ISO 19111:2003

INTERNATIONAL ISO
STANDARD 19111
First edition
2003-02-15
Geographic information — Spatial
referencing by coordinates
Information géographique — Système de références spatiales par
coordonnées
Reference number
©
ISO 2003
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ii © ISO 2003 — All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Conformance requirements . 1
3 Normative references . 1
4 Terms and definitions. 1
5 Conventions . 5
5.1 Symbols and abbreviated terms. 5
5.2 UML notation . 6
6 Definition of the conceptual schema for coordinate reference systems . 7
6.1 Introduction . 7
6.2 Coordinate reference system. 7
6.2.1 Type of coordinate reference system . 7
6.2.2 Single coordinate reference system . 8
6.2.3 Compound coordinate reference system. 8
6.3 Datum . 9
6.3.1 Types of datums. 9
6.3.2 Datum description. 10
6.3.3 Prime meridian . 10
6.3.4 Ellipsoid . 11
6.4 Coordinate system. 11
6.5 Coordinate operation — coordinate conversion and coordinate transformation. 12
6.5.1 General. 12
6.5.2 Coordinate conversion (including map projection) . 13
6.5.3 Coordinate transformation. 14
6.5.4 Requirements for describing a coordinate operation. 14
6.5.5 Concatenated coordinate operation . 16
6.6 Citations. 17
6.7 Accuracy and precision of coordinates, coordinate operations, and parameters. 18
6.8 Attributes to describe a coordinate reference system. 19
Annex A (normative) Conformance . 22
Annex B (normative) UML schemas . 24
Annex C (informative) Decision trees. 27
Annex D (informative) Geodetic relationships. 29
Annex E (informative) Examples. 35

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 19111 was prepared by Technical Committee ISO/TC 211, Geographic information/Geomatics.
iv © ISO 2003 — All rights reserved

Introduction
Geographic information contains spatial references which relate the features represented in the data to
positions in the real world. Spatial references fall into two categories:
 those using coordinates;
 those based on geographic identifiers.
This International Standard deals only with spatial referencing by coordinates. Spatial referencing by
geographic identifiers is the subject of ISO 19112, Geographic information ― Spatial referencing by
geographic identifiers.
Coordinates are unambiguous only when the coordinate reference system to which those coordinates are
related has been fully defined. A coordinate reference system is a coordinate system which has a reference to
the Earth. This International Standard describes the elements that are necessary to define fully various types
of coordinate systems and coordinate reference systems applicable to geographic information. The subset of
elements required is partially dependent upon the type of coordinates. This International Standard also
includes optional fields to allow for the inclusion of non-essential coordinate reference system information. The
elements are intended to be both machine and human readable. A set of coordinates on the same coordinate
reference system requires one coordinate reference system description.
In addition to describing a coordinate reference system, this International Standard provides for the
description of a coordinate transformation or coordinate conversion between two different coordinate
reference systems. With such information, geographic data referred to different coordinate reference systems
can be merged together for integrated manipulation. Alternatively, an audit trail of coordinate reference system
manipulations can be maintained.

INTERNATIONAL STANDARD ISO 19111:2003(E)

Geographic information — Spatial referencing by coordinates
1 Scope
This International Standard defines the conceptual schema for the description of spatial referencing by
coordinates. It describes the minimum data required to define one-, two- and three-dimensional coordinate
reference systems. It allows additional descriptive information to be provided. It also describes the information
required to change coordinate values from one coordinate reference system to another.
This International Standard is applicable to producers and users of geographic information. Although it is
applicable to digital geographic data, its principles can be extended to many other forms of geographic data
such as maps, charts, and text documents.
2 Conformance requirements
This International Standard defines two classes of conformance, Class A for conformance of coordinate
reference systems and Class B for coordinate operations between two coordinate reference systems. Any
coordinate reference system claiming conformance to this International Standard shall satisfy the
requirements given in Annex A, Clause A.1. Any coordinate operation claiming conformance to this
International Standard shall satisfy the requirements given in Annex A, Clause A.2.
3 Normative references
The following referenced documents are indispensable for the application 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 1000, SI units and recommendations for use of their multiples and of certain other units
1)
ISO/TS 19103:— , Geographic information — Conceptual schema language
ISO 19113:2002, Geographic information — Quality principles
1)
ISO 19114:— , Geographic information — Quality evaluation procedures
4 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
4.1
Cartesian coordinate system
coordinate system which gives the position of points relative to n mutually perpendicular axes

1)
To be published.
NOTE n is 1, 2 or 3 for the purposes of this International Standard.
4.2
compound coordinate reference system
coordinate reference system using two other independent coordinate reference systems to describe a position
EXAMPLE One coordinate reference system based on a two- or three-dimensional coordinate system and the other
coordinate reference system based on a gravity-related height system.
4.3
coordinate
one of a sequence of n numbers designating the position of a point in n-dimensional space
NOTE 1 In a coordinate reference system, the numbers must be qualified by units.
NOTE 2 A coordinate operation is performed on coordinates in a source system resulting in coordinates in a target
system.
4.4
coordinate conversion
change of coordinates, based on a one-to-one relationship, from one coordinate system to another based
on the same datum
EXAMPLE Between geodetic and Cartesian coordinate systems or between geodetic coordinates and projected
coordinates, or change of units such as from radians to degrees or feet to metres.
NOTE A coordinate conversion uses parameters which have constant values.
4.5
coordinate operation
change of coordinates, based on a one-to-one relationship, from one coordinate reference system to
another
NOTE Supertype of coordinate transformation and coordinate conversion.
4.6
coordinate reference system
coordinate system that is related to the real world by a datum
NOTE For geodetic and vertical datums, it will be related to the Earth.
4.7
coordinate system
set of mathematical rules for specifying how coordinates are to be assigned to points
4.8
coordinate transformation
change of coordinates from one coordinate reference system to another coordinate reference system
based on a different datum through a one-to-one relationship
NOTE A coordinate transformation uses parameters which are derived empirically by a set of points with known
coordinates in both coordinate reference systems.
4.9
datum
parameter or set of parameters that serve as a reference or basis for the calculation of other parameters
NOTE A datum defines the position of the origin, the scale, and the orientation of the axes of a coordinate system.
2 © ISO 2003 — All rights reserved

4.10
easting
E
distance in a coordinate system, eastwards (positive) or westwards (negative) from a north-south reference
line
4.11
ellipsoid
surface formed by the rotation of an ellipse about a main axis
NOTE In this International Standard, ellipsoids are always oblate, meaning that the axis of rotation is always the
minor axis.
4.12
ellipsoidal height
geodetic height
h
distance of a point from the ellipsoid measured along the perpendicular from the ellipsoid to this point
positive if upwards or outside of the ellipsoid
NOTE Only used as part of a three-dimensional geodetic coordinate system and never on its own.
4.13
engineering datum
local datum
datum describing the relationship of a coordinate system to a local reference
NOTE Engineering datum excludes both geodetic and vertical datums.
EXAMPLE A system for identifying relative positions within a few kilometres of the reference point.
4.14
flattening
f
ratio of the difference between the semi-major (a) and semi-minor axis (b) of an ellipsoid to the semi-major
axis: f = (a − b)/a
NOTE Sometimes inverse flattening 1/f = a/(a − b) is given instead; 1/f is also known as reciprocal flattening.
4.15
geodetic coordinate system
ellipsoidal coordinate system
coordinate system in which position is specified by geodetic latitude, geodetic longitude and (in the three-
dimensional case) ellipsoidal height
4.16
geodetic datum
datum describing the relationship of a coordinate system to the Earth
NOTE In most cases, the geodetic datum includes an ellipsoid definition.
4.17
geodetic latitude
ellipsoidal latitude
ϕ
angle from the equatorial plane to the perpendicular to the ellipsoid through a given point, northwards treated
as positive
4.18
geodetic longitude
ellipsoidal longitude
λ
angle from the prime meridian plane to the meridian plane of a given point, eastward treated as positive
4.19
geoid
level surface which best fits mean sea level either locally or globally
NOTE “Level surface” means an equipotential surface of the Earth’s gravity field which is everywhere perpendicular
to the direction of gravity.
4.20
gravity-related height
H
height dependent on the Earth’s gravity field
NOTE In particular, orthometric height or normal height, which are both approximations of the distance of a point
above the mean sea level.
4.21
Greenwich meridian
meridian that passes through the position of the Airy Transit Circle at the Royal Observatory Greenwich,
United Kingdom
NOTE Most geodetic datums use the Greenwich meridian as the prime meridian. Its precise position differs slightly
between different datums.
4.22
height
h, H
distance of a point from a chosen reference surface along a line perpendicular to that surface
NOTE 1 See ellipsoidal height and gravity-related height.
NOTE 2 Height of a point outside the surface treated as positive; negative height is also called depth.
4.23
map projection
coordinate conversion from a geodetic coordinate system to a plane
4.24
mean sea level
average level of the surface of the sea over all stages of tide and seasonal variations
NOTE 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. Mean sea level in a global context differs from a global geoid by not
more than 2 m.
4.25
meridian
intersection of an ellipsoid by a plane containing the semi-minor axis of the ellipsoid
NOTE This term is often used for the pole-to-pole arc rather than the complete closed figure.
4.26
northing
N
distance in a coordinate system, northwards (positive) or southwards (negative) from an east-west reference
line
4 © ISO 2003 — All rights reserved

4.27
polar coordinate system
coordinate system in which position is specified by distance and direction from the origin
NOTE In three dimensions also called spherical coordinate system.
4.28
prime meridian
zero meridian
meridian from which the longitudes of other meridians are quantified
4.29
projected coordinate system
two-dimensional coordinate system resulting from a map projection
4.30
semi-major axis
a
longest radius of an ellipsoid
NOTE For an ellipsoid representing the Earth, it is the radius of the equator.
4.31
semi-minor axis
b
shortest radius of an ellipsoid
NOTE For an ellipsoid representing the Earth, it is the distance from the centre of the ellipsoid to either pole.
4.32
spatial reference
description of position in the real world
NOTE This may take the form of a label, code or set of coordinates.
4.33
vertical datum
datum describing the relation of gravity-related heights to the Earth
NOTE In most cases the vertical datum will be related to a defined mean sea level based on water level observations
over a long time period. Ellipsoidal heights are treated as related to a three-dimensional ellipsoidal coordinate system
referenced to a geodetic datum. Vertical datums include sounding datums (used for hydrographic purposes), in which
case the heights may be negative heights or depths.
5 Conventions
5.1 Symbols and abbreviated terms
a semi-major axis
b semi-minor axis
CCRS Compound coordinate reference system
E easting
h ellipsoidal height
N northing
SC Spatial referencing by Coordinates
SI le Système International d’unités
UML Unified Modeling Language
λ geodetic longitude
ϕ geodetic latitude
x, y, z Cartesian coordinates in a geodetic datum
i, j, k Cartesian coordinates in a engineering datum
r, Ω, θ spherical polar coordinates
5.2 UML notation
The diagrams that appear in this International Standard are presented using the Unified Modeling Language
(UML) static structure diagram with the ISO Interface Definition Language (IDL) basic type definitions and the
UML Object Constraint Language (OCL) as the conceptual schema language. The UML notations used in this
International Standard are described in Figure 1.

Figure 1 — UML notation
6 © ISO 2003 — All rights reserved

6 Definition of the conceptual schema for coordinate reference systems
6.1 Introduction
Location or position on or near the Earth's surface may be described using coordinates. Coordinates are
unambiguous only when the coordinate reference system to which those coordinates are related has been
fully defined. Each position shall be described by a set of coordinates in a coordinate reference system.
Coordinates supplied in a dataset shall belong to the same coordinate reference system. A description of this
coordinate reference system shall be supplied with the dataset. Coordinate data shall be accompanied by
information sufficient to make the coordinates unambiguous. This information varies by coordinate system
type and datum type.
In the clauses below, attributes are given a requirement status:
Requirement Definition Comment
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 following tables indicates the maximum number of occurrences of
attribute values that are permissible, with N indicating no upper limit. The conceptual schema for describing
coordinate reference systems is modelled with the Unified Modeling Language (UML) in Annex B. In case of
inconsistency between the metadata textual description and the UML model (re: Annex B), the textual
description shall prevail. The basic data types are defined in ISO/TS 19103.
6.2 Coordinate reference system
6.2.1 Type of coordinate reference system
A coordinate reference system may be either single or compound. A single coordinate reference system is
defined in 6.2.2 and a compound coordinate reference system is defined in 6.2.3. The requirements for
describing the type of coordinate reference system shall be in accordance with Table 1.
Table 1 — Requirements for describing the type of coordinate reference system
Element name UML Data type Obligation Maximum Description
identifier occurrence
Coordinate reference typeCode SC_TypeCode M 1 Code denoting the type of coordinate
system type code reference system:
1 — a single coordinate reference
system
2 — a compound coordinate reference
system
Coordinate reference remarks CharacterString O 1 Comments on the coordinate
system remarks reference system including source
information.
To determine whether the coordinate reference system is compound or single, decision tree 1 in Annex C may
be used (see Figure C.1).
6.2.2 Single coordinate reference system
A position of a feature can be given by a set of coordinates. Coordinates are unambiguous if the coordinate
reference system to which those coordinates are related has been fully defined.
A coordinate reference system is realized by a set of coordinates. The realization is sometimes known as a
reference frame.
A coordinate reference system shall be defined by one datum and by one coordinate system; see Figure 2.

Figure 2 — Coordinate reference system
For the purposes of this International Standard, a coordinate reference system shall not change with time.
When a reference frame changes with time, a new datum and coordinate reference system shall be created,
with date of realization of the datum and coordinate reference system included in their names or identifiers.
The requirements for describing a coordinate reference system shall be in accordance with Table 2.
Table 2 — Requirements for describing a coordinate reference system
Element name UML Data type Obligation Maximum Description
identifier occurrence
Coordinate reference CRSID RS_Identifier M 1 Identifier of the coordinate reference
system identifier system.
Coordinate reference alias CharacterString O N Alternative name or identifier by which
system alias this coordinate reference system is
known.
Coordinate reference validArea EX_Extent O 1 Area for which the coordinate
system valid area reference system is valid.
Coordinate reference scope CharacterString O N Application for which the coordinate
system scope reference system is valid.

6.2.3 Compound coordinate reference system
The horizontal and vertical components of a description of position in three dimensions may sometimes come
from different coordinate reference systems rather than through a single three-dimensional coordinate
reference system. This is always the case for positions where vertical coordinates are related to mean sea
level. This shall be handled through a compound coordinate reference system (CCRS) which identifies the two
coordinate reference systems utilized, see Figure 3. Vertical datum and gravity-related height are an example
of a datum and coordinate system for coordinate reference system 2.
8 © ISO 2003 — All rights reserved

Figure 3 — Compound coordinate reference system
The requirements for describing a compound coordinate reference system shall be in accordance with Table 3.
Each of the two coordinate reference systems shall then be described in the normal way.
The compound coordinate reference system identifier may be a concatenation of the coordinate reference
system identifiers for the component coordinate reference systems.
Table 3 — Requirements for describing a compound coordinate reference system
Element name UML Data type Obligation Maximum Description
identifier occurrence
Compound coordinate CCRSID RS_Identifier O 1 Identifier of the compound coordinate
reference system reference system.
identifier
Compound coordinate alias CharacterString O N Alternative name or identifier by which
reference system alias this compound coordinate reference
system is known.
Compound coordinate validArea EX_Extent O 1 Area for which the compound
reference system valid coordinate reference system is valid.
area
Compound coordinate scope CharacterString O N Application for which the compound
reference system coordinate reference system is valid.
scope
6.3 Datum
6.3.1 Types of datums
A datum is geodetic, vertical or engineering. A geodetic datum gives the relationship of a coordinate system to
the Earth and is used as the basis for two- or three-dimensional systems. In most cases, it shall require an
ellipsoid definition. A vertical datum gives the relationship of gravity-related heights to a surface known as the
geoid. The geoid is a surface close to mean sea level. In this International Standard, a datum shall be
engineering if it is neither geodetic nor vertical.
For geographic information purposes it is necessary to identify a datum, but the definition of the datum itself is
optional.
If the type of coordinate reference system is not known, decision tree 2 in Annex C may be used in the
determination of the datum type (see Figure C.2).
6.3.2 Datum description
If a coordinate reference system citation is not supplied, then a datum description in accordance with Table 4
shall be supplied.
Table 4 — Requirements for describing a datum
Element name UML Data type Obligation Maximum Description
identifier occurrence
Datum identifier datumID RS_Identifier M 1 Identifier of the datum.
Datum alias alias CharacterString O N Alternative name or names by which
this datum is known.
Datum type type CharacterString O 1 Type of datum. Valid values are:
— geodetic,
— vertical, or
— engineering
Datum anchor point point CharacterString O 1 Description including coordinates of the
point or points used to anchor the
datum to the Earth.
Datum realization realization Date O 1 Epoch of realization of the datum.
epoch Epoch
Datum valid area validArea EX_Extent O 1 Area for which the datum is valid.
Datum scope scope CharacterString O N Application for which the datum is valid.
Datum remarks remarks CharacterString O 1 Comments on the datum including
source information.
When the datum type is geodetic, then certain prime meridian and ellipsoid attributes as described below shall
be mandatory regardless of whether a value for datum type has been provided or not.
6.3.3 Prime meridian
A prime meridian defines the origin from which longitude values are specified. Most geodetic datums use
Greenwich as their prime meridian.
A prime meridian description shall be mandatory if the datum type is geodetic and its prime meridian is not
Greenwich and if neither coordinate reference system citation nor datum citation is supplied.
The requirements for describing a prime meridian shall be in accordance with Table 5.
Table 5 — Requirements for describing a prime meridian
Element name UML Data type Obligation Maximum Description
identifier occurrence
Prime meridian meridianID RS_Identifier M 1 Identifier of the prime meridian.
identifier
Prime meridian Greenwich Angle M 1 Longitude of the prime meridian
Greenwich Longitude measured from the Greenwich meridian,
longitude positive eastward.
If the datum type is geodetic and the
prime meridian name is not supplied,
then the prime meridian name is taken to
be “Greenwich” and the prime meridian
Greenwich longitude is taken to be “0°”.
Prime meridian remarks CharacterString O 1 Comments on the prime meridian
remarks including source information.

10 © ISO 2003 — All rights reserved

6.3.4 Ellipsoid
An ellipsoid description is not required if the datum type is
a) vertical,
b) engineering, or
c) geodetic,
and any of the following circumstances apply:
 the coodinate reference system citation is supplied;
 the datum citation is supplied;
 the coordinate system type is Cartesian.
The requirements for describing an ellipsoid shall be in accordance with Table 6.
Table 6 — Requirements for describing an ellipsoid
Element name UML Data type Obligation Maximum Description
identifier occurrence
Ellipsoid identifier ellipsoidID RS_Identifier M 1 Identifier of the ellipsoid for the datum.
Ellipsoid alias alias CharacterString O N Alternative name or names of the ellipsoid.
Ellipsoid semi- semiMajor Length M 1 Length of the semi-major axis of the
major axis Axis ellipsoid.
Ellipsoid shape ellipsoid boolean M 1 boolean = TRUE when the reference
Shape surface is an ellipsoid,
FALSE when the reference
surface is a sphere.
Ellipsoid inverse inverse SC_inverse C 1 Inverse flattening of the ellipsoid. Unitless.
flattening Flattening Flattening Condition 1 (cd 1): Mandatory if ellipsoid
shape is true.
Ellipsoid remarks remarks CharacterString O 1 Comments on or information about the
ellipsoid.
6.4 Coordinate system
A coordinate system is described by the name, the units, the direction and sequence of the axes. Coordinates
in a set are listed according to this sequence. Coordinates based on a projected coordinate reference system
are the result of a coordinate conversion which is described in 6.5.
The coordinate system description shall be mandatory if a coordinate reference system citation is not supplied.
The requirements for describing a coordinate system shall be in accordance with Tables 7 and 8.
Table 7 — Requirements for describing a coordinate system
Element name UML Data type Obligation Maximum Description
identifier occurrence
Coordinate CSID RS_Identifier M 1 Identifier of the coordinate system.
system identifier
Coordinate type SC_Coordinate M 1 Type of the coordinate system. The
system type SystemType most commonly used entries are:
— Cartesian
— geodetic
— projected
— polar
— gravity-related
Do not use Cartesian if the system is
projected.
Coordinate dimension Integer M 1 Number of coordinates {3,2,1} in the set.
system dimension
Coordinate remarks CharacterString O 1 Comments on or information about the
system remarks coordinate system.

Each coordinate system axis shall be described, the order of each axis description following the order of the
coordinates in the data set. The elements for each coordinate system axis, as described in Table 8, shall be
kept together (as in a data block), and the number of data blocks shall be equal to the value provided for
coordinate system dimension in Table 7.
Table 8 — Requirements for describing a coordinate system axis
Element name UML Data type Obligation Maximum Description
identifier occurrence
Coordinate axisName CharacterString M 1 Name of the coordinate system axis.
system axis name
Coordinate axisDirection CharacterString M 1 Direction of the coordinate system axes
system axis (or, in the case of Cartesian or
direction projected coordinates, the direction of
the coordinate system axis at the
origin).
Examples: north, east, up
Coordinate axisUnitID UnitOf Measure M 1 Identifier of the unit for the coordinate
system axis unit system axis.
identifier
If the type of coordinate reference system is not known, decision tree 2 in Annex C may be used in the
determination of the coordinate system type (see Figure C.2).
6.5 Coordinate operation — coordinate conversion and coordinate transformation
6.5.1 General
This subclause describes coordinate operations to change coordinate values from one coordinate reference
system to coordinate values based on another coordinate reference system. Coordinate operation information
may be given if datasets having coordinates using different coordinate reference systems are to be merged.
12 © ISO 2003 — All rights reserved

Generally the description of a coordinate operation is not required for the unambiguous identification of
coordinates. However, projected coordinates are the result of a coordinate conversion applied to geodetic
coordinates; in this special case, a coordinate operation description must be part of the coordinate reference
system description.
In this International Standard, two types of coordinate operations shall be recognized:
1) A coordinate conversion changes coordinates from one coordinate system to another based on the
same datum. In a coordinate conversion, the parameter values are exact.
2) A coordinate transformation changes coordinates from a coordinate reference system based on one
datum to a coordinate reference system based on a second datum. A coordinate transformation
differs from a coordinate conversion in that the coordinate transformation parameter values are
derived empirically: therefore there may be several different estimations (or realizations).
Once the parameter values are obtained, both coordinate conversion and coordinate transformation use
similar mathematical processes.
6.5.2 Coordinate conversion (including map projection)
A coordinate conversion is a one-to-one mapping of coordinates based on one coordinate reference system to
another coordinate reference system on the same datum. These coordinate conversions are widely used to
provide mapping projections of ellipsoidal coordinates to two-dimensional Cartesian coordinates. Other
coordinate conversions include conversion of the units of measure or shifting the origin of coordinate system.
In this International Standard, coordinate conversions (see Figure 4) shall be distinguished from coordinate
transformations (see Figure 5). Coordinate conversions do not change the underlying datum since they use
analytical mathematical functions which do not alter the fundamental accuracy of the coordinate values.
Coordinate conversions include
 map projections, which is a method using mathematical functions to convert ellipsoidal coordinates
(excluding height) to two-dimensional Cartesian coordinates, or vice-versa;
 coordinate conversions of ellipsoidal coordinates (including ellipsoidal height) to three-dimensional
Cartesian coordinates, or vice-versa;
 unit changes by application of a multiplication factor (for example, metres to feet) or an algorithm (for
example, radians to degrees, minutes and seconds);
 shifting the origin of a plane to make a local grid.

Figure 4 — Coordinate conversion
The map projection is a special coordinate conversion of coordinate systems from the ellipsoid to the plane.
For the description of coordinates belonging to a projected coordinate system, the provision of a coordinate
operation description shall be mandatory.
6.5.3 Coordinate transformation
Coordinates may be transformed by changing them to another datum. The coordinate systems shall be of the
same type (for example, both geodetic or both Cartesian). A coordinate transformation is performed through a
method which has an algorithm. Each algorithm has a set of related parameters. Because their values are
empirically derived, they depend upon the measurements used and include measurement errors. Different
sets of measurements will result in multiple sets of parameter values of a coordinate transformation between
two datums.
Figure 5 — Coordinate transformation
A coordinate transformation description is not necessary for describing a coordinate reference system.
However, it may sometimes be useful to describe a coordinate transformation that has already been applied to
the coordinates, or a coordinate transformation from that system to a user-defined coordinate reference
system.
6.5.4 Requirements for describing a coordinate operation
The requirements for describing a coordinate operation and connected terms shall be in accordance with
Tables 9 and 10.
A coordinate operation description shall also be given if the coordinate system type is projected and neither a
coordinate reference system citation nor a coordinate system citation has been supplied.
14 © ISO 2003 — All rights reserved

Table 9 — Requirements for describing a coordinate operation and connected terms
Element name UML Data type Obligation Maximum Description
identifier occurrence
Coordinate Coordinate RS_Identifier M 1 Identifier of the coordinate operation.
operation identifier OperationID
Coordinate validArea EX_Extent O 1 Area for which the coordinate
operation valid area operation is valid.
Coordinate scope CharacterString O N Application for which the coordinate
operation scope operation is valid.
Source coordinate sourceID RS_Identifier C 1 Identifier of the source coordinate
reference system reference system.
identifier Condition 2 (cd 2): Mandatory if
describing a coordinate
transformation.
Target coordinate targetID RS_Identifier C 1 Identifier of the target coordinate
reference system reference system.
identifier cd 2
Coordinate version CharacterString C 1 Version of the coordinate operation
operation version between the source coordinate
reference system and the target
coordinate reference system.
cd 2
Coordinate methodName CharacterString C 1 Name of the algorithm used for the
operation method coordinate operation.
name
Example (in case of coordinate
transformation):
— Abridged Molodenski
— Similarity transformation
Example (in case of coordinate
conversion):
— Cartesian into ellipsoidal
— Universal Transverse Mercator
— Mercator
— Lambert Conformal Conic
— Albers equal area
— Stereographic
— metres to feet
— radians to degrees
Condition 3 (cd 3):
Mandatory either (i) if describing a
projected coordinate system and none
of coordinate reference system
citation, coordinate system citation, or
coordinate operation citation is
supplied, or (ii) if describing a single
coordinate conversion or a coordinate
transformation.
Coordinate methodName CharacterString O N Alternative name or names of the
operation method Alias coordinate operation method identifier.
name alias
Coordinate formula CharacterString M 1 Formula(s) used by the coordinate
operation method operation method.
formula(s) This may be a reference to a
publication.
Coordinate numberOf- Integer M 1 Number of parameters required by this
operation method Parameters coordinate operation method.
number of
parameters
Coordinate remarks CharacterString O 1 Comments on or information about the
operation method coordinate operation method.
remarks
It is often useful to include an
example. This may define a time
dependent parameter such as epoch.
Coordinate operation parameters shall be described following the order of coordinate operation parameters in
the data set.
When several coordinate operation parameters are being described, the elements for each parameter as
detailed in Table 10 shall be kept together in a data block, and the number of data blocks shall be the same as
the value given by coordinate operation method parameters number.
Table 10 — Requirements for describing coordinate operation parameters
Element name UML Data type Obligation Maximum Description
identifier occurrence
Coordinate operation name CharacterString M 1 Identifier of the coordinate operation
parameter name parameter that is defined or used with
this coordinate operation method. The
parameters differ among coordinate
operation methods.
Example (in case of coordinate
transformation):
— geocentric x translation
— geocentric y translation
— geocentric z translation
Example (in case of coordinate
conversion):
— latitude of origin
— longitude of origin
— scale factor
— false easting
— false northing
Coordinate operation value Measure M 1 Value of the coordinate operation
parameter value parameter.
Coordinate operation remarks CharacterString O 1 Comments on or information about the
parameter remarks coordinate operation parameter.

6.5.5 Concatenated coordinate operation
The change of coordinates from one coordinate reference system to another coordinate reference system
may follow from a series of coordinate operations consisting of one or more coordinate transformations and/or
one or more coordinate conversions. This is called a concatenated coordinate operation. Figure 6 shows a
two-step concatenated coordinate operation. There is no upper limit to the number of steps a concatenated
coordinate operation may have. Each step is a coordinate operation described in the normal way.

Figure 6 — Concatenated coordinate operation
16 © ISO 2003 — All rights reserved

The requirements for describing a concatenated coordinate operation are given in Table 11. Individual
coordinate operations shall be described according to Table 9 and coordinate operation parameters for
individual coordinate operations shall be described according to Table 10. The order of individual coordinate
operations is significant and follows the order in which the steps are performed. The number of individual
coordinate operations described shall b
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