Geo::Coordinates::OSGB::Background(3)

NAME
Geo::Coordinates::OSGB::Background - Background and extended description

VERSION
2.16

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.

  Upgrading from V2.09 or earlier
These modules suffered a major overhaul in V2.10 which changed the semantics and interface. The motivation for the change was to simplify the interface, to make WGS84 the default model for latitude and longitude, and to speed up conversions. This section explains what you might have to change to get your old code to work with V2.10 and above.

•   The module structure changed from V2.09 to V2.10 so it would probably be a good idea to clean up any old installation you have and re-install a new version so that none of the old files hangs about. The simplest way to do this is to use the "cpanm" application.

•   "OSGB.pm" no longer exports an ":all" tag. This means you must explicitly name all the functions that you want to use. Fortunately there are rather fewer functions now.

•   The parsing and formatting routines have been split into a separate module, so before V2.10 you could do this:

       use Geo::Coordinates::OSGB ':all'; # Don't do this any more

    now you need to do this:

       use Geo::Coordinates::OSGB       qw(grid_to_ll ll_to_grid);
       use Geo::Coordinates::OSGB::Grid qw(parse_grid format_grid);

•   The old "OSTN02.pm" module has been removed. The data and the functions it provided are now integrated into "OSGB.pm". The old "OSGB36_to_ETRS89" and "ETRS89_to_OSGB36" functions are now built into "ll_to_grid" and "grid_to_ll"; the OSTN02 data is used automatically when needed.

•   The main conversion routines now assume the WGS84 model for lat/lon coordinates. If you still want to work with the OSGB36 model for lat/lon, then you need to add the "{ shape => 'OSGB36' }" option when converting. In other words if you did this before:

       my ($e, $n) = ll_to_grid(52.5, -2);

    then to get exactly the same results, you now need to do this

       my ($e, $n) = ll_to_grid(52.5, -2, { shape => 'OSGB36' } );

    On the other hand, if you are working with the WGS84 model, then before V2.10 you had to do this:

       # Old and exact
       my ($e, $n) = ETRS89_to_OSGB36(ll_to_grid(52.5, -2));

    or this:

       # Old and approximate
       my ($e, $n) = ll_to_grid(shift_ll_from_WGS84(52.5, -2));

    now you can just do:

       my ($e, $n) = ll_to_grid(52.5, -2);

    This means that the most likely use case is the default. In other words you can take lat/lon from your GPS and get correct grid references by default

•   The functions "shift_ll_into_WGS84" and "shift_ll_from_WGS84" are no longer provided. They were not very accurate, and they were confusing. "ll_to_grid" and "grid_to_ll" now do the necessary adjustments for you automatically.

•   The functions to parse and format grid references are moved to "OSGB/Grid.pm". They have also been radically simplified, so that there are now only two functions: "parse_grid" and "format_grid". The old names like "parse_GPS_grid" and "format_grid_trad" are retained with their former meanings, but they are just synonyms for the two core functions.

•   The functions to parse and format lat/lon are removed. They were not really a core function of this module. There are other modules on CPAN to deal with latitude and longitude. I also give an example of how to format decimal degrees as degrees, minutes, and seconds in "examples/bng_to_ll.pl".

  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 semi-axes of the solid, and a ratio between them called the flattening. There are three ellipsoid models that are relevant to the UK:

OSGB36
    The OSGB36 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 120 metres away to the south-east, in the park.

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 49N, 2W 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 PDF picture supplied with this package in the "examples" folder. The effect of moving the False Point of Origin to the far south west is to make all grid references positive.

Strictly speaking, 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 TV

SW 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. However general use rarely demands such precision, so the OSGB recommendation 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.

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 on WGS84. It is perhaps possible that future editions of OS maps will show WGS84 latitude and longitude along the edges instead of OSGB36.

The differences between the models are only important if you are working at a fairly small scale. The average differences on the ground vary from about -67 metres to + 124 meters depending on where you are in the country.

    Square                 Easting difference           Northing difference
    --------------------   -------------------------    ------------------
                 HP                        109                          66
              HT HU                    100 106                      59  62
        HW HX HY                73  83  93                  51  48  47
     NA NB NC ND            61  65  81  89              40  39  38  40
     NF NG NH NJ NK         57  68  79  92  99          30  29  28  26  26
     NL NM NN NO            56  66  79  91              18  17  15  15
        NR NS NT NU             66  77  92 100               3   2   1   0
        NW NX NY NZ             70  77  92 103              -9  -8 -10 -13
           SC SD SE TA              77  93 104 112             -19 -22 -23 -24
           SH SJ SK TF TG           79  91 103 114 124         -35 -34 -35 -38 -40
        SM SN SO SP TL TM       72  80  90 101 113 122     -49 -47 -46 -46 -46 -47
           SS ST SU TQ TR           80  90 101 113 121         -57 -56 -57 -57 -59
        SW SX SY SZ TV          71  79  90 100 113         -67 -64 -62 -62 -62

The chart above shows the mean difference in each grid square. A positive easting difference means the WGS84 Lat/Lon is to the east of OSGB36; a positive northing difference means it is to the north of OSGB36.

The transformation from WGS84 to OSGB36 is called OSTN02 and consists of a large data set 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. To show you an OSGB grid reference, your GPS 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 satellites.

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 this is not used in this module, so it is omitted from the table of data in order to save space.

The table of data supplied by the Orndance Survey contains 876951 rows with entries for each km intersection between (0,0) and (700000, 1250000). However 567472 of these entries 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 as stored in my implementation. The last 21 rows (north of Shetland) are all zeroes, so these are omitted as well.

My implementation of the OSTN02 data is included in the "OSGB.pm" module after the "__DATA__" line, and is read using the magic "<DATA>" file handle. In 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 up to 701 pairs of shift data encoded as pairs of 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. The integer pairs are all coded in a home-grown version of base32 using the character set "0123456789:;<=>?@ABDEFGHIJKLMNO". This allows integers up to 32767 to be stored in three bytes of printable ASCII. It would (obviously) be possible to compress the data more by storing them as binary integers but I need them to be printable ASCII so that they can be stored inside "OSGB.pm". Decoding them is very slightly slower than decoding hex strings, but using 3 bytes integer instead of 4 reduces the memory and loading time by 25%.

Earlier versions of the module had the data in a hash, but this was much too slow to load. Later versions stored keys with the data and use a binary search to find the right data, but the current method with no keys requires less space and runs much faster.

The "build" directory includes a script called "pack_ostn_data" which creates the 1229 rows of data in the correct form from the "OSTN02_OSGM02_GB.txt" file which is available from the Ordnance Survey.

  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.

Accuracy of readings from GPS devices

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.

To get really accurate readings you have to invest in some serious professional or military grade surveying equipment.

How big is 0.000001 of a degree?

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 longitude 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

How accurate are the conversions?

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 is 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 software that produced the test data.

Translating from the given grid coordinates to WGS84 latitude and longitude coordinates is very slightly less accurate, but in all of England, Wales, Scotland and the Isle of Man this software produces WGS84 lat/lon coordinates from the given grid references that are within a few mm of the OSGB data.

I have also run extensive `round trip' testing by generating random grid references, converting them to WGS84 latitude and longitude and then converting them back to grid easting and northing. In all places between 6W and 2E (the whole of the UK mainland, plus Orkney and Shetland) the round trip error is less than 1 mm. This is the design point of the OS formulae. So far so good. However, west of 6W (that is in the Scilly Isles and the Hebrides), the round trip error creeps up slowly as you go further west; the furthest west you can go on the OS grid is St Kilda (at about 8.57W) and here the round trip error is about 4 mm. As far as I can tell, this is just a limitation of the OS formulae as designed in 2001 when OSTN02 was published.

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 up to about +/-5 metres, in the vicinity of the British Isles.

You can also use this transformation 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 metres and +5.1 metres 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.

How fast are the conversions?

If you can live with a 5 m level of uncertainty, then you will find that the Helmert routines are a bit faster at translating from grid to latitude and longitude. A typical bench mark 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                  41677        0.024
    ll_to_grid_helmert          42145        0.024
    grid_to_ll                  18131        0.055
    grid_to_ll_helmert          35793        0.028

None of the routines is really slow, since even "grid_to_ll" averages under 60 microseconds per call, but "grid_to_ll_helmert" runs about twice as fast as "grid_to_ll". Using a locally compiled version of Perl 5.22, I see a small improvement on the Helmert routines, but the OSTN02 routines are about the same. `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 difference to the round trip precision of the routines. Since the formulae were specifically designed for ordinary double precision arithmetic, Perl's default arithmetic is more than adequate.

  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 in the list of keys as "A:47", 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:OL1W".

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 Scilly Isles, or for promontories 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 as supplied through the "Maps.pm" interface.

The source files for the map data are in the "maps" directory. The script that builds "Maps.pm" from the source files is "build/make_maps_code".

It's probably fair to say that everything about the maps is still (2016) experimental and may change in the future.

For each series there are (currently) two files

•   "catalogue" which defines: an index number for each map (unique to this project); the sheet number for the map - not necessarily an integer; the sheet title as a UTF8 string; the current ISBN number for maps that have them.

•   "polygons" which has: index numbers that match the corresponding "catalogue"; sheet numbers that match the corresponding "catalogue"; a flag - an integer to show the status of the entry (no formal meaning); followed by a MULTIPOLYGON in WKT format.

There is one line in each of the files for each map in the series. The data format for the sheet polygons is Well Known Text (as defined in Wikipedia). The set of polygons for each map is defined as a MULTIPOLYGON; a list of POLYGONS. There are no holes in any of the polygons. A missing polygon list is recorded as "EMPTY" (this is part of WKT).

The units are metres from the false point of origin (which is some miles west of the Scilly Isles). So the south west corner of Landranger sheet 204, which has traditional grid reference SW 720 140 is defined in this data as "172000 14000". This is essentially what "parse_grid" returns. No leading zeros needed.

The polygon starts at the south west corner of the sheet and is recorded anticlockwise. WKT insists that the first pair is repeated at the end to close the polygon. So a simple 40km square Landranger sheet with no insets or extensions, such as Sheet 152, whose SW corner is at SP 530 300 is recorded as

    (((453000 230000, 493000 230000, 493000 270000, 453000 270000, 453000 230000)))

If the sheet boundary is more complicated than a square, the polygon consists of a coordinate pair for each corner. Extensions - where the coloured printing spills over the neat edge - are included as part of the main polygon in the appropriate place, always moving anticlockwise. Extensions don't have to be rectilinear but they are made up of straight lines. Extensions consisting of administrative boundaries and labels are ignored. If in doubt, use common sense.

If an inset is drawn on the map sheet with its own grid margin then it is recorded as a separate polygon following the WKT format, even if it overlaps the main sheet.

On OS Landranger maps, the first (and last) pair should always be the SW corner, if an extension affects the SW corner, start and end with the regular corner pair even if they are technically redundant. This allows me to find the SW corners currently defined for the Landranger maps easily. In other series, start somewhere near the SW and go anticlockwise.