# Geohash

Last updated

Geohash is a public domain geocode system invented in 2008 by Gustavo Niemeyer [1] and (similar work in 1966) G.M. Morton, [2] which encodes a geographic location into a short string of letters and digits. It is a hierarchical spatial data structure which subdivides space into buckets of grid shape, which is one of the many applications of what is known as a Z-order curve, and generally space-filling curves.

## Contents

Geohashes offer properties like arbitrary precision and the possibility of gradually removing characters from the end of the code to reduce its size (and gradually lose precision). Geohashing guarantees that the longer a shared prefix between two geohashes is, the spatially closer they are together. The reverse of this is not guaranteed, as two points can be very close but have a short or no shared prefix.

## History

The core part of the Geohash algorithm and the first initiative to similar solution was documented in a report of G.M. Morton in 1966, "A Computer Oriented Geodetic Data Base and a New Technique in File Sequencing". [2] The Morton work was used for efficient implementations of Z-order curve, like in this modern (2014) Geohash-integer version, based on direct interleave 64-bits integers. But his geocode proposal was not human-readable and was not popular.

Apparently, in the late 2000s, G. Niemeyer still didn't know about Morton's work, and reinvented it, adding the use of base32 representation. In February 2008, together with the announcement of the system, [1] he launched the website  http://geohash.org , which allows users to convert geographic coordinates to short URLs which uniquely identify positions on the Earth, so that referencing them in emails, forums, and websites is more convenient.

Many variations has been developed, as the OpenStreetMap's short link [3] (using base64 instead of base32) in 2009, the 64-bit Geohash [4] in 2014, the exotic Hilbert-Geohash [5] [6] in 2016, and others.

## Typical and main usages

To obtain the Geohash, the user provides an address to be geocoded, or latitude and longitude coordinates, in a single input box (most commonly used formats for latitude and longitude pairs are accepted), and performs the request.

Besides showing the latitude and longitude corresponding to the given Geohash, users who navigate to a Geohash at geohash.org are also presented with an embedded map, and may download a GPX file, or transfer the waypoint directly to certain GPS receivers. Links are also provided to external sites that may provide further details around the specified location.

For example, the coordinate pair 57.64911,10.40744 (near the tip of the peninsula of Jutland, Denmark) produces a slightly shorter hash of u4pruydqqvj.

The main usages of Geohashes are:

• As a unique identifier.
• To represent point data, e.g. in databases.

Geohashes have also been proposed to be used for geotagging.

When used in a database, the structure of geohashed data has two advantages. First, data indexed by geohash will have all points for a given rectangular area in contiguous slices (the number of slices depends on the precision required and the presence of geohash "fault lines"). This is especially useful in database systems where queries on a single index are much easier or faster than multiple-index queries. Second, this index structure can be used for a quick-and-dirty proximity search: the closest points are often among the closest geohashes.

## Technical description

A formal description for Computational and Mathematical views.

### Textual representation

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".

 Decimal Base 32 Decimal Base 32 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 b c d e f g 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 h j k m n p q r s t u v w x y z

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 = ${\displaystyle [(e\times B^{4})+(z\times B^{3})+(s\times B^{2})+(4\times B^{1})+(2\times B^{0})]_{32ghs}}$
= ${\displaystyle [e]_{32ghs}\times B^{4}+[z]_{32ghs}\times B^{3}+[s]_{32ghs}\times B^{2}+[4]_{32ghs}\times B^{1}+[2]_{32ghs}\times B^{0}}$
= ${\displaystyle [13]_{10}\times B^{4}+[31]_{10}\times B^{3}+[24]_{10}\times B^{2}+[4]_{10}\times B^{1}+[2]_{10}\times B^{0}}$
= ${\displaystyle 13\times 1048576+31\times 32768+24\times 1024+4\times 32+2\times 1}$ = ${\displaystyle 13631488+1015808+24576+128+2=14672002}$

### Geometrical representation

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).

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).

### Algorithm and example

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 0110111111110000010000010. 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.

Latitude code 101111001001
bit positionbit valueminmidmaxmean valuemaximum error
01-90.0000.00090.00045.00045.000
100.00045.00090.00022.50022.500
210.00022.50045.00033.75011.250
3122.50033.75045.00039.3755.625
4133.75039.37545.00042.1882.813
5139.37542.18845.00043.5941.406
6042.18843.59445.00042.8910.703
7042.18842.89143.59442.5390.352
8142.18842.53942.89142.7150.176
9042.53942.71542.89142.6270.088
10042.53942.62742.71542.5830.044
11142.53942.58342.62742.6050.022
Longitude code 0111110000000
bit positionbit valueminmidmaxmean valuemaximum error
00-180.0000.000180.000-90.00090.000
11-180.000-90.0000.000-45.00045.000
21-90.000-45.0000.000-22.50022.500
31-45.000-22.5000.000-11.25011.250
41-22.500-11.2500.000-5.6255.625
51-11.250-5.6250.000-2.8132.813
60-5.625-2.8130.000-4.2191.406
70-5.625-4.219-2.813-4.9220.703
80-5.625-4.922-4.219-5.2730.352
90-5.625-5.273-4.922-5.4490.176
100-5.625-5.449-5.273-5.5370.088
110-5.625-5.537-5.449-5.5810.044
120-5.625-5.581-5.537-5.6030.022

(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.

### Digits and precision in km

geohash lengthlat bitslng bitslat errorlng errorkm error
123±23±23±2500
255±2.8±5.6±630
378±0.70±0.70±78
41010±0.087±0.18±20
51213±0.022±0.022±2.4
61515±0.0027±0.0055±0.61
71718±0.00068±0.00068±0.076
82020±0.000085±0.00017±0.019

## Limitations when used for deciding proximity

### Edge cases

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 .

### Non-linearity

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.

## Similar indexing systems

An alternative to storing Geohashes as strings in a database are Locational codes, which are also called spatial keys and similar to QuadTiles. [11] [12]

In some geographical information systems and Big Data spatial databases, a Hilbert curve based indexation can be used as an alternative to Z-order curve, like in the S2 Geometry library. [13]

## Licensing

The Geohash algorithm was put in the public domain by its inventor in a public announcement on February 26, 2008. [14]

While comparable algorithms have been successfully patented [15] and had copyright claimed upon, [16] [17] GeoHash is based on an entirely different algorithm and approach.

## Related Research Articles

Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography to provide equivalent security.

A hash function is any function that can be used to map data of arbitrary size to fixed-size values. The values returned by a hash function are called hash values, hash codes, digests, or simply hashes. The values are used to index a fixed-size table called a hash table. Use of a hash function to index a hash table is called hashing or scatter storage addressing.

A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

The reflected binary code (RBC), also known just as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit.

In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square. Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.

The Ordnance Survey National Grid reference system is a system of geographic grid references used in Great Britain, distinct from latitude and longitude. It is often called British National Grid (BNG).

A geocode is a code that represents a geographic entity. It is a unique identifier of the entity, to distinguish it from others in a finite set of geographic entities. In general the geocode is a human-readable and short identifier.

Base32 is one of several base 32 transfer encodings. Base32 uses a 32-character set comprising the twenty-six upper-case letters A–Z, and the digits 2–7.

In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It is named after Guy Macdonald Morton, who first applied the order to file sequencing in 1966. The z-value of a point in multidimensions is simply calculated by interleaving the binary representations of its coordinate values. Once the data are sorted into this ordering, any one-dimensional data structure can be used such as binary search trees, B-trees, skip lists or hash tables. The resulting ordering can equivalently be described as the order one would get from a depth-first traversal of a quadtree.

Many letters of the Latin alphabet, both capital and small, are used in mathematics, science, and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, or physical entities. Certain letters, when combined with special formatting, take on special meaning.

The Universal Transverse Mercator (UTM) is a system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.

The shapefile format is a geospatial vector data format for geographic information system (GIS) software. It is developed and regulated by Esri as a mostly open specification for data interoperability among Esri and other GIS software products. The shapefile format can spatially describe vector features: points, lines, and polygons, representing, for example, water wells, rivers, and lakes. Each item usually has attributes that describe it, such as name or temperature.

A Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890.

World Meteorological Organization squares or WMO squares is a system of geocodes that divides a chart of the world with latitude-longitude gridlines into grid cells of 10° latitude by 10° longitude, each with a unique, 4-digit numeric identifier. On the plate carrée projection the grid cells appear square, however if the Mercator projection is used, the grid cells appear 'stretched' vertically nearer the tops and bottoms of the map. On the actual surface of the Globe, the cells are approximately "square" only adjacent to the equator, and become progressively narrower and tapered as they approach the poles, and cells adjoining the poles are unique in possessing three faces rather than four.

C-squares is a system of spatially unique, location-based identifiers (geocodes) for areas on the surface of the earth, represented as cells from a Discrete Global Grid at a hierarchical set of resolution steps. The identifiers incorporate literal values of latitude and longitude in an interleaved notation, together with additional digits that support intermediate grid resolutions of 5, 0.5, 0.05 degrees, etc. The system was initially designed to represent data "footprints" or spatial extents in a more flexible manner than a standard minimum bounding rectangle, and to support "lightweight", text-based spatial querying; it can also provide a set of identifiers for grid cells used for assembly, storage and analysis of spatially organised data. Dataset extents expressed in c-squares notation can be visualised using a web-based utility, the c-squares mapper, an online instance of which is currently provided by CSIRO Oceans and Atmosphere in Australia. C-squares codes and associated published software are free to use and the software is released under version 2 of the GNU General Public License (GPL), a licence of the Free Software Foundation.

In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominos, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares.

Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.

The Geohash-36 geocode is an opensource compression algorithm for world coordinate data. It was developed as a variation of the OpenPostcode format developed as a candidate geolocation postcode for the Republic of Ireland. It is calculated differently and uses a more concise base 36 representation rather than other geocodes that adopted base 32.

A Discrete Global Grid (DGG) is a mosaic which covers the entire Earth's surface. Mathematically it is a space partitioning: it consists of a set of non-empty regions that form a partition of the Earth's surface. In a usual grid-modeling strategy, to simplify position calculations, each region is represented by a point, abstracting the grid as a set of region-points. Each region or region-point in the grid is called a cell.

The Open Location Code (OLC) is a geocode system for identifying an area anywhere on the Earth. It was developed at Google's Zürich engineering office, and released late October 2014. Location codes created by the OLC system are referred to as "plus codes".

## References

1. G. M. Morton (1966)"A Computer Oriented Geodetic Data Base and a New Technique in File Sequencing". Report in IBM Canada.
2. The "Geohash binary 64 bits" have classic solutions, as yinqiwen/geohash-int, and optimized solutions, as mmcloughlin/geohash-assembly.
3. Tibor Vukovic (2016), "Hilbert-Geohash - Hashing Geographical Point Data Using the Hilbert Space-Filling Curve". https://pdfs.semanticscholar.org/d23c/996f44b1443fca76276ce8d37239fb8fd6f9.pdf
4. "S2 Geometry Library" for optimized spatial indexation, https://s2geometry.io