LogoLogo
OS Docs HomeOS NGDOS APIsOS Download ProductsMore than MapsContact Us
  • More than Maps
  • Geographic Data Visualisation
    • Guide to cartography
      • Introduction to cartography
      • Types of maps
      • Symbology
      • Colour
      • Text on maps
      • Generalisation
      • Coordinate reference systems
      • Projections
      • Scale
      • Map legends
      • Map layout
      • Relief representation
      • North arrows
    • Guide to data visualisation
      • Introduction to data visualisation
      • GeoDataViz design principles
      • Types of visualisation
      • Thematic mapping techniques
      • Data visualisation critique
      • Accessible data visualisation
      • Ethical data visualisation
      • Software
      • Data
    • GeoDataViz assets
      • GeoDataViz basemaps
      • Stylesheets
      • GeoDataViz virtual gallery
      • Equal area cartograms
      • How did I make that?
        • Apollo 11 Landing
        • North York Moors National Park, 70 years
        • Snowdonia National Park, 70 years
        • Great Britain's National Parks
        • Great Britain's Islands
        • Great Britain's AONB's and National Scenic Areas
        • Famous shipwrecks of Pembrokeshire
        • Trig pillars today
        • Britain's most complex motorway junctions
      • #30DayMapChallenge
  • Data in Action
    • Examples
  • Demonstrators
    • 🆕Product Viewer
    • Addressing & location demonstrators
      • Address Portfolio overview
      • Which address product should you use?
      • AddressBase
      • AddressBase Core
      • AddressBase Plus
      • AddressBase Premium
      • Address Classifications
      • Addressing Lifecycle
      • OS Emergency Services Gazetteer
      • What are Vertical Streets?
      • Why are there differences in boundaries?
    • Contextual demonstrators
    • Customer best practice
      • Channel Shift
      • Data Management and OS Data Hub
      • End User Licence vs Contractor Licence
      • 🆕 IDs vs Spatial Relationships
      • Why we should capture good quality addresses at source
      • Why we Snap and Trace
    • Network Demonstrators
      • OS Detailed Path Network
      • OS Multi Modal Routing Network
        • OS Multi Modal Routing Network
      • Water Networks overview
      • OS MasterMap Highways Network and OS NGD Speeds
      • OS MasterMap® Highways Network and OS Open Roads™
    • OS MasterMap Generation APIs
      • Using the OS Features API
      • Using the OS Features API Archive
      • Using the OS Downloads API
      • Using OS APIs in ESRI Software
    • 🆕OS NGD (National Geographic Database)
      • OS NGD Address
      • OS NGD Boundaries
      • 🆕OS NGD Buildings
        • 🆕Building and Building Access Feature Types
        • Building Part and Building Line Feature Types
      • 🆕OS NGD Geographical Names
      • OS NGD Land
      • OS NGD Land Cover enhancements
      • 🆕OS NGD Land Use
      • OS NGD Land Use enhancements
      • 🆕OS NGD Structures
        • 🆕OS NGD Structures
        • Field Boundaries
      • 🆕OS NGD Transport Features
      • 🆕OS NGD Transport Network
      • OS NGD Transport RAMI
      • OS NGD Water Features
      • OS NGD Water Network
      • OS NGD API - Features
      • Ordering OS NGD data
      • Change only updates
      • OS NGD Versioning
      • Creating a topographic map from OS NGD Data
      • Analytical styling for OS NGD data
    • OS MasterMap® demonstrators
    • 🆕Product & API Comparisons
      • 🆕Comparison of Water Network Products
  • Tutorials
    • GeoDataViz
      • Thematic Mapping Techniques
      • Downloading and using data from the OS Data Hub
      • How to download and use OS stylesheets
      • How to use the OS Maps API
      • Creating a bespoke style in Maputnik
    • GIS
      • Analysing pavement widths
      • Basic routing with OS Open Data and QGIS
      • Walktime analysis using OS Multi-modal Routing Network and QGIS
      • Creating 3D Symbols for GIS Applications
      • Constructing a Single Line Address using a Geographic Address
      • Creating a Digital Terrain Model (DTM)
      • Visualising a road gradient using a Digital Terrain Model
      • Visualising a road gradient using OSMM Highways
    • 🆕APIs
      • 🆕Using OS APIs with EPC API
      • 🆕OS APIs and ArcGIS
  • Deep Dive
    • Introduction to address matching
    • Guide to routing for the Public Sector
      • Part 1: Guide to routing
      • Part 2: Routing software and data options
      • Part 3: Building a routable network
    • Unlocking the Power of Geospatial Data
    • Using Blender for Geospatial Projects
    • A Guide to Coordinate Systems in Great Britain
      • Myths about coordinate systems
      • The shape of the Earth
      • What is position?
        • Types of coordinates
        • We need a datum
        • Position summary
      • Modern GNSS coordinate systems
        • Realising WGS84 with a TRF
        • The WGS84 broadcast TRF
        • The International Terrestrial Reference Frame (ITRF)
        • The International GNSS Service (IGS)
        • European Terrestrial Reference System 1989 (ETRS89)
      • Ordnance Survey coordinate systems
        • ETRS89 realised through OS Net
        • National Grid and the OSGB36 TRF
        • Ordnance Datum Newlyn
        • The future of British mapping coordinate systems
        • The future of British mapping coordinate systems
      • From one coordinate system to another: geodetic transformations
        • What is a geodetic transformation?
        • Helmert datum transformations
        • National Grid Transformation OSTN15 (ETRS89–OSGB36)
        • National Geoid Model OSGM15 (ETRS89-Orthometric height)
        • ETRS89 to and from ITRS
        • Approximate WGS84 to OSGB36/ODN transformation
        • Transformation between OS Net v2001 and v2009 realisations
      • Transverse Mercator map projections
        • The National Grid reference convention
      • Datum, ellipsoid and projection information
      • Converting between 3D Cartesian and ellipsoidal latitude, longitude and height coordinates
      • Converting between grid eastings and northings and ellipsoidal latitude and longitude
      • Helmert transformation worked example
      • Further information
  • Code
    • Ordnance Survey APIs
    • Mapping
    • Routing with pgRouting
      • Getting started with OS MasterMap Highways and pgRouting
      • Getting started with OS MasterMap Highways Network - Paths and pgRouting
      • Getting started with OS NGD Transport Theme and pgRouting
      • Getting started with OS NGD Transport Path features and pgRouting
  • RESOURCES
    • 🆕Data Visualisation External Resources
Powered by GitBook

Website

  • Ordnance Survey

Data

  • OS Data Hub
On this page

Was this helpful?

  1. Deep Dive
  2. A Guide to Coordinate Systems in Great Britain
  3. From one coordinate system to another: geodetic transformations

What is a geodetic transformation?

PreviousFrom one coordinate system to another: geodetic transformationsNextHelmert datum transformations

Last updated 4 months ago

Was this helpful?

A geodetic transformation is a mathematical operation that takes the coordinates of a point in one coordinate system and returns the coordinates of the same point in a second coordinate system. Hopefully we will also have an ‘inverse transformation’ to get back to the first coordinate system from given coordinates in the second. Many types of mathematical operation are used to accomplish this task. The term ‘geodetic transformation’ does not indicate a particular type of formula.

To properly understand geodetic transformations, it is essential to understand the relationship between a datum and the TRF that realises the datum. Having looked at several examples of real TRFs in and , we know that they are of widely variable quality. Unless a TRF is perfect, the datum it realises is ambiguous. We can think about this in the following way: the given three-dimensional coordinates of any three ground points in the TRF imply the location of the coordinate origin, and the orientation of the coordinate axes, relative to those three points. If we select a different set of three ground points, the origin and orientation implied will be slightly different. This is due to observational errors in the determination of the coordinates of the ground points. Taking the TRF as a whole, the datum that it implies has some degree of uncertainty, because the origin positions implied by different selections of TRF ground points are never in perfect agreement with each other.

The usual situation in geodesy is that we have two TRFs realising two different datums, with a large number of points in common between the two TRFs. That is, a particular ground point will be included in both TRFs, but it will have different coordinates in each. Because the TRFs are not perfect realisations of their respective datums, it is impossible to say exactly what the transformation between the two datums is. We can only give a best estimate of the transformation with a statistical measure of the quality of the estimate.

To clarify this idea, let’s look at an example. To keep things simple, we’ll consider one‑dimensional coordinate systems (The is an instance of a real one-dimensional coordinate system). Imagine we have two coordinate systems: A and B, each of which state coordinates for three points called White, Grey and Black. So here we have 100% overlap between the TRFs of the two coordinate systems – all three points appear in both TRFs.

Point
Coordinate system A
Coordinate system B

White

4.1

7.3

Grey

9.8

12.7

Black

12.2

15.0

We want to work out a transformation from the datum of coordinate system A to the datum of coordinate system B. This transformation will be a single number t that can be added to any coordinate in system A to give the equivalent coordinate in system B. We don’t have any theoretical datum definitions of these two systems to work from, and even if we did we would not use them (in this example, we don’t even know what the coordinates represent – it doesn’t matter). We want a practical transformation that will convert any coordinate in coordinate system A to coordinate system B. So, we use points that have known coordinates in both systems – in other words, we relate the two datums by using their TRFs.

In this example, we have three points that are coordinated in both TRFs – White, Grey and Black. The best estimate of the transformation t between the frames is simply the average of the transformation at each of these points:

However, this ‘best estimate’ value for t doesn’t actually agree with the translation at any of the three known points. We need to compute a statistic that represents the ‘amount of disagreement’ with this estimate among the known points. This is the standard deviation of the three individual t values from the average t value:

So, our estimate of t would be written t = 2.97 ±0.17, that is, t could quite probably be anything from 2.80 to 3.14 based on the information we have available. It is common to state a range of three standard deviations as a likely range within which the true figure lies, that is, between 2.46 and 3.48.

So, what is the transformation between datum A and datum B? We don’t know exactly, and we can never know exactly. But we have a best estimate and a measure of the likely error in the estimate.

The point of all this is that since TRFs aren’t perfect, neither are transformations between the datums they realise. In practice, this means that no exact transformation exists between two geodetic coordinate systems. People often assume, quite reasonably, that the transformation between two ellipsoidal datums can be exactly specified, in the way that a map projection can. In theory, this is true, but what use would it be? What you really want to know is what the transformation is for real points on the ground – so you are actually asking a question about TRFs. And the answer to such a question is always an approximation.

The degree of error in a geodetic transformation will depend on the patterns of errors present in the TRFs A and B (which are often characteristic of the methods used to establish the TRF in the first place), and also on how carefully the transformation has been designed to take account of those errors. For example, TRFs established by triangulation generally contain significant errors in the overall scale of the network and often, this scale error varies in different parts of the network. Therefore, a real transformation is likely to represent not only the difference between geodetic datums, but also the difference between the TRFs that realise those datums due to errors in the original observations.

twhite=whiteB−whiteA=3.2tgrey=greyB−greyA=2.9tblack=blackB−blackA=2.8so,t=twhite+tgrey+tblack3=2.97t_{white}=white_B-white_A=3.2\\ t_{grey}=grey_B-grey_A=2.9\\ t_{black}=black_B-black_A=2.8\\ so, t = \frac{t_{white}+t_{grey}+t_{black}}{3} = 2.97twhite​=whiteB​−whiteA​=3.2tgrey​=greyB​−greyA​=2.9tblack​=blackB​−blackA​=2.8so,t=3twhite​+tgrey​+tblack​​=2.97
σt=(twhite−t)2+(tgrey−t)2+(tblack−t)23=0.17σ_t= \sqrt{\frac{(t_{white}-t)^2+(t_{grey}-t)^2+(t_{black}-t)^2}{3}} = 0.17σt​=3(twhite​−t)2+(tgrey​−t)2+(tblack​−t)2​​=0.17
Modern GNSS coordinate system
Ordnance Survey coordinate systems
Ordnance Datum Newlyn page
An example of two one-dimensional coordinate systems with three points appearing on both TRFs
An example of two one-dimensional coordinate systems with three points (white, grey, and black) appearing on both TRFs