Geohash

For exact latitude and longitude translations Geohash is a spatial index of base 4, because it transforms the continuous latitude and lontitude space coordinates into a hierarchical discrete grid, using a recurrent four-partition of the space. To be a compact code it uses base 32 and represents its values by the following alphabet, that is the "standard textual representation".

The "Geohash alphabet" (32ghs) uses all digits 0-9 and almost all lower case letters except "a", "i", "l" and "o".

For example, using the table above and the constant ${\displaystyle B=32}$, the Geohash ezs42 can be converted to a decimal representation by ordinary positional notation:

[ezs42]_32ghs = ${\displaystyle [(e\times B^{4})+(z\times B^{3})+(s\times B^{2})+(4\times B^{1})+(2\times B^{0})]_{32ghs}}$

= ${\displaystyle [e]_{32ghs}\times B^{4}+[z]_{32ghs}\times B^{3}+[s]_{32ghs}\times B^{2}+[4]_{32ghs}\times B^{1}+[2]_{32ghs}\times B^{0}}$

= ${\displaystyle [13]_{10}\times B^{4}+[31]_{10}\times B^{3}+[24]_{10}\times B^{2}+[4]_{10}\times B^{1}+[2]_{10}\times B^{0}}$

= ${\displaystyle 13\times 1048576+31\times 32768+24\times 1024+4\times 32+2\times 1}$ = ${\displaystyle 13631488+1015808+24576+128+2=14672002}$

The geometry of the Geohash have a mixed spatial representation:

• Geohashes with 2, 4, 6, ... e digits (even digits) are represented by Z-order curve in a "regular grid" where decoded pair (latitude,longitude) has uniform uncertainty, valid as Geo URI.
• Geohashes with 1, 3, 5, ... d digits (odd digits) are represented by "И-order curve". Latitude and longitude of the decoded pair has different uncertainty (longitude is truncated).

Merging cells of the "next level" to obtain a geometrical representation of the "odd-digit Geohash".

Is possible to build the "И-order curve" from Z-order by merging neighboring cells and indexing the result rectangular grid by the function j=floor(i ÷ 2). The side illustration shows how to obtain the grid of 32 rectangular cells from the grid of 64 square cells.

The most important property of Geohash for humans is that it preserves spatial hierarchy in the code prefixes.
For example, in the "1 Geohash digit grid" illustration of 32 rectangles, above, the spatial region of the code e (rectangle of greyish blue circle at position 4,3) is preserved with prefix e in the "2 digit grid" of 1024 rectangles (scale showing em and greyish green to blue circles at grid).

Using the hash ezs42 as an example, here is how it is decoded into a decimal latitude and longitude. The first step is decoding it from textual "base 32ghs", as showed above, to obtain the binary representation:

${\displaystyle [e]_{32ghs}=[13]_{10}=[01101]_{2}}$

${\displaystyle [z]_{32ghs}=[31]_{10}=[11111]_{2}}$

${\displaystyle [s]_{32ghs}=[24]_{10}=[11000]_{2}}$

${\displaystyle [4]_{32ghs}=[4]_{10}=[00100]_{2}}$

${\displaystyle [2]_{32ghs}=[2]_{10}=[00010]_{2}}$.

This operation results in the bits 01101 11111 11000 00100 00010. Assuming that counting starts at 0 in the left side, the even bits are taken for the longitude code (0111110000000), while the odd bits are taken for the latitude code (101111001001).

Each binary code is then used in a series of divisions, considering one bit at a time, again from the left to the right side. For the latitude value, the interval -90 to +90 is divided by 2, producing two intervals: -90 to 0, and 0 to +90. Since the first bit is 1, the higher interval is chosen, and becomes the current interval. The procedure is repeated for all bits in the code. Finally, the latitude value is the center of the resulting interval. Longitudes are processed in an equivalent way, keeping in mind that the initial interval is -180 to +180.

For example, in the latitude code 101111001001, the first bit is 1, so we know our latitude is somewhere between 0 and 90. Without any more bits, we'd guess the latitude was 45, giving us an error of ±45. Since more bits are available, we can continue with the next bit, and each subsequent bit halves this error. This table shows the effect of each bit. At each stage, the relevant half of the range is highlighted in green; a low bit selects the lower range, a high bit selects the upper range.

The column “mean value” shows the latitude, simply the mean value of the range. Each subsequent bit makes this value more precise.

(The numbers in the above table have been rounded to 3 decimal places for clarity)

Final rounding should be done carefully in a way that

${\displaystyle \min \leq \mathrm {round} (value)\leq \max }$

So while rounding 42.605 to 42.61 or 42.6 is correct, rounding to 43 is not.

Geohashes can be used to find points in proximity to each other based on a common prefix. However, edge case locations close to each other but on opposite sides of the 180 degree meridian will result in Geohash codes with no common prefix (different longitudes for near physical locations). Points close to the North and South poles will have very different geohashes (different longitudes for near physical locations).

Two close locations on either side of the Equator (or Greenwich meridian) will not have a long common prefix since they belong to different 'halves' of the world. Put simply, one location's binary latitude (or longitude) will be 011111... and the other 100000...., so they will not have a common prefix and most bits will be flipped. This can also be seen as a consequence of relying on the Z-order curve (which could more appropriately be called an N-order visit in this case) for ordering the points, as two points close by might be visited at very different times. However, two points with a long common prefix will be close by.

In order to do a proximity search, one could compute the southwest corner (low geohash with low latitude and longitude) and northeast corner (high geohash with high latitude and longitude) of a bounding box and search for geohashes between those two. This search will retrieve all points in the z-order curve between the two corners, which can be far too many points. This method also breaks down at the 180 meridians and the poles. Solr uses a filter list of prefixes, by computing the prefixes of the nearest squares close to the geohash .

Since a geohash (in this implementation) is based on coordinates of longitude and latitude the distance between two geohashes reflects the distance in latitude/longitude coordinates between two points, which does not translate to actual distance, see Haversine formula.

Example of non-linearity for latitude-longitude system:

• At the Equator (0 Degrees) the length of a degree of longitude is 111.320 km, while a degree of latitude measures 110.574 km, an error of 0.67%.
• At 30 Degrees (Mid Latitudes) the error is 110.852/96.486 = 14.89%
• At 60 Degrees (High Arctic) the error is 111.412/55.800 = 99.67%, reaching infinity at the poles.

Note that these limitations are not due to geohashing, and not due to latitude-longitude coordinates, but due to the difficulty of mapping coordinates on a sphere (non linear and with wrapping of values, similar to modulo arithmetic) to two dimensional coordinates and the difficulty of exploring a two dimensional space uniformly. The first is related to Geographical coordinate system and Map projection, and the other to Hilbert curve and z-order curve. Once a coordinate system is found that represents points linearly in distance and wraps up at the edges, and can be explored uniformly, applying geohashing to those coordinates will not suffer from the limitations above.

While it is possible to apply geohashing to an area with a Cartesian coordinate system, it would then only apply to the area where the coordinate system applies.

Despite those issues, there are possible workarounds, and the algorithm has been successfully used in Elasticsearch,[7] MongoDB,[8] HBase, Redis,[9] and Accumulo[10] to implement proximity searches.