracks: you will see that they do not coincide.
The accuracy of the readings you get will be affected by cloud cover, tree cover, the exact positions of the satellites relative to you (which are constantly changing as the earth rotates), how close you are to sources of interference, like buildings or electricity installations, not to mention the ambient temperature and the state of your rechargeable batteries.
Having said that, it's still worthwhile considering what errors this software might introduce.
First some sizes. In the British Isles the distance along a meridian between two points that are one degree of latitude apart is about 110 km or just under 70 miles. This is the distance as the crow flies from, say, Swindon to Walsall. So a tenth of a degree is about 11 km or 7 miles, a hundredth is just over 1km, 0.001 is about 110m, 0.0001 about 11m and 0.00001 just over 1 m. If you think in minutes, and seconds, then one minute is about 1840 m (and it's no coincidence that this happens to be approximately the same as 1 nautical mile). One second is a bit over 30m, 0.1 seconds is about 3 m, and 0.0001 second is about 3mm.
Degrees Minutes Seconds * LATITUDE *
1 = 110 km 1 = 1.8 km 1 = 30 m
0.1 = 11 km 0.1 = 180 m 0.1 = 3 m
0.01 = 1.1 km 0.01 = 18 m 0.01 = 30 cm
0.001 = 110 m 0.001 = 2 m 0.001 = 3 cm
0.0001 = 11 m 0.0001 = 20 cm 0.0001 = 3 mm
0.00001 = 1.1 m 0.00001 = 2 cm
0.000001 = 11 cm 0.000001 = 2 mm
0.0000001 = 1 cm Degrees of latitude get very slightly longer as you go further north but not by much. In contrast degrees of longitude, which represent the same length on the ground as latitude at the equator, get significantly smaller in northern latitudes. In southern England one degree of latitude represents about 70 km or 44 miles, in northern Scotland it's less than 60 km or about 35 miles. Scaling everything down means that the fifth decimal place of a degree of longitude represents about 60-70cm on the ground.
Degrees Minutes Seconds * LONGITUDE *
1 = 60-70 km 1 = 1.0-1.2 km 1 = 17-20 m
0.1 = 6-7 km 0.1 = 100-120 m 0.1 = 2 m
0.01 = 600-700 m 0.01 = 10-12 m 0.01 = 20 cm
0.001 = 60-70 m 0.001 = 1 m 0.001 = 2 cm
0.0001 = 6-7 m 0.0001 = 10 cm 0.0001 = 2 mm
0.00001 = 60-70 cm 0.00001 = 1 cm The OS supply test data with OSTN02 that comes from various fixed stations around the country and that form part of the definition of the transformation. If you look in the test file 06osdata.t you can see how it it used for testing these modules.
In all cases translating from the WGS84 coordinates to the national grid is accurate to the millimetre, so these modules are at least as accurate as the OSGB's own software that produced the test data.
Translating from the given grid coordinates to WGS84 latitude and longitude coordinates is less accurate. This appears to be related to the iterative approch required but there is no obvious reason. Moreover there is a pattern to the errors. In all of England and Wales, this software produces WGS lat/lon coordinates from the given grid references that are within a few mm of the OSGB data. However in Scotland, and especially for those stations that are further north and west the latitude coordinates produced are upto 12cm out, while the longitude coordinates are about 1cm out. This may be because the OS test data has some bias, or it may be a feature of the mathematics. Either way that's what's achievable with the OS test data, and your results are likely to be similar, provided you have access to such accurate data readings in the first place.
Outside the area covered by OSTN02 this module uses the small Helmert transformation recommended by the OS. The OS state, that with the parameters they provide, this transformation will be accurate upto about 5m within the immediate vicinity of the British Isles.
You can also use this tranformation within the OSTN02 polygon by calling the grid_to_ll_helmert and ll_to_grid_helmert routines. If you compare the output from these routines to the output from the more accurate routines that use OSTN02 you will find that the differences are between about -3.6 m and +5.1 m depending on where you are in the country. In the South East both easting and northing are underestimated, in northern Scotland they tend to be overestimated.
If you can live with this level of uncertainty, then you will find that the Helmert routines are a bit faster. A typical benchmarking run on my development machine (an old Mac Mini server) using the Apple-supplied Perl 5.16 gave:
Subroutine calls per sec ms per call
----------------------------------------------
ll_to_grid 26750 0.037
ll_to_grid_helmert 35208 0.028
grid_to_ll 9916 0.101
grid_to_ll_helmert 28386 0.035 None of the routines is really slow, since even grid_to_ll averages about 0.1 ms per call, but ll_to_grid_helmert runs about 30% faster than ll_to_grid, and grid_to_ll_helmert nearly 3 times faster than grid_to_ll. Using a locally compiled version of Perl 5.22, I see an improvement of about 10%-15% in general on these numbers.
`Your mileage may vary', of course.
The routines have been tested with various versions of Perl, including recent versions with the uselongdouble option enabled. On my system, long doubles slow everything down by about 10%, and make no significant difference to the round trip precision of the routines. Perl's default arithmetic is more than adequate for the calculations required. The loss of precision in round trip testing appears to be inherent in the formulae published by the OS.
Maps
Since Version 2.09 these modules have included a set of map sheet definitions so that you can find which paper maps your coordinates are on.
See Geo::Coordinates::OSGB::Maps for details of the series included. The first three series are OS maps:
A - OS Landranger maps at 1:50000 scale;
B - OS Explorer maps at 1:25000;
C - the old OS One-Inch maps at 1:63360. Landranger sheet 47 appears as "A:47" and Explorer sheet 161 as "B:161" and so on. As of 2015, the Explorer series of incorporates the Outdoor Leisure maps, so (for example) the two sheets that make up the map "Outdoor Leisure 1" appear as "B:OL1E" and "B:0L1W".
Thanks to the marketing department at the OS and their ongoing re-branding exercise several Explorer sheets have been "promoted" to Outdoor Leisure status. So (for example) Explorer sheet 364 has recently become "Explorer sheet Outdoor Leisure 39". Maps like this are listed with a combined name, thus: 'B:395/OL54'.
Many of the Explorer sheets are printed on both sides. In these cases each side is treated as a separate sheet and distinguished with suffixes. The pair of suffixes used for a map will either be N and S, or E and W. So for example there is no Explorer sheet 'B:271', but you will find sheets 'B:271N' and 'B:271S'. The suffixes are determined automatically from the layout of the sides, so in a very few cases it might not match what is printed on the sheet but it should still be obvious which side is which. Where the map has a combined name the suffix only appears at the end. For example: "B:386/OL49E" and "B:386/OL49W".
Several sheets also have insets, for islands, like Lundy or The Scillies, or for promontaries like Selsey Bill or Spurn Head. Like the sides, these insets are also treated as additional sheets (albeit rather smaller). They are named with an alphabetic suffix so Spurn Head is on an inset on Explorer sheet 292 and this is labelled "B:292.a". Where there is more than one inset on a sheet, they are sorted in descending order of size and labelled ".a", ".b" etc. On some sheets the insets overlap the area of the main sheet, but they are still treated as separate map sheets.
Some maps have marginal extensions to include local features - these are simply included in the definition of the main sheets. There are, therefore, many sheets that are not regular rectangles. Nevertheless, the module is able to work out when a point is covered by one of these extensions.
In the examples folder there is an extended example showing how to work with the map data.