![Helmert (seven-parameter) transformation](https://upload.wikimedia.org/wikipedia/commons/2/2d/Helmert_%28seven-parameter%29_transformation.svg)
The Helmert transformation (named after
Friedrich Robert Helmert
Friedrich Robert Helmert (31 July 1843 – 15 June 1917) was a German geodesist and statistician with important contributions to the theory of errors.
Career
Helmert was born in Freiberg, Kingdom of Saxony. After schooling in Freiberg and D ...
, 1843–1917) is a
geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often ...
method within a
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
.
It is frequently used in
geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
to produce
datum transformations between
datums.
The Helmert transformation is also called a seven-parameter transformation and is a
similarity transformation.
Definition
It can be expressed as:
:
where
* is the transformed vector
* is the initial vector
The
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s are:
* –
translation vector. Contains the three
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
along the coordinate axes
* –
scale factor
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
, which is unitless; if it is given in
ppm, it must be divided by 1,000,000 and added to 1.
* –
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end ...
. Consists of three axes (small
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s around each of the three coordinate axes) , , . The rotation matrix is an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
. The angles are given in either
degrees or
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s.
Variations
A special case is the two-dimensional Helmert transformation. Here, only four parameters are needed (two translations, one scaling, one rotation). These can be determined from two known points; if more points are available then checks can be made.
Sometimes it is sufficient to use the five parameter transformation, composed of three translations, only one rotation about the Z-axis, and one change of scale.
Restrictions
The Helmert transformation only uses one scale factor, so it is not suitable for:
* The manipulation of measured drawings and
photographs
A photograph (also known as a photo, image, or picture) is an image created by light falling on a photosensitive surface, usually photographic film or an electronic image sensor, such as a CCD or a CMOS chip. Most photographs are now created ...
* The comparison of paper deformations while
scanning old plans and maps.
In these cases, a more general
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
is preferable.
Application
The Helmert transformation is used, among other things, in
geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
to transform the coordinates of the point from one coordinate system into another. Using it, it becomes possible to convert regional
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
points into the
WGS84
The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
locations used by
GPS
The Global Positioning System (GPS), originally Navstar GPS, is a Radionavigation-satellite service, satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of t ...
.
For example, starting with the
Gauss–Krüger coordinate, and , plus the height, , are converted into 3D values in steps:
# Undo the
map projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
: calculation of the ellipsoidal latitude, longitude and height (, , )
# Convert from
geodetic coordinates
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a ''reference ellipsoid''.
They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as geo ...
to
geocentric coordinates
The Earth-centered, Earth-fixed coordinate system (acronym ECEF), also known as the geocentric coordinate system, is a cartesian spatial reference system that represents locations in the vicinity of the Earth (including its surface, interior, ...
: Calculation of , and relative to the
reference ellipsoid
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
of surveying
# 7-parameter transformation (where , and almost always change by a few hundred metres at most, and distances by a few mm per km).
# Because of this, terrestrially measured positions can be compared with GPS data; these can then be brought into the surveying as new points – transformed in the opposite order.
The third step consists of the application of a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end ...
, multiplication with the
scale factor
(with a value near 1) and the addition of the three translations, , , .
The coordinates of a reference system B are derived from reference system A by the following formula (position vector transformation convention and very small rotation angles simplification):
:
or for each single parameter of the coordinate:
:
For the reverse transformation, each element is multiplied by −1.
The seven parameters are determined for each region with three or more "identical points" of both systems. To bring them into agreement, the small inconsistencies (usually only a few cm) are
adjusted using the method of
least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
– that is, eliminated in a statistically plausible manner.
Standard parameters
:''Note: the rotation angles given in the table are in
arcseconds
A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The na ...
and must be converted to
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s before use in the calculation.''
These are standard parameter sets for the 7-parameter transformation (or data transformation) between two datums. For a transformation in the opposite direction, inverse transformation parameters should be calculated or inverse transformation should be applied (as described in paper "On geodetic transformations"
[On geodetic transformations, Bo-Gunnar Reit, 2009 https://www.lantmateriet.se/contentassets/4a728c7e9f0145569edd5eb81fececa7/rapport_reit_eng.pdf]). The translations , , are sometimes described as , , , or , , . The rotations ''r''
''x'', ''r''
''y'', and ''r''
''z'' are sometimes also described as
,
and
. In the United Kingdom the prime interest is the transformation between the OSGB36 datum used by the Ordnance survey for Grid References on its Landranger and Explorer maps to the WGS84 implementation used by GPS technology. The
Gauss–Krüger coordinate system
The transverse Mercator map projection (TM, TMP) is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Merca ...
used in Germany normally refers to the
Bessel ellipsoid
The Bessel ellipsoid (or Bessel 1841) is an important reference ellipsoid of geodesy. It is currently used by several countries for their national geodetic surveys, but will be replaced in the next decades by modern ellipsoids of satellite geod ...
. A further datum of interest was
ED50
ED50 ("European Datum 1950", EPSG:4230) is a geodetic datum which was defined after World War II for the international connection of geodetic networks.
Background
Some of the important battles of World War II were fought on the borders of Germ ...
(European Datum 1950) based on the
Hayford ellipsoid
The Hayford ellipsoid is a geodetic reference ellipsoid, named after the US geodesist John Fillmore Hayford (1868–1925), which was introduced in 1910. The Hayford ellipsoid was also referred to as the International ellipsoid 1924 after it had be ...
. ED50 was part of the fundamentals of the
NATO
The North Atlantic Treaty Organization (NATO, ; french: Organisation du traité de l'Atlantique nord, ), also called the North Atlantic Alliance, is an intergovernmental military alliance between 30 member states – 28 European and two No ...
coordinates up to the 1980s, and many national coordinate systems of Gauss–Krüger are defined by ED50.
The earth does not have a perfect ellipsoidal shape, but is described as a
geoid
The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
. Instead, the geoid of the earth is described by many ellipsoids. Depending upon the actual location, the "locally best aligned ellipsoid" has been used for surveying and mapping purposes. The standard parameter set gives an accuracy of about for an OSGB36/WGS84 transformation. This is not precise enough for surveying, and the Ordnance Survey supplements these results by using a lookup table of further translations in order to reach accuracy.
Estimating the parameters
If the transformation parameters are unknown, they can be calculated with reference points (that is, points whose coordinates are known before and after the transformation. Since a total of seven parameters (three translations, one scale, three rotations) have to be determined, at least two points and one coordinate of a third point (for example, the Z-coordinate) must be known. This gives a system with seven equations and seven unknowns, which can be solved.
For transformations between
conformal map projection
In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere or an ellipsoid) is preserved in the image of the projection, i.e. the projection is a conformal map in the mathema ...
s near an arbitrary point, the Helmert transformation parameters can be calculated exactly from the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
of the transformation function.
In practice, it is best to use more points. Through this correspondence, more accuracy is obtained, and a statistical assessment of the results becomes possible. In this case, the calculation is adjusted with the Gaussian
least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
method.
A numerical value for the accuracy of the transformation parameters is obtained by calculating the values at the reference points, and weighting the results relative to the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
of the points.
While the method is mathematically rigorous, it is entirely dependent on the accuracy of the parameters that are used. In practice, these parameters are computed from the inclusion of at least three known points in the networks. However the accuracy of these will affect the following transformation parameters, as these points will contain observation errors. Therefore, a "real-world" transformation will only be a best estimate and should contain a statistical measure of its quality.
See also
*
Geographic coordinate conversion
In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of convers ...
*
Procrustes analysis
In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name ''Procrustes'' ( el, Προκρούστης) refers to a bandit from Greek mythology who made his victims fi ...
*
Surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
References
External links
Helmert transformin
PROJ
PROJ (formerly PROJ.4) is a library for performing conversions between cartographic projections. The library is based on the work of Gerald Evenden at the United States Geological Survey (USGS), but since 2019-11-26 is an Open Source Geospatial F ...
coordinate transformation software
Computing Helmert Transformations
{{DEFAULTSORT:Helmert Transformation
Geodesy
Transformation (function)