Foreword
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This document was prepared by Technical Committee ISO/TC 211 Geographic information/Geomatics, in close collaboration with the Open Geospatial Consortium (OGC).
This third edition cancels and replaces the second edition (ISO 19111:2007), which has been technically revised. This document also incorporates the provisions of ISO 191112:2009, which is cancelled.
The changes in this edition compared to the previous edition are:
 inclusion of applicable modern geodetic terminology;
 extension to describe dynamic geodetic reference frames;
 extension to describe geoidbased vertical coordinate reference systems;
 extension to allow triaxial ellipsoid for planetary applications;
 extension to describe threedimensional projected coordinate reference systems;
 addition of ‘datum ensembles’ to allow grouping of related realizations of a reference frame where for lower accuracy applications the differences are insignificant;
 clarification in the modelling of derived coordinate reference systems;
 remodelling of the metadata elements scope and extent;
 addition of requirements to describe coordinate metadata and the relationship between spatial coordinates.
 additional modelling of temporal coordinate reference system components sufficient for spatiotemporal coordinate referencing;
 consolidation of the provisions of ISO 191112:2009 (Spatial referencing by coordinates  Extension for parametric values) into this document;
 change in name from ‘Spatial referencing by coordinates’ to ‘Referencing by coordinates’, due to the inclusion of the nonspatial coordinate reference system subtypes of parametric (from 191112) and temporal.
 the correction of minor errors.
Further details are given in Annex G.
In accordance with the ISO/IEC Directives, Part 2, 2018, Rules for the structure and drafting of International Standards, in International Standards the decimal sign is a comma on the line. However the General Conference on Weights and Measures (Conférence Générale des Poids et Mesures) at its meeting in 2003 passed unanimously the following resolution:
“The decimal marker shall be either a point on the line or a comma on the line.”
In practice, the choice between these alternatives depends on customary use in the language concerned. In the technical areas of geodesy and geographic information it is customary for the decimal point always to be used, for all languages. That practice is used throughout this document.
Introduction
Geographic information is inherently fourdimensional and includes time. The spatial component relates the features represented in geographic data to positions in the real world. Spatial references fall into two categories:
 those using coordinates;
 those based on geographic identifiers.
Spatial referencing by geographic identifiers is defined in ISO 19112^{[5]}. This document describes the data elements, relationships and associated metadata required for spatial referencing by coordinates, expanded from a strictly spatial context to include time. The temporal element is restricted to temporal coordinate systems having a continuous axis. The temporal element excludes calendars and ordinal reference systems due to their complexities in definition and in transformation. The context is shown in Figure 1.
Certain scientific communities use threedimensional systems where horizontal position is combined with a nonspatial parameter. In these communities, the parameter is considered to be a third, vertical, axis. The parameter, although varying monotonically with height or depth, does not necessarily vary in a simple manner. Thus conversion from the parameter to height or depth is nontrivial. The parameters concerned are normally absolute measurements and the datum is taken with reference to a direct physical measurement of the parameter. These nonspatial parameters and parametric coordinate reference system modelling constructs were previously described in ISO 191112:2009 but have been incorporated into this revision because the modelling constructs are identical to the other coordinate reference system types included in this document.
This document describes the elements that are necessary to fully define various types of coordinate reference systems applicable to geographic information. The subset of elements required is partially dependent upon the type of coordinates. This document also includes optional fields to allow for the inclusion of metadata about the coordinate reference systems. The elements are intended to be both machine and human readable.
In addition to describing a coordinate reference system, this document provides for the description of a coordinate operation between two different coordinate reference systems or a coordinate operation to account for crustal motion over time. With such information, spatial data referenced to different coordinate reference systems can be referenced to one specified coordinate reference system at one specified time. This facilitates spatial data integration. Alternatively, an audit trail of coordinate manipulations can be maintained.
1 Scope
This document defines the conceptual schema for the description of referencing by coordinates. It describes the minimum data required to define coordinate reference systems. This document supports the definition of:
 spatial coordinate reference systems where coordinate values do not change with time. The system may:
 be geodetic and apply on a national or regional basis, or
 apply locally such as for a building or construction site, or
 apply locally to an image or image sensor;
 be referenced to a moving platform such as a car, a ship, an aircraft or a spacecraft. Such a coordinate reference system may be related to a second coordinate reference system which is referenced to the Earth through a transformation that includes a time element;
 spatial coordinate reference systems in which coordinate values of points on or near the surface of the earth change with time due to tectonic plate motion or other crustal deformation. Such dynamic systems include time evolution, however they remain spatial in nature;
 parametric coordinate reference systems which use a nonspatial parameter that varies monotonically with height or depth;
 temporal coordinate reference systems which use dateTime, temporal count or temporal measure quantities that vary monotonically with time;
 mixed spatial, parametric or temporal coordinate reference systems.
The definition of a coordinate reference system does not change with time, although in some cases some of the defining parameters may include a rate of change of the parameter. The coordinate values within a dynamic and in a temporal coordinate reference system may change with time.
This document also describes the conceptual schema for defining the information required to describe operations that change coordinate values.
In addition to the minimum data required for the definition of the coordinate reference system or coordinate operation, the conceptual schema allows additional descriptive information  coordinate reference system metadata  to be provided.
This document is applicable to producers and users of geographic information. Although it is applicable to digital geographic data, the principles described in this document can be extended to many other forms of spatial data such as maps, charts and text documents.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content constitutes requirements of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 8601, Data elements and interchange formats — Information interchange — Representation of dates and times
ISO 19103, Geographic information — Conceptual schema language
ISO 191151:2014, Geographic information — Metadata — Part 1: Fundamentals
3 Terms, definitions, symbols and abbreviated terms
3.1 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http://www.electropedia.org/
— ISO Online browsing platform: available at https://www.iso.org/obp
3.1.1
affine coordinate system
coordinate system in Euclidean space with straight axes that are not necessarily mutually perpendicular
3.1.2
Cartesian coordinate system
coordinate system in Euclidean space which gives the position of points relative to n mutually perpendicular straight axes all having the same unit of measure
Note 1 to entry: n is 2 or 3 for the purposes of this document.
Note 2 to entry: A Cartesian coordinate system is a specialisation of an affine coordinate system.
3.1.3
compound coordinate reference system
coordinate reference system using at least two independent coordinate reference systems
Note 1 to entry: Coordinate reference systems are independent of each other if coordinate values in one cannot be converted or transformed into coordinate values in the other.
3.1.4
concatenated operation
coordinate operation consisting of the sequential application of multiple coordinate operations
3.1.5
coordinate
one of a sequence of numbers designating the position of a point
Note 1 to entry: In a spatial coordinate reference system, the coordinate numbers are qualified by units.
3.1.6
coordinate conversion
coordinate operation that changes coordinates in a source coordinate reference system to coordinates in a target coordinate reference system in which both coordinate reference systems are based on the same datum
Note 1 to entry: A coordinate conversion uses parameters which have specified values.
EXAMPLE 1 A mapping of ellipsoidal coordinates to Cartesian coordinates using a map projection.
EXAMPLE 2 Change of units such as from radians to degrees or from feet to metres.
3.1.7
coordinate epoch
epoch to which coordinates in a dynamic coordinate reference system are referenced
3.1.8
coordinate operation
process using a mathematical model, based on a onetoone relationship, that changes coordinates in a source coordinate reference system to coordinates in a target coordinate reference system, or that changes coordinates at a source coordinate epoch to coordinates at a target coordinate epoch within the same coordinate reference system
Note 1 to entry: Generalization of coordinate conversion, coordinate transformation and point motion operation.
3.1.9
coordinate reference system
coordinate system that is related to an object by a datum
Note 1 to entry: Geodetic and vertical datums are referred to as reference frames.
Note 2 to entry: For geodetic and vertical reference frames, the object will be the Earth. In planetary applications, geodetic and vertical reference frames may be applied to other celestial bodies.
3.1.10
coordinate set
collection of coordinate tuples referenced to the same coordinate reference system and if that coordinate reference system is dynamic also to the same coordinate epoch
3.1.11
coordinate system
set of mathematical rules for specifying how coordinates are to be assigned to points
3.1.12
coordinate transformation
coordinate operation that changes coordinates in a source coordinate reference system to coordinates in a target coordinate reference system in which the source and target coordinate reference systems are based on different datums
Note 1 to entry: A coordinate transformation uses parameters which are derived empirically. Any error in those coordinates will be embedded in the coordinate transformation and when the coordinate transformation is applied the embedded errors are transmitted to output coordinates.
Note 2 to entry: A coordinate transformation is colloquially sometimes referred to as a ‘datum transformation’. This is erroneous. A coordinate transformation changes coordinate values. It does not change the definition of the datum. In this document coordinates are referenced to a coordinate reference system. A coordinate transformation operates between two coordinate reference systems, not between two datums.
3.1.13
coordinate tuple
tuple composed of coordinates
Note 1 to entry: The number of coordinates in the coordinate tuple equals the dimension of the coordinate system; the order of coordinates in the coordinate tuple is identical to the order of the axes of the coordinate system.
3.1.14
cylindrical coordinate system
threedimensional coordinate system in Euclidean space in which position is specified by two linear coordinates and one angular coordinate
3.1.15
datum
reference frame
parameter or set of parameters that realize the position of the origin, the scale, and the orientation of a coordinate system
3.1.16
datum ensemble
group of multiple realizations of the same terrestrial or vertical reference system that, for approximate spatial referencing purposes, are not significantly different
Note 1 to entry: Datasets referenced to the different realizations within a datum ensemble may be merged without coordinate transformation.
Note 2 to entry: ‘Approximate’ is for users to define but typically is in the order of under 1 decimetre but may be up to 2 metres.
EXAMPLE “WGS 84” as an undifferentiated group of realizations including WGS 84 (TRANSIT), WGS 84 (G730), WGS 84 (G873), WGS 84 (G1150), WGS 84 (G1674) and WGS 84 (G1762). At the surface of the Earth these have changed on average by 0.7m between the TRANSIT and G730 realizations, a further 0.2m between G730 and G873, 0.06m between G873 and G1150, 0.2m between G1150 and G1674 and 0.02m between G1674 and G1762).
3.1.17
depth
distance of a point from a chosen vertical reference surface downward along a line that is perpendicular to that surface
Note 1 to entry: The line direction may be straight, or be dependent on the Earth’s gravity field or other physical phenomena.
Note 2 to entry: A depth above the vertical reference surface will have a negative value.
3.1.18
derived coordinate reference system
coordinate reference system that is defined through the application of a specified coordinate conversion to the coordinates within a previously established coordinate reference system
Note 1 to entry: The previously established coordinate reference system is referred to as the base coordinate reference system.
Note 2 to entry: A derived coordinate reference system inherits its datum or reference frame from its base coordinate reference system.
Note 3 to entry: The coordinate conversion between the base and derived coordinate reference system is implemented using the parameters and formula(s) specified in the definition of the coordinate conversion.
3.1.19
dynamic coordinate reference system
coordinate reference system that has a dynamic reference frame
Note 1 to entry: Coordinates of points on or near the crust of the Earth that are referenced to a dynamic coordinate reference system may change with time, usually due to crustal deformations such as tectonic motion and glacial isostatic adjustment.
Note 2 to entry: Metadata for a dataset referenced to a dynamic coordinate reference system should include coordinate epoch information.
3.1.20
dynamic reference frame
dynamic datum
reference frame in which the defining parameters include time evolution
Note 1 to entry: The defining parameters that have time evolution are usually a coordinate set.
3.1.21
easting
E
distance in a coordinate system, eastwards (positive) or westwards (negative) from a northsouth reference line
3.1.22
ellipsoid
reference ellipsoid
<geodesy> geometric reference surface embedded in 3D Euclidean space formed by an ellipse that is rotated about a main axis
Note 1 to entry: For the Earth the ellipsoid is biaxial with rotation about the polar axis. This results in an oblate ellipsoid with the midpoint of the foci located at the nominal centre of the Earth.
3.1.23
ellipsoidal coordinate system
geodetic coordinate system
coordinate system in which position is specified by geodetic latitude, geodetic longitude and (in the threedimensional case) ellipsoidal height
3.1.24
ellipsoidal height
geodetic height
h
distance of a point from the reference ellipsoid along the perpendicular from the reference ellipsoid to this point, positive if upwards or outside of the reference ellipsoid
Note 1 to entry: Only used as part of a threedimensional ellipsoidal coordinate system or as part of a threedimensional Cartesian coordinate system in a threedimensional projected coordinate reference system, but never on its own.
3.1.25
engineering coordinate reference system
coordinate reference system based on an engineering datum
EXAMPLE 1 System for identifying relative positions within a few kilometres of the reference point, such as a building or construction site.
EXAMPLE 2 Coordinate reference system local to a moving object such as a ship or an orbiting spacecraft.
EXAMPLE 3 Internal coordinate reference system for an image. This has continuous axes. It may be the foundation for a grid.
3.1.26
engineering datum
local datum
datum describing the relationship of a coordinate system to a local reference
Note 1 to entry: Engineering datum excludes both geodetic and vertical reference frames.
3.1.27
epoch
<geodesy> point in time
Note 1 to entry: In this document an epoch is expressed in the Gregorian calendar as a decimal year.
EXAMPLE 20170325 in the Gregorian calendar is epoch 2017.23.
3.1.28
flattening
f
ratio of the difference between the semimajor axis (a) and semiminor axis (b) of an ellipsoid to the semimajor axis: f=(a – b)/a
Note 1 to entry: Sometimes inverse flattening 1/f = a/(a  b) is given instead; 1/f is also known as reciprocal flattening.
3.1.29
frame reference epoch
epoch of coordinates that define a dynamic reference frame
3.1.30
geocentric latitude
angle from the equatorial plane to the direction from the centre of an ellipsoid through a given point, northwards treated as positive
3.1.31
geodetic coordinate reference system
threedimensional coordinate reference system based on a geodetic reference frame and having either a threedimensional Cartesian or a spherical coordinate system
Note 1 to entry: In this document a coordinate reference system based on a geodetic reference frame and having an ellipsoidal coordinate system is geographic.
3.1.32
geodetic latitude
ellipsoidal latitude
j
angle from the equatorial plane to the perpendicular to the ellipsoid through a given point, northwards treated as positive
3.1.33
geodetic longitude
ellipsoidal longitude
l
angle from the prime meridian plane to the meridian plane of a given point, eastward treated as positive
3.1.34
geodetic reference frame
reference frame or datum describing the relationship of a two or threedimensional coordinate system to the Earth
Note 1 to entry: In the data model described in this document, the UML class GeodeticReferenceFrame includes both modern terrestrial reference frames and classical geodetic datums.
3.1.35
geographic coordinate reference system
coordinate reference system that has a geodetic reference frame and an ellipsoidal coordinate system
3.1.36
geoid
equipotential surface of the Earth’s gravity field which is perpendicular to the direction of gravity and which best fits mean sea level either locally, regionally or globally
3.1.37
gravityrelated height
H
height that is dependent on the Earth’s gravity field
Note 1 to entry: This refers to, amongst others, orthometric height and Normal height, which are both approximations of the distance of a point above the mean sea level, but also may include Normalorthometric heights, dynamic heights or geopotential numbers.
Note 2 to entry: The distance from the reference surface may follow a curved line, not necessarily straight, as it is influenced by the direction of gravity.
3.1.38
height
distance of a point from a chosen reference surface positive upward along a line perpendicular to that surface
Note 1 to entry: A height below the reference surface will have a negative value.
Note 2 to entry: Generalisation of ellipsoidal height (h) and gravityrelated height (H).
3.1.39
linear coordinate system
onedimensional coordinate system in which a linear feature forms the axis
EXAMPLE 1 Distances along a pipeline.
EXAMPLE 2 Depths down a deviated oil well bore.
3.1.40
map projection
coordinate conversion from an ellipsoidal coordinate system to a plane
3.1.41
mean sea level
MSL
<geodesy> average level of the surface of the sea over all stages of tide and seasonal variations
Note 1 to entry: Mean sea level in a local context normally means mean sea level for the region calculated from observations at one or more points over a given period of time. To meet IHO standards that period should be one full lunar cycle of 19 years. Mean sea level in a global context differs from a global geoid by not more than 2 m.
3.1.42
meridian
intersection of an ellipsoid by a plane containing the shortest axis of the ellipsoid
Note 1 to entry: This term is generally used to describe the poletopole arc rather than the complete closed figure.
3.1.43
northing
N
distance in a coordinate system, northwards (positive) or southwards (negative) from an eastwest reference line
3.1.44
parameter reference epoch
epoch at which the parameter values of a timedependent coordinate transformation are valid
Note 1 to entry: The transformation parameter values first need to be propagated to the epoch of the coordinates before the coordinate transformation can be applied.
3.1.45
parametric coordinate reference system
coordinate reference system based on a parametric datum
3.1.46
parametric coordinate system
onedimensional coordinate system where the axis units are parameter values which are not inherently spatial
3.1.47
parametric datum
datum describing the relationship of a parametric coordinate system to an object
Note 1 to entry: The object is normally the Earth.
3.1.48
point motion operation
coordinate operation that changes coordinates within one coordinate reference system due to the motion of the point
Note 1 to entry: The change of coordinates is from those at an initial epoch to those at another epoch.
Note 2 to entry: In this document the point motion is due to tectonic motion or crustal deformation.
3.1.49
polar coordinate system
twodimensional coordinate system in Euclidean space in which position is specified by one distance coordinate and one angular coordinate
Note 1 to entry: For the threedimensional case, see spherical coordinate system.
3.1.50
prime meridian
meridian from which the longitudes of other meridians are quantified
3.1.51
projected coordinate reference system
coordinate reference system derived from a geographic coordinate reference system by applying a map projection
Note 1 to entry: May be two or threedimensional, the dimension being equal to that of the geographic coordinate reference system from which it is derived.
Note 2 to entry: In the threedimensional case the horizontal coordinates (geodetic latitude and geodetic longitude coordinates) are projected to northing and easting and the ellipsoidal height is unchanged.
3.1.52
reference frame
datum
parameter or set of parameters that realize the position of the origin, the scale, and the orientation of a coordinate system
3.1.53
semimajor axis
a
semidiameter of the longest axis of an ellipsoid
3.1.54
semiminor axis
b
semidiameter of the shortest axis of an ellipsoid
3.1.55
sequence
finite, ordered collection of related items (objects or values) that may be repeated
3.1.56
spatial reference
description of position in the real world
Note 1 to entry: This may take the form of a label, code or coordinate tuple.
3.1.57
spatioparametric coordinate reference system
compound coordinate reference system in which one constituent coordinate reference system is a spatial coordinate reference system and one is a parametric coordinate reference system
Note 1 to entry: Normally the spatial component is “horizontal” and the parametric component is “vertical”.
3.1.58
spatioparametrictemporal coordinate reference system
compound coordinate reference system comprised of spatial, parametric and temporal coordinate reference systems
3.1.59
spatiotemporal coordinate reference system
compound coordinate reference system in which one constituent coordinate reference system is a spatial coordinate reference system and one is a temporal coordinate reference system
3.1.60
spherical coordinate system
threedimensional coordinate system in Euclidean space in which position is specified by one distance coordinate and two angular coordinates
Note 1 to entry: Not to be confused with an ellipsoidal coordinate system based on an ellipsoid ‘degenerated’ into a sphere.
3.1.61
static coordinate reference system
coordinate reference system that has a static reference frame
Note 1 to entry: Coordinates of points on or near the crust of the Earth that are referenced to a static coordinate reference system do not change with time.
Note 2 to entry: Metadata for a dataset referenced to a static coordinate reference system does not require coordinate epoch information.
3.1.62
static reference frame
static datum
reference frame in which the defining parameters exclude time evolution
3.1.63
temporal coordinate reference system
coordinate reference system based on a temporal datum
3.1.64
temporal coordinate system
<geodesy> onedimensionalcoordinate system where the axis is time
3.1.65
temporal datum
datum describing the relationship of a temporal coordinate system to an object
Note 1 to entry: The object is normally time on the Earth.
3.1.66
terrestrial reference system
TRS
set of conventions defining the origin, scale, orientation and time evolution of a spatial reference system corotating with the Earth in its diurnal motion in space
Note 1 to entry: The abstract concept of a TRS is realised through a terrestrial reference frame that usually consists of a set of physical points with precisely determined coordinates and optionally their rates of change. In this document terrestrial reference frame is included within the geodetic reference frame element of the data model.
3.1.67
transformation reference epoch
epoch at which the parameter values of a timespecific coordinate transformation are valid
Note 1 to entry: Coordinates first need to be propagated to this epoch before the coordinate transformation is applied. This is in contrast to a parameter reference epoch where the transformation parameter values first need to be propagated to the epoch of the coordinates before the coordinate transformation is applied.
3.1.68
tuple
ordered list of values
[SOURCE: ISO 19136:2007, 4.1.63]
3.1.69
unit
defined quantity in which dimensioned parameters are expressed
Note 1 to entry: In this document, the subtypes of units are length units, angular units, scale units, parametric quantities and time quantities.
3.1.70
vertical coordinate reference system
onedimensional coordinate reference system based on a vertical reference frame
3.1.71
vertical coordinate system
onedimensional coordinate system used for gravityrelated height or depth measurements
3.1.72
vertical reference frame
vertical datum
reference frame describing the relation of gravityrelated heights or depths to the Earth
Note 1 to entry: In most cases, the vertical reference frame will be related to mean sea level. Vertical datums include sounding datums (used for hydrographic purposes), in which case the heights may be negative heights or depths.
Note 2 to entry: Ellipsoidal heights are related to a threedimensional ellipsoidal coordinate system referenced to a geodetic reference frame.
3.1.73
vertical reference system
VRS
set of conventions defining the origin, scale, orientation and time evolution that describes the relationship of gravityrelated heights or depths to the Earth
Note 1 to entry: The abstract concept of a VRS is realised through a vertical reference frame.
3.2 Symbols
a semimajor axis of ellipsoid
b semiminor axis of biaxial ellipsoid
E easting
f flattening
H gravityrelated height
h ellipsoidal height
N northing
l geodetic longitude
j geodetic latitude
E, N, [h] Cartesian coordinates in a projected coordinate reference system
X, Y, Z Cartesian coordinates in a geodetic coordinate reference system
i, j, [k] Cartesian coordinates in an engineering coordinate reference system
r, q polar coordinates in a 2D engineering coordinate reference system
r, W, q spherical coordinates in a 3D engineering coordinate reference system
Note: In this document W is the polar (zenith) angle and q is the azimuthal angle.
j,l, [h] ellipsoidal coordinates in a geographic coordinate reference system
3.3 Abbreviated terms
CC coordinate conversion
CCRS compound coordinate reference system
CRS coordinate reference system
CT coordinate transformation
MSL mean sea level
pixel a contraction of “picture element”, the smallest element of a digital image to which attributes are assigned
PMO point motion operation
SI le Système International d’unités (International System of Units)
UML Unified Modeling Language
URI Uniform Resource Identifier
1D onedimensional
2D twodimensional
3D threedimensional
4 Conformance requirements
This document defines
— two classes of conformance for relating coordinates to coordinate metadata; and
— twenty six classes of conformance for the definition of a coordinate reference system (CRS) or of a coordinate operation.
These are differentiated by type, as shown in Table 1. Implementations should indicate which conformance classes they comply with. Any implementations claiming conformance shall satisfy the requirements in Annex A.
Conformance class  Description  Conformance requirements given in 

Conformance for relating coordinates to coordinate metadata 
A.2 

1 2 
CRS with static reference frame CRS with dynamic reference frame 

Conformance of a CRS definition 
A.3 

3 4 5 
Geodetic CRS with static reference frame with dynamic reference frame derived geodetic CRS 

6 7 8 
Geographic CRS with static reference frame with dynamic reference frame derived geographic CRS 

9 10 
Projected CRS derived projected CRS 

11 12 13 
Vertical CRS with static reference frame with dynamic reference frame derived vertical CRS 

14 15 
Parametric CRS derived parametric CRS 

16 17 
Engineering CRS derived engineering CRS 

18 19 20 21 
Temporal CRS dateTime temporal count temporal measure derived temporal CRS 

22 
CRS with datum ensemble 

23 
Compound CRS 
A.3 
Conformance of a coordinate operation definition 
A.4 

24 25 26 27 28 
Coordinate conversion Coordinate transformation Point motion operation Concatenated operation Passthrough operation 

The requirements classes for the definition of a coordinate reference system or a coordinate operation are described in this document through tables grouped by UML package. The requirements are then brought together in the conformance classes in Annex A. This retains the packagebased layout for describing requirements used in previous versions of this document.
5 Conventions
5.1 Unified Modeling Language notation
In this document, the conceptual schema for describing coordinate reference systems and coordinate operations are presented in the Unified Modeling Language (UML). ISO 19103 Conceptual schema languagepresents the specific profile of UML used in this document.
In the UML diagrams in this document, a grey background surround to boxes indicates classes from other standards.
5.2 Attribute and association status
In this document the conceptual schema is described in Clauses 6 to 12 through tables. In these tables:
· attributes and associations are given an obligation status:
Obligation  Definition  Meaning 

M 
mandatory 
This attribute shall be supplied. 
C 
conditional 
This attribute shall be supplied if the condition (given in the attribute description) is true. It may be supplied if the condition is false. 
O 
optional 
This attribute may be supplied. 
The Maximum Occurrence column in the tables indicates the maximum number of occurrences of attribute values that are permissible, with N indicating no upper limit.
· nonnavigable associations are not included in the UML diagrams or tables.
In the event of any discrepancies between the UML diagrams and text, the UML shall prevail.
6 Referencing by coordinates  Data model overview
The specification for referencing by coordinates is described in this document in the form of a UML model with supplementary text. The UML model contains six UML packages, as shown in Figure 2. Each box represents a package, and contains the package name. Each arrowed line shows the dependency of one package upon another package (at the head of the arrow).
Coordinates require metadata that fully specifies the coordinate reference system to which they are referenced; without this CRS reference the description of position is ambiguous. The UML package for coordinates and their metadata is described in Clause 7. This includes aspects of coordinate operations required to change coordinate values when the coordinate reference system is changed.
A coordinate reference system is usually comprised of two components, one coordinate system and one datum. In modern geodetic terminology the datum is referred to as a reference frame. Some geodetic concepts underpinning spatial referencing by coordinates are given in Annex B. The information required to fully specify a coordinate reference system is described in Clauses 9 to 11, with attributes common to all three packages described in Clause 8.
Some coordinate reference systems have a third component, a defining coordinate conversion from another preexisting CRS. In this document a CRS having this third component is a derived CRS. The specification for describing coordinate operations, including a defining coordinate conversion, is described in Clause 12.
Further context for the requirements of Clauses 8 to 12 is given in Annexes C and D. Examples illustrating how the specifications of this document can be applied when defining a coordinate reference system or a coordinate operation are given in Annex E. Recommendations for referencing to classes defined in this document are given in Annex F. Changes between this document and the previous version ISO 19111:2007 are described in Annex G.
7 Coordinates package
7.1 Relationship between coordinates and coordinate reference system
In this document, a coordinate is one of n scalar values that define the position of a single point. In other contexts, the term ordinate is used for a single value and coordinate for multiple ordinates. Such usage is not part of this document.
A coordinate tuple is an ordered list of coordinates that define the position of a single point. The coordinates within a coordinate tuple are mutually independent. The number of coordinates in a tuple is equal to the dimension of the coordinate space.
A coordinate set is a collection of coordinate tuples referenced to the same coordinate reference system. For a coordinate set, one CRS identification or definition may be associated with the coordinate set and then all coordinate tuples in that coordinate set inherit that association. If only one point is being described, the association between coordinate tuple and coordinate reference system is direct.
The concepts of dynamic and static coordinate reference systems are outlined in B.3. If the coordinate reference system is dynamic, operations on the geometry of the tuples within the coordinate set are valid only if all tuples are referenced to the same coordinate epoch. In this document all coordinate tuples in a spatial coordinate set are referenced to one specified coordinate epoch.
Together the coordinate reference system and the coordinate epoch are the coordinate metadata.
Coordinate sets referenced to one CRS may be referenced to another CRS through the application of a coordinate operation. A coordinate operation operates on coordinates, not on coordinate reference systems. A coordinate operation may be single or concatenated: refer to Clause 12. The high level conceptual model for changing coordinates is shown in Figure 3.
Coordinate sets referenced to a dynamic CRS at a given coordinate epoch t_{1} may be converted to another coordinate epoch t_{2} through a point motion coordinate operation that includes time evolution, often described using velocities, as shown schematically in Figure 4.
It is also possible to change coordinates from being referenced to one dynamic CRS at one coordinate epoch to being referenced to another dynamic CRS at another coordinate epoch, or to change coordinates between a dynamic CRS and a static CRS or viceversa. Further information is in C.1 and C.5.
The description of quality of coordinates is covered by the provisions of ISO 19157^{[8]}.
7.2 Coordinate reference system identification
The elements required for the definition of coordinate reference systems and coordinate operations are described in Clauses 8 to 12.
CRS or coordinate operation identification may be through:
a) a full description, as defined in this document; or
b) reference to a full description in a register of geodetic parameters (the reference is made to the register and to the identifier of the object description within that register); or
c) both a full description and a reference to a full description in a register. If there is a conflict between the two, the object full description should prevail over the reference to a register.
a) and b) are alternative means of providing a full description. b) is recommended for simplicity, but if it is not available from a register the description is required to be given explicitly and in full. In both methods, the order of coordinates in each coordinate tuple is required to be as given in the coordinate reference system’s coordinate system description.
When using method b), reference to a register, applications that are required only to confirm the identification of a CRS or coordinate operation can do so through the register citation and the identifier from that register. They do not need to retrieve the elements that constitute the full description from the register unless there is a need to quote these or to perform a coordinate operation on the coordinate set.
7.3 Requirements for coordinate metadata
7.3.1 Requirements class: static CRS coordinate metadata
Requirement 1: All coordinate tuples in a coordinate set shall be referenced to the same coordinate reference system.
7.3.2 Requirements class: dynamic CRS coordinate metadata
CRS is described in Clause 9 and datum or reference frame in Clause 11. The following subtypes of CRS may have a dynamic reference frame and therefore may be dynamic CRSs: geodetic, geographic, vertical, projected and derived variants of these subtypes. Implementers are warned that CRSs of these subtypes are not necessarily dynamic; their reference frame attributes need to be examined to clarify this.
Requirement 2: When the coordinate reference system to which a coordinate set is referenced is dynamic, all coordinate tuples in the coordinate set shall be referenced to the same coordinate epoch.
7.4 UML schema for the Coordinates package
Figure 5 shows the UML class diagram for coordinate metadata. The definition of the classes in the package are provided in Tables 2 to 4.