Description of the WorldBiG Binary Grid

The creation of a binary grid originates from the need to illustrate botanical issues (drawing cartograms). Grid of this type can be used to virtually divide the area of the terrain and study on this basis the dynamics of changes in any parameters that can be analysed in the area. The issue is therefore quite general and may concern many areas of human activity. It is important, however, that the division of the area is the same (the same surface areas), so that it is possible to carry out statistical analysis of changes occurring in individual areas.
The aim of creating the WorldBiG binary grid was to enable its cooperation with GIS family programs, precise and simple determination of the field surface in any fixed division of the full range of the grid, natural way of labelling binary fields, ease of transfer between reference systems (WGS84 and Cartesian system), and finally, easier implementation of the grid in future utility applications "dividing" the surface with the help of the grid.

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The WorldBiG grid with dimensions of 1024×1024 km is developed on a tangential plane to the sphere (with radius R) in the central point of the grid (Xo, Yo)=(512, 512), and the applied mapping is based on mathematical formulas derived from the Lambert Azimuthal Equal Area Projection model. Reverse transformation f -1 determines the values of geographical coordinates in the WGS 84 reference system for the "meshes" of the grid in the rectangular coordinate system (the equivalent of squares are the fields determined after the transformation to the WGS 84 ellipsoid):
 $$\begin{array}{l}\\For\;transformation\;(x,\;y)=f\;(\lambda,\;\varphi):\\x=Xo+R\times p\times\cos\;\varphi\times\sin(\lambda-\beta)\\y=Yo+R\times p\times(\mathrm{co}s\;\alpha\times\sin\;\varphi-\sin\;\alpha\times\cos\;\varphi\times\cos(\lambda-\beta))\\\\For\;revers\;transformation\;(\lambda,\;\varphi)=f^{-1}(x,\;y):\\\varphi=arc\;\sin(\cos\;c\times\sin\;\alpha+(y-Yo)\times\sin\;c\times\frac{\cos\;\alpha}g)\\\lambda=\beta+arc\;tg((x-Xo)\times\frac{\sin\;c}{g\;\times\;\cos\;c\;\times\;\cos\;\alpha\;-\;(y-Yo)\;\times\;\sin\;c\;\times\;\sin\;\alpha})\\\\where:\\\boldsymbol p=\sqrt{2/(1+\sin\;\alpha\times\sin\;\varphi+\cos\;\varphi\times\cos\;\alpha\times\cos(\lambda-\beta))}\\\boldsymbol g=\sqrt{(x-Xo)^2+(y-Yo)^2}\;\;;\;\;\boldsymbol c=2\times arc\;\sin(\frac g{2\times R})\\\end{array}$$
For those interested I make available a simple MS Excel workbook where in two subsequent sheets I have formally prescribed the above transformation formulas (conversions inside the trigonometric functions are performed using the angle curve measurement, in the case of reverse functions we return to degrees). Thus, it is easy to check the conformity of the transformation in both directions, the conformity of the mid-point (Xo,Yo) = (512,512) for the geographical coordinates α = 52°N and β = 19°E (these are the exemplary values in the file) and also the distinguished property of the meridian β = 19°E, which always corresponds to the value of the Cartesian coordinate x = 512. The MS Excel file also allows you to convert geographical coordinates into Cartesian coordinates of the grid and vice versa in bulk quantities, just insert them into the corresponding sheet and convert them on a copy-paste basis.

WorldBiG grid labelling

The WorldBiG binary grid allows simple introduction of mesh labelling for any division. Large squares of the WorldBiG grid (64×64 km) can be labelled in small letters in line and column order (similarly to the matrix) only for better orientation in the geographical position of large binary fields. The general scheme of labelling is described in the work cited above - it is based on the numbering of 1,2,3,4 consecutive quarters of "thickened" squares in this binary way. The numbering is added in the next division to the label of the square of the grid on its right side. Look how it works. Large binary squares of 64×64 km are already divided into 64 "meshes" of 8×8 km after three consecutive divisions. Also 8×8 km squares will form 64 smaller "meshes" of 1×1 km after the next triple division. For both of these cases the label record is identical.
All basic information about the grid created using the program user data is in the KML file. This file is always generated by the WorldBiG software, along with information about the mesh's non-empty mesh labels and name as well as the number of objects contained in it. So there is no need to generate your own grids. This is done by a program that is already available. Nevertheless, it is worth informing the readers that QGIS has the right tools to generate identical grids in the poligon version. Below you can find some examples of such grids with instructions how to generate them.

EXAMPLE WorldBiG (field generated from QGIS):

Binary grid (1024 × 1024 km) - square 8×8 km
Binary grid (1024 × 1024 km) - square 4×4 km
Binary grid (part of Małopolska) - square 1×1 km
Binary grid (Kraków-Borek Fałęcki) - square 0,25×0,25 km

Decimal grid (1000 × 1000 km) - square 10×10 km
Decimal grid (1000 × 1000 km) - square 5×5 km

How to generate a WorldBiG grid using QGIS software ?
1. We install QGIS in the currently available version (description applies in 3.2 for Windows).
2. Go to Settings and then to User Coordinate System.
3. Create the definition of the coordinate system by pressing + in green.
4. Enter the name of the coordinate system: e.g. Polbig1024 to generate a grid of maximum 1024×1024 km.
5. Insert the following text into the "Parameters" window (Proj4 definition of the coordinate system):
+proj=laea +lat_1=52 +lat_2=52 +lat_0=52 +lon_0=19 +a=6371000 +b=6371000 +ellps=sphere +x_0=512000 +y_0=512000
or in a slightly shorter alternative notation, according to Proj4 and the definition of "own" coordinate system in the QGIS package:
+proj=laea +lat_1=52 +lat_2=52 +lat_0=52 +lon_0=19 +R=6371000 +x_0=512000 +y_0=512000
6. Confirm our definition of the created coordinate system with OK button and click on EPSG:4326, which is the current WGS 84 coordinate system - the button is located in the bottom right corner of the QGIS program window.
7. In the "Available coordinate systems" window, at the very bottom (you have to use the bar), drop down > "User coordinate systems" and select our defined system with Apply and then OK.
8. Now we can generate a grid in this defined layout. Go to the main menu, select Vector , then Research Tools and Create Grid.
9. In the resulting window (Create Grid), choose: Rectangle (polygon), grid extent (xmin, xmax, ymin, ymax). In our case, when we want to generate a full WorldBiG grid, it will be an entry: 0,1024000,0,1024000 because we enter the data in meters. Of course, you can enter other values, it will not change the grid itself but only its maximum size. However, it is important to specify such a range, so that the size of the "mesh of the grid" is completely within the given range.
10. Now, we select the dimension of the "meshes", i.e. Horizontal Distance and Vertical Distance. According to the above, these can be examples of sizes 16 km, 8 km, 4 km, 2 km, 1 km, 1/2 km, 1/4 km, 1/8 km etc. These sizes are given in meters, for example, enter 8000 for both fields. Note, the small size of "meshes" generates large files later on, because each "mesh (within the declared range) will be described. Then set the range to smaller.
11. Leave the rest unchanged, checking that the box is checked: Load the file after finishing.
12. Press the Run button in the background and then OK. The generated grid should appear as adjacent squares.
13. In the bar on the left side of the main window, in the so-called Layers, a poem entitled Grid will appear. You can now right-click the Grid context menu and select Export when expanded. Then choose Save feature as.
14. In the resulting window we choose the format ... for example, Keyhole Markup Language [KML] (allows you to see the grid in the popular G. Earth program), or ESRI Shapefile format (you should then point to an empty folder and enter your own file name - a set of files associated with this name will be generated in the folder). In the Coordinate System, below, you should choose the coordinate system you are interested in, so the default one - EPSG: 4326 - WGS 84. This will ensure that the grid "located on the map" drawn in this coordinate system will look correct.
15. Below we uncheck the option Add saved file to the map, it is important as this way we will still remain in our own "Cartesian" system, even though the file(s) themselves will already be generated in the global WGS 84 standard. So we will be able to continue to generate other binary grids remaining in the rectangular coordinate system.
16. Click OK , after a while a message at the top of the main program window with information about the saved file.

People who would like to generate other grids, e.g. 10×10 km, can of course do the same, but a different definition of their own coordinate system should be used. For example, if the whole grid is to have 1000×1000 km (slightly smaller than the binary one), then for such a size of grid there will be natural meshes of 10 km, 5 km, 2 km or 1 km side. Then we define our own layout as follows:
+proj=laea +lat_1=52 +lat_2=52 +lat_0=52 +lon_0=19 +a=6371000 +b=6371000 +ellps=sphere +x_0=500000 +y_0=500000
or:  +proj=laea +lat_1=52 +lat_2=52 +lat_0=52 +lon_0=19 +R=6371000 +x_0=500000 +y_0=500000
The only thing that has changed is the numerical value x_0 and y_0 (the grid has a smaller range calculated from the central point), all the rest of the parameters remained the same, which guarantees the generation of the appropriate grid (the same geographical location of the central point and the same value of parameter R).
It is worth noting that the WorldBiG grid, binary and decimal, are identical for kilometre and two-kilometre "meshes", but their labelling will be different.

Marek Verey, Kraków, November 2020