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https://github.com/JuliaGeo/Geodesy.jl

Work with points defined in various coordinate systems.
https://github.com/JuliaGeo/Geodesy.jl

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Work with points defined in various coordinate systems.

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README 

        
# Geodesy



[![Build Status](https://github.com/JuliaGeo/Geodesy.jl/workflows/CI/badge.svg)](https://github.com/JuliaGeo/Geodesy.jl/actions)



**Geodesy** is a Julia package for working with points in various world and

local coordinate systems. The primary feature of *Geodesy* is to define and

perform coordinate transformations in a convenient and safe framework,

leveraging the *CoordinateTransformations* [package](https://github.com/FugroRoames/CoordinateTransformations.jl).

Transformations are accurate and efficient and implemented in native Julia code

(with many functions being ported from Charles Karney's *GeographicLib*

[C++ library](http://geographiclib.sourceforge.net/)), and some common geodetic

datums are provided for convenience.



## Quick start



Lets define a 3D point by its latitude, longitude and altitude (LLA):

```julia

x_lla = LLA(-27.468937, 153.023628, 0.0) # City Hall, Brisbane, Australia

```



This can be converted to a Cartesian Earth-Centered-Earth-Fixed (ECEF)

coordinate simply by calling the constructor

```julia

x_ecef = ECEF(x_lla, wgs84)

```



Here we have used the WGS-84 ellipsoid to calculate the transformation, but other

datums such as `osgb36`, `nad27` and `grs80` are provided. All transformations

use the *CoordinateTransformations*' interface, and the above is short for

```julia

x_ecef = ECEFfromLLA(wgs84)(x_lla)

```

where `ECEFfromLLA` is a type inheriting from *CoordinateTransformations*'

`Transformation`. (Similar names `XfromY` exist for each of the

coordinate types.)



Often, points are measured or required in a *local* frame, such as the north-east-up

coordinates with respect to a given origin. The `ENU` type represents points in this

coordinate system and we may transform between ENU and globally referenced

coordinates using `ENUfromLLA`, etc.

```julia

origin_lla = LLA(-27.468937, 153.023628, 0.0) # City Hall, Brisbane, Australia

point_lla = LLA(-27.465933, 153.025900, 0.0)  # Central Station, Brisbane, Australia



# Define the transformation and execute it

trans = ENUfromLLA(origin_lla, wgs84)

point_enu = trans(point_lla)



# Equivalently

point_enu = ENU(point_lla, origin_lla, wgs84)

```



Similarly, we could convert to UTM/UPS coordinates, and two types are provided

for this - `UTM` stores 3D coordinates `x`, `y`, and `z` in an unspecified zone,

while `UTMZ` includes the `zone` number and `hemisphere` bool (where `true` =

northern, `false` = southern). To get the canonical zone for your coordinates,

simply use:

```julia

x_utmz = UTMZ(x_lla, wgs84)

```



If you are transforming a large number of points to or from a given zone, it may

be more effective to define the transformation explicitly and use the lighter

`UTM` storage type.

```julia

points_lla::Vector{LLA{Float64}}

utm_from_lla = UTMfromLLA(56, false, wgs84) # Zone 56-South

points_utm = map(utm_from_lla, points_lla) # A new vector of UTM coordinates

```



*Geodesy* becomes particularly powerful when you chain together transformations.

For example, you can define a single transformation from your data on disk in UTM

coordinates to a local frame in ENU coordinates. Internally, this will perform

UTM (+ zone) → LLA → ECEF → ENU via composing transformations with `∘` into a

`ComposedTransformation`:

```julia

julia> origin = LLA(-27.468937, 153.023628, 0.0) # City Hall, Brisbane, Australia

LLA(lat=-27.468937°, lon=153.023628°, alt=0.0)



julia> trans = ENUfromUTMZ(origin, wgs84)

(ENUfromECEF(ECEF(-5.046925124630393e6, 2.5689157252069353e6, -2.924416653602336e6), lat=-27.468937°, lon=153.023628°) ∘ (ECEFfromLLA(wgs84) ∘ LLAfromUTMZ(wgs84)))

```



This transformation can then be composed with rotations and translations in

*CoordinateTransformations* (or your own custom-defined `AbstractTransformation`

to define further reference frames. For example, in this way, a point measured

by a scanner on a moving vehicle at a particular time may be globally

georeferenced with a single call to the `Transformation`!



Finally, the Cartesian distance between world points can be calculated via

automatic transformation to a Cartesian frame:

```julia

x_lla = LLA(-27.468937, 153.023628, 0.0) # City Hall, Brisbane, Australia

y_lla = LLA(-27.465933, 153.025900, 0.0) # Central Station, Brisbane, Australia

euclidean_distance(x_lla, y_lla)                   # 401.54 meters

```

(assuming the `wgs84` datum, which can be configured in `distance(x, y, datum)`).



## Basic Terminology



This section describes some terminology and concepts that are relevant to

*Geodesy.jl*, attempting to define Geodesy-specific jargon where possible.  For

a longer, less technical discussion with more historical context, ICSM's

[Fundamentals of Mapping page](https://www.icsm.gov.au/education/fundamentals-mapping)

is highly recommended.



### Coordinate Reference Systems and Spatial Reference Identifiers



A position on the Earth can be given by some numerical coordinate values, but

those don't mean much without more information.  The extra information is called

the **Coordinate Reference System** or **CRS** (also known as a *Spatial

Reference System* or SRS).  A CRS tells you two main things:



* The measurement procedure: which real world objects were used to

  define the frame of reference or *datum* of the measurement?

* The *coordinate system*: how do coordinate numerical values relate to the

  reference frame defined by the datum?



The full specification of a CRS can be complex, so a short label called a

**Spatial Reference IDentifier** or **SRID** is usually used instead.  For

example, [EPSG:4326](http://epsg.io/4326) is one way to refer to the 2D WGS84

latitude and longitude you'd get from a mobile phone GPS device.  An SRID

is of the form `AUTHORITY:CODE`, where the code is a number and the authority is

the name of an organization maintaining a list of codes with associated CRS

information.  There are services where you can look up a CRS, for example,

 is a convenient interface to the SRIDs maintained by the

*European Petroleum Survey Group* (EPSG) authority.  Likewise,

 is an open registry to which anyone can

contribute.



When maintaining a spatial database, it's typical to define an internal list of

SRIDs (effectively making your organization the authority), and a mapping from

these to CRS information.  A link back to a definitive SRID from an external

authority should also be included where possible.



### Datums



In spatial measurement and positioning, a **datum** is a set of reference

objects with given coordinates, *relative to which* other objects may be

positioned.  For example, in traditional surveying a datum might comprise a

pair of pegs in the ground, separated by a carefully measured distance.  When

surveying the position of an unknown but nearby point, the angle back to the

original datum objects can be measured using a theodolite.  After this, the

relative position of the new point can be computed using simple triangulation.

Repeating this trick with any of the now three known points, an entire

triangulation network of surveyed objects can be extended outward.  Any point

surveyed relative to the network is said to be measured *in the datum* of the

original objects.  Datums are often named with an acronym, for example OSGB36 is

the Ordnance Survey of Great Britain, 1936.



In the era of satellite geodesy, coordinates are determined for an object

by timing signals from a satellite constellation (eg, the GPS satellites) and

computing position relative to those satellites.  Where is the datum here? At

first glance the situation seems quite different from the traditional setup

described above.  However, the satellite positions as a function of time

(*ephemerides*, in the jargon) must themselves be defined relative to some

frame. This is done by continuously observing the satellites from a set of

highly stable ground stations equipped with GPS receivers. It is the full set of

these ground stations and their assigned coordinates which form the datum.



Let's inspect the flow of positional information in both cases:

* For traditional surveying,

  ```

  datum object positions -> triangulation network -> newly surveyed point

  ```

* For satellite geodesy,

  ```

  datum object positions -> satellite ephemerides -> newly surveyed point

  ```



We see that the basic nature of a datum is precisely the same regardless of

whether we're doing a traditional survey or using a GPS receiver.



### Terrestrial reference systems and frames



Coordinates for new points are measured by transferring coordinates from the

datum objects, as described above.  However, how do we decide

on coordinates for the datum objects themselves?  This is purely a matter of

convention, consistency and measurement.



For example, the **International Terrestrial Reference System** (**ITRS**) is a

reference system that rotates with the Earth so that the average velocity of

the crust is zero. That is, in this reference system the only crust movement is

geophysical.  Roughly speaking, the *defining conventions* for the ITRS are:



* Space is modeled as a three-dimensional Euclidean affine space.

* The origin is at the center of mass of the Earth (it is *geocentric*).

* The z-axis is the axis of rotation of the Earth.

* The scale is set to 1 SI meter.

* The x-axis is orthogonal to the z-axis and aligns with the international

  reference meridian through Greenwich.

* The y-axis is set to the cross product of the z and x axes, forming a right

  handed coordinate frame.

* Various rates of change of the above must also be specified, for example, the

  scale should stay constant in time.



The precise conventions are defined in chapter 4 of the

[IERS conventions](https://www.iers.org/IERS/EN/Publications/TechnicalNotes/tn36.html)

published by the International Earth Rotation and Reference Service (IERS).

These conventions define an ideal reference *system*, but they're useless

without physical measurements that give coordinates for a set of real world

datum objects.  The process of measuring and computing coordinates for datum

objects is called *realizing* the reference system and the result is called a

*reference frame*.  For example, the **International Terrestrial Reference Frame

of 2014** (**ITRF2014**) realizes the ITRS conventions using raw measurement

data gathered in the 25 years prior to 2014.



To measure and compute coordinates, several space geodesy techniques are used to

gather raw measurement data; currently the IERS includes

[VLBI (very long baseline interferometry)](https://en.wikipedia.org/wiki/Very-long-baseline_interferometry) of distant astronomical radio sources,

[SLR (satellite laser ranging)](https://en.wikipedia.org/wiki/Satellite_laser_ranging),

[GPS (global positioning system)](https://en.wikipedia.org/wiki/Global_Positioning_System) and

[DORIS (gosh these acronyms are tiring)](https://en.wikipedia.org/wiki/DORIS_(geodesy)).

The raw data is not in the form of positions, but must be condensed down in a

large scale fitting problem, ideally by requiring physical and statistical

consistency of all measurements, tying measurements at different sites together

with physical models.



### Coordinate systems



In geometry, a **coordinate system**

[is a system](https://en.wikipedia.org/wiki/Coordinate_system)

which uses one or more numbers, or **coordinates** to uniquely

determine the position of a point in a mathematical space such as Euclidean

space.  For example, in geodesy a point is commonly referred to using geodetic

latitude, longitude and height relative to a given reference ellipsoid; this is

called a **geodetic coordinate system**.



An [**ellipsoid**](https://en.wikipedia.org/wiki/Ellipsoid) is chosen because

it's a reasonable model for the shape of the Earth and its gravitational field

without being overly complex; it has only a few parameters, and a simple

mathematical form.  The term [**spheroid**](https://en.wikipedia.org/wiki/Spheroid)

is also used because the ellipsoids in use today are rotationally symmetric

around the pole. Note that there's several ways to define

[latitude](https://en.wikipedia.org/wiki/Latitude) on an ellipsoid. The most

natural for geodesy is **geodetic latitude**, used by default because it's

physically accessible in any location as a good approximation to the angle

between the gravity vector and the equatorial plane.  (This type of latitude

is not an angle measured at the centre of the ellipsoid, which may be surprising

if you're used to spherical coordinates!)



There are usually several useful coordinate systems for the same space.  As well

as the geodetic coordinates mentioned above, it's common to see



* The x,y,z components in an Earth-Centred Cartesian coordinate system rotating

  with the Earth.  This is conventionally called an

  **Earth-Centred Earth-Fixed** (**ECEF**) coordinate system. This is a natural

  coordinate system in which to define coordinates for the datum objects

  defining a terrestrial reference frame.

* The east,north and up **ENU** components of a Cartesian coordinate frame at a

  particular point on the ellipsoid.  This coordinate system is useful as a

  local frame for navigation.

* Easting,northing and vertical components of a **projected coordinate system** or

  [**map projection**](http://www.icsm.gov.au/mapping/about_projections.html).

  There's an entire zoo of these, designed to represent the curved surface of an

  ellipsoid with a flat map.



Different coordinates systems provide different coordinates for the same point,

so it's obviously important to specify exactly which coordinate system you're

using.  In particular, you should specify which ellipsoid parameters are in

use if you deal with latitude and longitude, as in principle you could have more

than one ellipsoid.  This is a point of confusion, because a datum in geodesy

also comes with a reference ellipsoid as a very strong matter of convention

(thus being called a **geodetic datum**).



With its conventional ellipsoid, a geodetic datum also defines a conventional

geodetic coordinate system, thus bringing together concepts which are

interconnected but conceptually distinct.  To emphasize:



* A coordinate system is a mathematical abstraction allowing us to manipulate

  *geometric* quantities using numeric and algebraic techniques.  By itself,

  mathematical geometry is pure abstraction without a connection to the physical

  world.

* A datum is a set of physical objects with associated coordinates, thereby

  *defining* a reference frame in a way which is physically accessible.  A datum

  is the bridge which connects physical reality to the abstract ideal of

  mathematical geometry, via the algebraic mechanism of a coordinate system.



## The API



### Coordinate types



Geodesy provides several in-built coordinate storage types for convenience and

safety. The philosophy is to avoid carrying around raw data in generic containers

like `Vector`s with no concept of what coordinate system it is in.



##### `LLA{T}` - latitude, longitude and altitude



The global `LLA` type stores data in a lat-lon-alt order, where latitude and longitude

are expected in degrees (not radians). A keyword constructor, `LLA(lat=x, lon=y, alt=z)`,

is also provided to help with having to remember the storage order.



##### `LatLon{T}` - latitude and longitude



The 2D `LatLon` type stores data in a lat-lon order, where latitude and longitude

are expected in degrees (not radians). A keyword constructor, `LatLon(lat=x, lon=y)`,

is also provided. `LatLon` is currently the only supported 2D coordinate.



##### `ECEF{T}` - Earth-centered, Earth-fixed



The global `ECEF` type stores Cartesian coordinates `x`, `y`, `z`, according to the

[usual convention](https://en.wikipedia.org/wiki/ECEF). Being a Cartesian frame,

`ECEF` is a subtype of [StaticArrays](https://github.com/JuliaArrays/StaticArrays.jl)'

`StaticVector` and they can be added and subtracted with themselves and other

vectors.



##### `UTM{T}` - universal transverse-Mercator



The `UTM` type encodes the easting `x`, northing `y` and height `z` of a UTM

coordinate in an unspecified zone. This data type is also used to encode

universal polar-stereographic (UPS) coordinates (where the zone is `0`).



##### `UTMZ{T}` - universal transverse-Mercator + zone



In addition to the easting `x`, northing `y` and height `z`, the global `UTMZ` type

also encodes the UTM `zone` and `hemisphere`, where `zone` is a `UInt8` and

`hemisphere` is a `Bool` for compact storage. The northern hemisphere is

denoted as `true`, and the southern as `false`. Zone `0` corresponds to the UPS

projection about the corresponding pole, otherwise `zone` is an integer between

`1` and `60`.



##### `ENU{T}` - east-north-up



The `ENU` type is a local Cartesian coordinate that encodes a point's distance

towards east `e`, towards north `n` and upwards `u` with respect to an

unspecified origin. Like `ECEF`, `ENU` is also a subtype of `StaticVector`.



### Geodetic Datums



Geodetic datums are modelled as subtypes of the abstract type `Datum`.  The

associated ellipsoid may be obtained by calling the `ellipsoid()` function, for

example, `ellipsoid(NAD83())`.



There are several pre-defined datums.  Worldwide datums include



* `WGS84` - standard GPS datum for moderate precision work (representing both

  the latest frame realization, or if time is supplied a discontinuous dynamic

  datum where time looks up the frame implementation date in the broadcast

  ephemerides.)

* `WGS84{GpsWeek}` - specific realizations of the WGS84 frame.

* `ITRF{Year}` - Realizations of the International Terrestrial Reference System

  for high precision surveying.



National datums include



* `OSGB36` - Ordnance Survey of Great Britain of 1936.

* `NAD27`, `NAD83` - North American Datums of 1927 and 1983, respectively

* `GDA94` - Geocentric Datum of Australia, 1994.



Datums may also be passed to coordinate transformation constructors such as

transverse-Mercator and polar-stereographic projections in which case the

associated ellipsoid will be extracted to form the transformation.  For datums

without extra parameters (everything except `ITRF` and `WGS84{Week}`) there is a

standard instance defined to reduce the amount of brackets you have to type.

For example, `LLAfromECEF(NAD83())` and `LLAfromECEF(nad83)` are equivalent.



### Transformations and conversions



*Geodesy* provides two interfaces changing coordinate systems.



"Transformations" are based on *CoordinateTransformations* interface for defining

`AbstractTransformation`s and allow the user to apply them by calling them,

invert them with `inv()` and compose them with `compose()` or `∘`. The transformations

cache any possible pre-calculations for efficiency when the same transformation

is applied to many points.



"Conversions" are based on type-constructors, obeying simple syntax like `LLA(ecef, datum)`.

The `datum` or other information is *always* necessary, as no assumptions are

made by *Geodesy* for safety and consistency reasons. Similarly, `Base.convert`

is not defined because, without assumptions, it would require additional

information. The main drawback of this approach is that some calculations may not

be pre-cached (for instance, the origin of an ENU transformation).



#### Between `LLA` and `ECEF`



The `LLAfromECEF` and `ECEFfromLLA` transformations require an ellipsoidal datum

to perform the conversion. The exact transformation is performed in both directions,

using a port the ECEF → LLA transformation from *GeographicLib*.



Note that in some cases where points are very close to the centre of the ellipsoid,

multiple equivalent `LLA` points are valid solutions to the transformation problem.

Here, as in *GeographicLib*, the point with the greatest altitude is chosen.



#### Between `LLA` and `UTM`/`UTMZ`



The `LLAfromUTM(Z)` and `UTM(Z)fromLLA` transformations also require an

ellipsoidal datum to perform the conversion. The transformation retains a cache

of the parameters used in the transformation, which in the case of the

transverse-Mercator projection leads to a significant saving.



In all cases zone `0` corresponds to the UPS coordinate system, and the

polar-stereographic projection of *GeographicLib* has been ported to Julia to

perform the transformation.



An approximate, 6th-order expansion is used by default for the transverse-Mercator

projection and its inverse (though orders 4-8 are defined). The algorithm is a

native Julia port of that used in *GeographicLib*, and is accurate to nanometers

for up to several UTM zones away from the reference meridian. However, the

series expansion diverges at ±90° from the reference meridian. While the `UTMZ`-methods

will automatically choose the canonical zone and hemisphere for the input,

extreme care must be taken to choose an appropriate zone for the `UTM` methods.

(In the future, we implement the exact UTM transformation as a fallback —

contributions welcome!)



There is also `UTMfromUTMZ` and `UTMZfromUTM` transformations that are helpful

for converting between these two formats and putting data into the same `UTM`

zone.



#### To and from local `ENU` frames



The `ECEFfromENU` and `ENUfromECEF` transformations define the transformation

around a specific origin. Both the origin coordinates as an `ECEF` as well as

its corresponding latitude and longitude are stored in the transformation for

maximal efficiency when performing multiple `transform`s. The transformation can

be inverted with `inv` to perform the reverse transformation with respect to the

same origin.



#### Web Mercator support



We support the Web Mercator / Pseudo Mercator projection with the

`WebMercatorfromLLA` and `LLAfromWebMercator` transformations for

interoperability with many web mapping systems. The scaling of the

northing and easting is defined to be meters at the Equator, the same as how

proj handles this (see https://proj.org/operations/projections/webmerc.html ).



If you need to deal with web mapping tile coordinate systems (zoom levels and

pixel coordinates, etc) these could be added by composing another

transformation on top of the web mercator projection defined in this package.



#### Composed transformations



Many other methods are defined as convenience constructors for composed

transformations, to go between any two of the coordinate types defined here.

These include:



* `ECEFfromUTMZ(datum) = ECEFfromLLA(datum) ∘ LLAfromUTMZ(datum)`

* `UTMZfromECEF(datum) = UTMZfromLLA(datum) ∘ LLAfromECEF(datum)`

* `UTMfromECEF(zone, hemisphere, datum) = UTMfromLLA(zone, hemisphere, datum) ∘ LLAfromECEF(datum)`

* `ECEFfromUTM(zone, hemisphere, datum) = ECEFfromLLA(datum) ∘ LLAfromUTM(zone, hemisphere, datum)`

* `ENUfromLLA(origin, datum) = ENUfromECEF(origin, datum) ∘ ECEFfromLLA(datum)`

* `LLAfromENU(origin, datum) = LLAfromECEF(datum) ∘ ECEFfromENU(origin, datum)`

* `ECEFfromUTMZ(datum) = ECEFfromLLA(datum) ∘ LLAfromUTMZ(datum)`

* `ENUfromUTMZ(origin, datum)  = ENUfromLLA(origin, datum) ∘ LLAfromUTMZ(datum`

* `UTMZfromENU(origin, datum)  = UTMZfromLLA(datum) ∘ LLAfromENU(origin, datum)`

* `UTMfromENU(origin, zone, hemisphere, datum) = UTMfromLLA(zone, hemisphere, datum) ∘ LLAfromENU(origin, datum)`

* `ENUfromUTM(origin, zone, hemisphere, datum) = ENUfromLLA(origin, datum) ∘ LLAfromUTM(zone, hemisphere, datum)`



Constructor-based transforms for these are also provided, such as `UTMZ(ecef, datum)`

which converts to `LLA` as an intermediary, as above. When converting multiple

points to or from the *same* ENU reference frame, it is recommended to use the

transformation-based approach for efficiency. However, the other

constructor-based conversions should be similar in speed to their transformation

counterparts.



### Distance



Currently, the only defined distance measure is the straight-line or [Euclidean

distance](https://en.wikipedia.org/wiki/Euclidean_distance),

`euclidean_distance(x, y, [datum = wgs84])`, which works for all combinations

of types for `x` and `y` - except that the UTM zone and hemisphere must also be

provided for `UTM` types, as in `euclidean_distance(utm1, utm2, zone,

hemisphere, [datum = wgs84])` (the Cartesian distance for `UTM` types is not

approximated, but achieved via conversion to `ECEF`).



This is the only function currently in *Geodesy* which takes a default datum,

and *should* be relatively accurate for close points where Euclidean distances

are most important. Future work may focus on geodesics and related calculations

(contributions welcome!).