ISO/IEC 18026:2009

INTERNATIONAL ISO/IEC
STANDARD 18026
Second edition
2009-07-15
Information technology — Spatial
Reference Model (SRM)
Technologies de l'information — Modèle de référence spatial (SRM)

Reference number
©
ISO/IEC 2009
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INTERNATIONAL STANDARD
Information technology — Spatial Reference Model (SRM)
1 Scope
This International Standard specifies the Spatial Reference Model (SRM) defining relevant aspects of spatial
positioning and related information processing. The SRM allows precise and unambiguous specification of
geometric properties such as position (location), direction, and distance. The SRM addresses the needs of a
broad community of users, who have a range of accuracy and performance requirements in computationally
intensive applications.
Aspects of this International Standard apply to, but are not limited to:
a) mapping, charting, geodesy, and imagery;
b) topography;
c) location-based services;
d) oceanography;
e) meteorology and climatology;
f) interplanetary and planetary sciences;
g) embedded systems; and
h) modelling and simulation.
The application program interface supports more than 30 forms of position representat
...

8 Spatial reference frames
8.1 Introduction
A spatial coordinate system is a means of associating a unique coordinate with a point in object-space. It is
defined by binding an abstract CS to a normal embedding (see 8.2). A spatial reference frame is a
specification of a spatial coordinate system for a region of object-space (see 8.3). It is formed by the binding of
an abstract coordinate system to the normal embedding specified by an ORM for that object. A full
specification specifies the CS and the ORM and includes values for CS parameters, if any, and a specification
of the region of object-space. Some or all CS parameters may be bound by ORM parameters. In particular, a
CS based on an oblate ellipsoid (or sphere) must match the parameters of the oblate ellipsoid (or sphere) RD
of the ORM.
A spatial reference frame template is an abstraction of a collection of spatial reference frames that share the
same abstract coordinate system, coordinate system parameter binding rules, and similar ORMs that model
the same spatial object type (see 8.5). Spatial reference frames may be organized into specified sets so as to
form an atlas for a large region of space. This International Standard specifies a collection of spatial reference
frame templates, realizations of those templates, and sets of those realizations.
8.2 Spatial coordinate systems
If a normal embedding of position-space into object-space is defined, any abstract CS for a region of that
position-space can be used to specify a spatial CS that associates coordinates in coordinate-space to points
in object-space. This association is a binding of a CS via a normal embedding. The association is defined as:
p = E G c
( )
( )
where:
c is a coordinate in the CS domain,

G is the CS generating function,
E is the normal embedding function, and
p is the point in object-space associated with c.
CS generating function
G
v-axis
z-axis
z-axis
normal
embedding E
E( G ( c )) = p
G (c ) = (x , y, z)
c = ( u , v)
z z
y-axis
y-axis
y y
u-axis
origin
x x
x-axis
x-axis
position-space object-space
coordinate-space
Spatial CS association
Figure 8.1 — A spatial embedding of a surface CS
© ISO/IEC 2009 – All rights reserved
EXAMPLE  Figure 8.1 illustrates a spatial surface CS bound with a normal embedding of 3D position-space to the 3D
object-space. In this illustration, a surface coordinate (u, v) in coordinate-space is associated to a position (x, y, z) in the
abstract position-space. That position is then identified with a position in the space of an object via the normal embedding
of position-space. In this example, the normal embedding is determined by the selection of an origin and three unit points.
8.3 Spatial reference frame
8.3.1 Specification
A spatial reference frame (SRF) is a specification of a spatial coordinate system that is constructed from an
ORM and a compatible abstract CS, such that coordinates uniquely specify positions with respect to the
spatial object of the ORM. A specification of an SRF includes:
a) an ORM,
b) a CS compatible with the ORM,
c) a binding of all parameters of the spatial CS,
th
d) (optionally) k coordinate-component names,
e) (optionally) additional restrictions on the domain of valid coordinates in that spatial CS, and
f) (optionally) if the CS is of CS type 3D, a vertical coordinate-component identification (see 8.4).
An SRF implicitly specifies a spatial CS defined by the binding of the CS via the normal embedding associated
with the ORM.
Spatial CS compatibility and the other elements of the specification of an SRF are defined in the following
clauses.
8.3.2 SRF specification elements
8.3.2.1 ORM and CS compatibility
The compatible CS type of the CS element of an SRF depends on the dimension of the ORM. The dimension
of an ORM is defined as the dimension of the RD components of the specification of the ORM. The
compatible CS types by ORM dimension are specified in Table 8.1.
Table 8.1 — Compatible CS types
ORM dimension Compatible CS types
1D 1D CS
Curve CS
2D
2D CS
Curve CS
3D Surface CS
3D CS
The use of surface CSs or 3D CSs that are based on an oblate ellipsoid (or sphere) are restricted to ORMs
that are based on an oblate ellipsoid (or respectively, sphere) RD.
The surface CSs that are based on an oblate ellipsoid (or sphere) are:
© ISO/IEC 2009 – All rights reserved
a) surface geodetic,
b) surface planetodetic, and
c) all map projections.
The 3D CSs that are based on an oblate ellipsoid (or sphere) are:
a) geodetic 3D,
b) planetodetic 3D, and
c) all augmented map projections.
As a further restriction, some CSs are based on spheres only. CS OBLIQUE_MERCATOR_SPHERICAL has
this restriction.
An SRF may be described in terms of the properties and other characteristics of the CS that is specified by
the SRF. In particular, an SRF is said to be a 3D SRF, surface SRF, or 2D SRF if the CS of the SRF is of the
corresponding CS type. Similarly, the CS properties of linearity, orthogonality, and handedness may be used
as descriptors of an SRF corresponding to the properties of the CS that is specified by the SRF. Thus, an
SRF is said to be a linear SRF or a curvilinear SRF if the CS of the SRF has the respective linearity property.
Every 3D SRF in this International Standard is a right-handed SRF in consequence to the CS handedness
restriction imposed in 5.6.4.
8.3.2.2 CS parameter binding
All CS parameter values must be specified. In the case of a combination of a CS and an ORM based on an
oblate ellipsoid (or sphere), the major semi-axis and minor semi-axis (or equivalently, the inverse flattening)
(or respectively, sphere radius) of the ORM and CS shall match.
8.3.2.3 Coordinate-component names
A CS specification (see 5.9) includes the coordinate-component symbols with common names (if any). A
th
specification of an SRF may optionally assign SRF-specific names to the k coordinate-components. The
name assignment shall reflect the common use in the intended application domain.
th
EXAMPLE  For an equatorial spherical CS, the assignment of SRF-specific names to the k coordinate-components
of “right ascension” for λ, “declination” for θ , and “radius” for ρ.
8.3.2.4 Coordinate valid-region
A CS specification (see 5.9) includes the specification of the CS domain and CS range where the generating
function (or mapping equations) and its inverse(s) are defined. An SRF specification may further restrict the
CS domain. A valid-region is a restriction of the CS domain of the generating function (or mapping equations)
for a CS as used in an SRF. An extended valid-region is a second valid-region that contains the first valid-
region as a subset. The specification of these restrictions is important for several (SRF specific) reasons:
a) If the ORM is local, the restrictions are used to model, in coordinate-space, the local region of the
space of the object.
b) If the CS is a map projection or an augmented map projection, the restrictions are used to bound or
otherwise limit distortions (see 5.8.3.1).
© ISO/IEC 2009 – All rights reserved
c) The SRF may be used in conjunction with other SRFs to form an atlas for a large region (see 8.7 SRF
sets). In this case, the restrictions are used to control the pair-wise overlap of the spatial coverage of
members of the SRF collection.
d) If the CS generating function (or map projection mapping equations) or the inverse function(s) have
been implemented with a numerical approximation, the restrictions are used to control error bounds.
The extended valid-region is used primarily for overlapping regions in forming an atlas as in (c) above. Not all
properties of the SRF that are true in the valid-region will necessarily be true in the extended valid-region. In
particular, a distortion error bound that holds in the valid-region may not hold in the extended valid-region.
A valid-region may be described and/or specified. A valid-region description is a descriptive statement of the
region such as the spatial boundary of a named political entity.
EXAMPLE 1 “The German state of Baden-Wurttemberg” and “The Baltic Sea” are valid-region descriptions.
In this International Standard, a valid-region specification is a finite (or empty) list of coordinate-component
constraints of the form:
th
k coordinate-component belongs to a non-empty interval of real numbers I .
k
An extended valid-region specification is a finite (or empty) list of coordinate-component constraints of the
form:
th
k coordinate-component belongs to an interval of real numbers J , where I has been specified

k k
and J ⊇ I .
k k
Angular coordinate-component intervals shall be evaluated modulo 2π to represent an interval of the unit
circle. Thus, 4π 3,2π 3 representes the angular interval 4π 3,2π ∪ 0,2π 3 .
[ ] [ ) [ ]
In the case of an SRF with an oblate ellipsoid (or sphere) based ORM, celestiodetic coordinates may be
similarly constrained. In particular, valid-region specifications for a map projection based SRF may specify
coordinate-component constraints for easting, northing, latitude, and/or longitude. Celestiodetic longitude
intervals shall be evaluated modulo 2π. In particular, if the interval limits satisfyλ >λ , then:
1 2
λ ,λ = λ ,π ∪ (−π,λ ,
[ ] [ ] ]
1 2 1 2
λ ,λ = λ ,π ∪ −π,λ ,
( ] ( ] ( ]
1 2 1 2
λ ,λ = λ ,π ∪ −π,λ , and
[ ) [ ] ( )
1 2 1 2
λ ,λ = λ ,π ∪ −π,λ .
( ) ( ] ( )
1 2 1 2
EXAMPLE 2 The SRF is based on a transverse Mercator map projection (see SRFT TRANSVERSE_MERCATOR).
Valid-region specification:  167 000 ≤ u ≤ 833 000, 0 ≤ v ≤ 9 500 000
Extended valid-region specification: 0 < u,  -100 < v
In this example, I = 167 000,833 000 and I = 0,9 500 000 are closed bounded intervals, and
[ ] [ ]
Easting Northing
J = 0,+∞ and J = −100,+∞ are open semi-bounded intervals that are further constrained by the CS
( ) ( )
Easting Northing
domain.
EXAMPLE 3 The SRF is based on a transverse Mercator map projection (see SRFT TRANSVERSE_MERCATOR).
Valid-region specification:  -78º ≤ λ < -72º, 0º ≤ ϕ < 84º
Extended valid-region specification: -78,5º ≤ λ < -71,5º
In this example, I = −78⋅ π 180 , −72⋅ π 180 and I = 0,84⋅ π 180 are left-closed, right-open
[ ( ) ( )) [ ( ))
Longitude Latitude
bounded intervals, as is J = −78,5⋅ π 180 , −71,5⋅ π 180 . J is not specified. This indicates that there
[ ( ) ( ))
Longitude Latitude
are no constraints for latitude (except for the CS domain definition) in the extended valid-region specification.
© ISO/IEC 2009 – All rights reserved
8.4 SRF induced surface spatial reference frame
In the case of an SRF specified with the combination of a 3D ORM and a 3D CS, the 3D CS induces a surface
rd
CS on each coordinate-component surface (see 5.5.2). An SRF specification may optionally identify the 3
coordinate-component as the vertical coordinate-component for the SRF. In that case, the surface CS induced
on the zero-value vertical coordinate-component surface is the induced surface SRF for the specification. The
vertical coordinate-component is optionally specified in the coordinate-component name specification element
of the SRF.
rd
The CS GEODETIC and the CS PLANETODETIC 3 coordinate-components (h: ellipsoidal height), and the
rd
3 coordinate-component of any augmented map projection CS (h: ellipsoidal height) are identified in this
International Standard as the vertical coordinate-component. When an SRF is specified with any of these 3D
CSs, the h = 0 coordinate-component surface coincides with the surface of the oblate ellipsoid (or sphere) RD
of the ORM. Any SRF based on these CSs intrinsically specifies the corresponding surface CS on the oblate
ellipsoid (or sphere) RD surface.
In an SRF realized from the SRF template LOCAL_TANGENT_SPACE_EUCLIDEAN specification (see 8.5.6)
rd
or the SRF template LOCAL_TANGENT_SPACE_CYLINDRICAL specification (see 8.5.8), the 3 coordinate-
component, height, is specified as the vertical coordinate-component. In these cases, the zero-value vertical
coordinate-component surface is a plane parallel to the tangent plane at the SRF tangent point. SRF
templates are defined in 8.5.
rd
The zero-value 3 coordinate-component surface of an SRF realized from the 3D CS SRF template
LOCAL_TANGENT_SPACE_AZIMUTHAL_SPHERICAL specification (see 8.5.7) induces a lococentric
surface azimuthal CS on the tangent plane of the SRF. For the purpose of specifying an induced surface
reference frame, the 3rd coordinate-component θ, depression/elevation angle, is specified as a vertical
coordinate. The zero-value vertical coordinate-component surface is a plane parallel to the tangent plane at
the SRF tangent point.
SRF templates that are based on surface CSs that can be induced by a zero-value vertical coordinate-
component surface of an SRF based on a 3D CS are not separately specified. The induced surface CS is
noted in the corresponding 3D CS based SRF template specification.
NOTE  Starting with a 3D SRF, this International Standard identifies surface SRFs on coordinate-component
surfaces. The relationship between a surface CS and the 3D CS which induces it is functionally similar to, but conceptually
different from, the ISO 19111 concept of compound coordinate reference frame. A compound coordinate reference frame
synthesizes a 3D reference frame from a surface and a vertical system. (See also 5.8.6.1 and Clause 9.)
8.5 SRF templates
8.5.1 Introduction
A spatial reference frame template (SRFT) is an abstraction of a collection of SRFs that share the same
abstract CS, coordinate component names, CS parameter binding rules, and similar ORMs that model the
same spatial object type. An SRF template allows for a consistent derivation of SRFs. It is not necessary that
an appropriate SRFT be defined in order to define a new SRF; however in this International Standard all SRFs
are derived from SRFTs. The specification elements for SRFTs are defined in Table 8.2.
Table 8.2 — SRFT specification elements
Element Definition
SRFT label The label of the SRF template (see 13.2.2).
The code of the SRF template (see 13.2.3).
SRFT code
Code 0 (UNSPECIFIED) is reserved.
© ISO/IEC 2009 – All rights reserved
Element Definition
A short name as published or as commonly known and an
Short name and description
optional description.
One or more of: abstract, physical, Earth, planet, satellite, and
Object or object type
Sun; and, optionally, additional restrictions.
ORM constraint Criteria for allowable ORMs.
CS label The label of a CS of compatible type.
th
SRF-specific names and/or symbols for the k coordinate-
component names and/or symbols. If all coordinate-
CS coordinate-component component names and symbols are the same as the CS, the
names and/or symbols phrase “Same as the CS.” shall be used. The vertical
coordinate-component shall be designated in this specification
element if applicable.
CS and RD parameters, if any, and/or SRF parameters that
Template parameters
are not specified by a CS parameter binding rule.
A set of rules for binding for CS parameters and ORM
CS parameter binding rules
component RD parameters, if any, and/or SRF parameters.
Optional restriction of the domain of the CS to a valid-region. If
a valid-region is specified, optionally an extended valid-region.
Coordinate valid-region
If both are unspecified, then there are no additional constraints
on coordinate validity.
Optional, additional, non-normative information such as a
Notes description of the SRF structure, modelled region, intended
use, and/or application domain.
References The references (see 13.2.5).

Coordinates in a given SRF may be represented in a variety of formats or encodings if the coordinate-
component values are sufficiently identified in the representation scheme. In particular, a representation
scheme for coordinates of an SRF:
1. shall identify the coordinate-components by name and/or symbol, or
2. shall identify coordinate-components of an encoding scheme in terms of the coordinate-components
specified in the SRF, or
3. shall define the ordering of a coordinate-component-tuple representation in terms of the coordinate-
components specified in the SRF.

The API (see 11) provides coordinate value encoding schemes in the form of data records with field names
that correspond to coordinate-component names. Where coordinate-component-tuples appear in the API, the
ordering is the order specified in the corresponding CS specification table.
This International Standard specifies a collection of SRFTs as identified in Table 8.3. Additional SRFTs may
be registered in accordance with Clause 13. Registered SRFs shall be derived only from standardized or
registered SRFTs.
© ISO/IEC 2009 – All rights reserved
Table 8.3 — SRFT directory
CS type Short name SRFT label
Celestiocentric CELESTIOCENTRIC
Local space rectangular 3D LOCAL_SPACE_RECTANGULAR_3D
Celestiodetic CELESTIODETIC
Planetodetic PLANETODETIC
Local tangent space Euclidean LOCAL_TANGENT_SPACE_EUCLIDEAN
Local tangent space azimuthal
LOCAL_TANGENT_SPACE_AZIMUTHAL_SPHERICAL
spherical
Local tangent space cylindrical LOCAL_TANGENT_SPACE_CYLINDRICAL
Lococentric Euclidean 3D LOCOCENTRIC_EUCLIDEAN_3D
3D
Celestiomagnetic CELESTIOMAGNETIC
Equatorial inertial EQUATORIAL_INERTIAL
Solar ecliptic SOLAR_ECLIPTIC
Solar equatorial SOLAR_EQUATORIAL
Solar magnetic ecliptic SOLAR_MAGNETIC_ECLIPTIC
Solar magnetic SOLAR_MAGNETIC_DIPOLE
Heliospheric Aries ecliptic HELIOSPHERIC_ARIES_ECLIPTIC
Heliospheric Earth ecliptic HELIOSPHERIC_EARTH_ECLIPTIC
Heliospheric Earth equatorial HELIOSPHERIC_EARTH_EQUATORIAL
Mercator MERCATOR
Surface (map
Oblique Mercator spherical OBLIQUE_MERCATOR_SPHERICAL
projection)
Transverse Mercator TRANSVERSE_MERCATOR
and 3D
(augmented
Lambert conformal conic LAMBERT_CONFORMAL_CONIC
map
Polar stereographic POLAR_STEREOGRAPHIC
projection)
Equidistant cylindrical EQUIDISTANT_CYLINDRICAL
Surface celestiodetic (induced) CELESTIODETIC
Surface planetodetic (induced) PLANETODETIC
Local tangent plane Euclidean
LOCAL_TANGENT_SPACE_EUCLIDEAN
(induced)
Surface
Local tangent plane azimuthal
LOCAL_TANGENT_SPACE_AZIMUTHAL_SPHERICAL
(induced)
Local tangent plane polar
LOCAL_TANGENT_SPACE_CYLINDRICAL
(induced)
Local space rectangular 2D LOCAL_SPACE_RECTANGULAR_2D
2D Local space azimuthal LOCAL_SPACE_AZIMUTHAL_2D
Local space polar LOCAL_SPACE_POLAR_2D

© ISO/IEC 2009 – All rights reserved
8.5.2 Celestiocentric SRFT
Celestiocentric SRFs shall be derived from the SRFT specified in Table 8.4.
Table 8.4 — Celestiocentric SRFT
Element Specification
SRFT label CELESTIOCENTRIC
SRFT code 1
celestiocentric SRFT
Short name and description The generalization of geocentric spatial reference frames to include non-
Earth objects.
Object type physical
ORM constraint Shall be derived from any 3D ORM.
CS label EUCLIDEAN_3D
CS coordinate-component The same as the CS.
names and/or symbols
Template parameters none
CS parameter binding rules None (no CS parameters).
Coordinate valid-region No additional restrictions.
Notes When the object is Earth, this SRFT is referred to as a geocentric SRFT.
References [EDM]
8.5.3 Local space rectangular 3D SRFT
Local space rectangular SRFs shall be derived from the SRFT specified in Table 8.5.
Table 8.5 — Local space rectangular 3D SRFT
Element Specification
SRFT label LOCAL_SPACE_RECTANGULAR_3D
SRFT code 2
local space rectangular 3D SRFT
Short name and description
A 3D Euclidean spatial reference frame for an abstract 3D space.
Object type 3D abstract object.
ORM constraint Shall be an ORM for a 3D abstract object.
CS label LOCOCENTRIC_EUCLIDEAN_3D
CS coordinate-component The same as the CS.
names and/or symbols
r = vector direction of forward (forward axis).
Template parameters
s = vector direction of up (up axis).
© ISO/IEC 2009 – All rights reserved
Element Specification
q =0,
r and s,select from:
+e positive primary axis,
+e positive secondary axis,
+e positive tertiary axis,
−e negative primary axis,
−e negative secondary axis, or
CS parameter binding rules
−e negative tertiary axis,
subject to: s ≠ ±r,
where:
 1 0 0
     
e = 0 , e = 1 , and e = 0 .
1 2 3
     
     
0 0 1
     
Coordinate valid-region No additional restrictions.
Notes CAD/CAM and other engineering applications.
References [EDM]
8.5.4 Celestiodetic SRFT
Celestiodetic SRFs shall be derived from the SRFT specified in Table 8.6.
Table 8.6 — Celestiodetic SRFT
Element Specification
SRFT label CELESTIODETIC
SRFT code 3
celestiodetic SRFT
Short name and description The generalization of geodetic SRFs to include other planets and ellipsoidal
bodies.
Object type
physical
Shall be derived from:
ORM constraint ORMT OBLATE_ELLIPSOID, OBLATE_ELLIPSOID_ORIGIN,
SPHERE, or SPHERE_ORIGIN.
CS label GEODETIC
CS coordinate-component The same as the CS.
names and/or symbols The vertical coordinate-component is ellipsoidal height (h).
Template parameters none
© ISO/IEC 2009 – All rights reserved
Element Specification
CS parameters match RD values.
-1
Oblate ellipsoid RD case with major semi-axis a and inverse flattening f :
a = a
CS parameter binding rules
b = a 1− f
( )
Sphere RD case with radius r :
a = b = r.
Coordinate valid-region No additional restrictions.
1) The SURFACE_GEODETIC CS is induced on the oblate ellipsoid (or
sphere) RD surface.
Notes
2) When the object is Earth, this SRFT is referred to as a geodetic SRFT.
References [HEIK]
8.5.5 Planetodetic SRFT
Planetodetic SRFs shall be derived from the SRFT specified in Table 8.7.
Table 8.7 — Planetodetic SRFT
Element Specification
SRFT label PLANETODETIC
SRFT code 4
Short name and planetodetic SRFT
description Similar to celestiodetic SRFT with reversed direction for longitude.
Object type planet
Shall be derived from:
ORM constraint ORMT OBLATE_ELLIPSOID, OBLATE_ELLIPSOID_ORIGIN,
SPHERE, or SPHERE_ORIGIN.
CS label PLANETODETIC
CS coordinate names The same as the CS.
and/or symbols The vertical coordinate-component is ellipsoidal height (h).
Template parameters none
CS parameters match RD values:
-1
Oblate ellipsoid RD case with major semi axis a and inverse flattening f :
a = a
CS parameter binding rules
b = a 1− f
( )
Sphere RD case with radius r :
a = b = r.
Coordinate valid region No additional restrictions
Notes Planetary science applications
References [RIIC]
© ISO/IEC 2009 – All rights reserved
8.5.6 Local tangent space Euclidean SRFT
Local tangent space Euclidean SRFs shall be derived from the SRFT specified in Table 8.8. The case with
template parameters α = 0 and h = 0 is illustrated in Figure 8.2.
Table 8.8 — Local tangent space Euclidean SRFT
Element Specification
SRFT label LOCAL_TANGENT_SPACE_EUCLIDEAN
SRFT code 5
local tangent space Euclidean SRFT
rd
Short name and description Euclidean 3D spatial CS with 3 coordinate-component surfaces that are
parallel to a plane tangent to the oblate ellipsoid RD.
Object type physical
Shall be derived from:
ORM constraint ORMT OBLATE_ELLIPSOID, OBLATE_ELLIPSOID_ORIGIN,
SPHERE, or SPHERE_ORIGIN.
CS label LOCOCENTRIC_EUCLIDEAN_3D
u: x (x)
CS coordinate-component
v: y (y)
names and/or symbols
w: height (h) is the vertical coordinate-component.
(λ,ϕ)= surface geodetic coordinate of the tangent point
α = azimuth (v-axis azimuth from north)
x = false origin x
Template parameters
F
y = false origin y
F
h = offset height
 1 0
   
r =T 0 , s =T 1 , and q = q − x r − y s
0 F F
   
   
0 0
   
where:
 
R ϕ +h cos ϕ cos λ
( ( ) ) ( ) ( )
 
N 0
 
 
 
q = R (ϕ)+h cos(ϕ)sin(λ) ,
CS parameter binding rules ( )
0 N 0
 
 
 
 b 
R ϕ +h sin ϕ
( ) ( )
 
 2 N 0 
a
 
 
a and b match the oblate ellipsoid (or sphere) RD values, and
−sinλ −cosλ sinϕ cosλ cosϕ cosα sinα 0
  
  
T = cosλ −sinλ sinϕ sinλ cosϕ −sinα cosα 0 .
  
  
0 cosϕ sinϕ 0 0 1
  
Coordinate valid-region No additional restrictions.
© ISO/IEC 2009 – All rights reserved
Element Specification
1) The LOCOCENTRIC_SURFACE_EUCLIDEAN CS is induced on the
tangent plane surface.
2) The w = -h coordinate-component plane is tangent to the oblate
Notes
ellipsoid RD at the point with surface celestiodetic coordinate (λ,ϕ).
3) α is the geodetic azimuth of the v-axis (see Figure 8.2).
4) h is the ellipsoidal height of the CS origin.
References [EDM]
w-axis
v-axis
u-axis
(λ,ϕ)
Figure 8.2 — Local tangent space Euclidean SRFT

8.5.7 Local tangent space azimuthal spherical SRFT
Local tangent space azimuthal spherical SRFs shall be derived from the SRFT specified in Table 8.9.
Table 8.9 — Local tangent space azimuthal spherical SRFT
Element Specification
SRFT label LOCAL_TANGENT_SPACE_AZIMUTHAL_SPHERICAL
SRFT code 6
In ISO 19111 terminology, the tangent plane is an engineering datum.
© ISO/IEC 2009 – All rights reserved
Element Specification
local tangent space azimuthal spherical SRFT
rd
Azimuthal spherical spatial CS with the zero 3 coordinate-component
Short name and description
surface that is tangent to the oblate ellipsoid RD and with CS natural origin at
the tangent point.
Object type physical
Shall be derived from:
ORM constraint ORMT OBLATE_ELLIPSOID, OBLATE_ELLIPSOID_ORIGIN,
SPHERE, or SPHERE_ORIGIN.
CS label LOCOCENTRIC_AZIMUTHAL_SPHERICAL
The same as the CS.
CS coordinate-component
names and/or symbols
θ: depression/elevation angle, is the vertical coordinate-component.
(λ,ϕ)= surface geodetic coordinate of the tangent point
α = azimuth (v-axis azimuth from north)
Template parameters
h = offset height
 
R (ϕ)+h cos(ϕ)cos(λ)
( ) 
N 0
 
 
 
q = R ϕ +h cos ϕ sin λ
( ( ) ) ( ) ( )
N 0
 
 
 
 b 
R ϕ +h sin ϕ
( ) ( )
 
 N 0 
a
   
 1
 
r =T 0
 
 
CS parameter binding rules
 
0
 
s =T 1
 
 
 
where:
a and b match the oblate ellipsoid (or sphere) RD values, and
−sinλ −cosλ sinϕ cosλ cosϕ cosα sinα 0
  
  
T = cosλ −sinλ sinϕ sinλ cosϕ −sinα cosα 0
  
  
0 cosϕ sinϕ 0 0 1
  
Coordinate valid-region No additional restrictions.
1) Used in radar localization.
Notes 2) h is the ellipsoidal height of the CS origin.
3) α is the geodetic azimuth of the v-axis (see Figure 8.2).
References [EDM]
8.5.8 Local tangent space cylindrical SRFT
Local tangent space cylindrical SRFs shall be derived from the SRFT specified in Table 8.10.
© ISO/IEC 2009 – All rights reserved
Table 8.10 — Local tangent space cylindrical SRFT
Element Specification
SRFT label LOCAL_TANGENT_SPACE_CYLINDRICAL
SRFT code 7
Short name and local tangent space cylindrical SRFT
rd
description Cylindrical spatial CS with 3 coordinate-component surfaces that are parallel
to a plane tangent to the oblate ellipsoid RD.
Object type physical
ORM constraint Shall be derived from:
ORMT OBLATE_ELLIPSOID, OBLATE_ELLIPSOID_ORIGIN,
SPHERE, or SPHERE_ORIGIN.
CS label LOCOCENTRIC_CYLINDRICAL
ρ : unchanged
CS coordinate-component
θ : unchanged
names and/or symbols
ζ : height (h) is the vertical coordinate
(λ,ϕ)= surface geodetic coordinate of the tangent point
α = azimuth (v-axis azimuth from north)
Template parameters
h = offset height
 
R ϕ +h cos ϕ cos λ
( ( ) ) ( ) ( )
N 0
 
 
 
q = R ϕ +h cos ϕ sin λ
( ( ) ) ( ) ( )
N 0
 
 
 b  
R ϕ +h sin ϕ
( ) ( )
 N 0 
 2 
a
 
 
 
 
r =T 0
 
CS parameter binding
 
 
rules
 
 
s =T 1
 
 
 
where:
a and b match the oblate ellipsoid (or sphere) RD values, and
−sinλ −cosλ sinϕ cosλ cosϕ cosα sinα 0
  
  
T = cosλ −sinλ sinϕ sinλ cosϕ −sinα cosα 0
  
  
0 cosϕ sinϕ 0 0 1
  
Coordinate valid-region No additional restrictions.
© ISO/IEC 2009 – All rights reserved
Element Specification
1) The LOCOCENTRIC_SURFACE_POLAR CS is induced on the tangent
plane surface.
2) The w = -h coordinate-component plane is tangent to the oblate
Notes
ellipsoid RD at the point with surface celestiodetic coordinate (λ,ϕ).
3) α is the geodetic azimuth of the v-axis (see Figure 8.2).
4) h is the ellipsoidal height of the CS origin.
References [EDM]
8.5.9 Lococentric Euclidean 3D SRFT
Lococentric Euclidean 3D SRFs shall be derived from the SRFT specified in Table 8.11.
Table 8.11 — Lococentric Euclidean 3D SRFT
Element Specification
SRFT label LOCOCENTRIC _EUCLIDEAN_3D
SRFT code 8
Lococentric Euclidean 3D SRFT
Short name and description
Euclidean 3D spatial CS with a localised origin and axes orientations
Object type Any 3D object
ORM constraint Shall be derived from any 3D ORM.
CS label LOCOCENTRIC_EUCLIDEAN_3D
CS coordinate-component The same as the CS.
names and/or symbols
Localization parameters:
q: the lococentric origin,
r: primary axis direction, and
Template parameters
s: secondary axis direction.
Constraints:
r and s are orthonormal vectors.
CS parameter binding rules The template parameters are the CS parameters
Coordinate valid-region No additional restrictions.
1) A CELESTIOCENTRIC SRFT is special case of an instance of this
SRFT with q = 0 0 0 , r = 1 0 0 , s = 0 1 0 , and a physical
( ) ( ) ( )
object.
2) A LOCAL_SPACE_RECTANGULAR_3D SRFT is special case of an
instance of this SRFT with q = 0 0 0 , and an abstract object.
( )
Notes
3) A LOCAL_TANGENT_SPACE_EUCLIDEAN SRFT is special case of an
instance of this SRFT with q, r, s, satisfying the SRFT
LOCAL_TANGENT_SPACE_EUCLIDEAN CS parameter binding rules
and ORM constraint.
4) This SRTF is required for the SRM treatment of directions (see 10.5)
© ISO/IEC 2009 – All rights reserved
Element Specification
References [EDM]
8.5.10 Celestiomagnetic SRFT
Celestiomagnetic SRFs shall be derived from the SRFT specified in Table 8.12.
Table 8.12 — Celestiomagnetic SRFT
Element Specification
SRFT label CELESTIOMAGNETIC
SRFT code 9
celestiomagnetic SRFT
Short name and description An equatorial spherical CS based SRFT aligned with the magnetic dipole of a
celestial object.
A planet or rotating satellite in a solar system with a magnetic dipole axis
Object type
distinct from its rotational axis.
ORM constraint Based on ORMT BI_AXIS_ORIGIN_3D and OBRS CELESTIOMAGNETIC.
CS label EQUATORIAL_SPHERICAL
CS coordinate-component
The same as the CS.
names and/or symbols
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
1) See 7.5.8.
2) When the object is Earth, this SRFT is referred to as a geomagnetic
Notes SRFT.
3) These SRFs are typically used at radii where the magnetic field is
approximated by a dipole.
References [CRUS]
8.5.11 Equatorial inertial SRFT
Equatorial inertial SRFs shall be derived from the SRFT specified in Table 8.13.
Table 8.13 — Equatorial inertial SRFT
Element Specification
SRFT label EQUATORIAL_INERTIAL
SRFT code 10
© ISO/IEC 2009 – All rights reserved
Element Specification
equatorial Inertial SRFT
Short name and description An equatorial spherical CS based SRF aligned with the equator of a planet
and the direction to the Sun at the vernal equinox (at a specified epoch).
A planet in the solar system for which the ecliptic plane is distinct from the
Object type
equatorial plane.
Based on ORMT BI_AXIS_ORIGIN_3D and
ORM constraint
OBRS EQUATORIAL_INERTIAL.
CS label
EQUATORIAL_SPHERICAL
λ : right ascension (ra)
CS coordinate-component
θ : declination (dec)
names and/or symbols
ρ : radius or range(r)
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
1) See 7.5.2.
Notes
2) Star catalogues use right ascension and declination to specify directions.
References [SEID]
8.5.12 Solar ecliptic SRFT
Solar ecliptic SRFs shall be derived from the SRFT specified in Table 8.14.
Table 8.14 — Solar ecliptic SRFT
Element Specification
SRFT label SOLAR_ECLIPTIC
SRFT code 11
solar ecliptic SRFT
Short name and description An equatorial spherical CS based SRF aligned with the ecliptic plane of a
planet and the direction of the Sun.
Object type A planet in the solar system.
ORM constraint Based on ORMT BI_AXIS_ORIGIN_3D and OBRS SOLAR_ECLIPTIC.
CS label EQUATORIAL_SPHERICAL
CS coordinate-component
The same as the CS.
names and/or symbols
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
Notes See 7.5.3.
References [HAPG]
© ISO/IEC 2009 – All rights reserved
8.5.13 Solar equatorial SRFT
Solar equatorial SRFs shall be derived from the SRFT specified in Table 8.15.
Table 8.15 — Solar equatorial SRFT
Element Specification
SRFT label SOLAR_EQUATORIAL
SRFT code 12
solar equatorial SRFT
Short name and description An equatorial spherical CS based planet centred SRF aligned with the ecliptic
plane and the rotational axis of the Sun.
A planet in the solar system for which the ecliptic plane is distinct from the
Object type
equatorial plane.
ORM constraint Based on ORMT BI_AXIS_ORIGIN_3D and OBRS SOLAR_EQUATORIAL.
CS label EQUATORIAL_SPHERICAL
CS coordinate-component
The same as the CS.
names and/or symbols
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
Notes See 7.5.4.
References [CRUS]
8.5.14 Solar magnetic ecliptic SRFT
Solar magnetic ecliptic SRFs shall be derived from the SRFT specified in Table 8.16.
Table 8.16 — Solar magnetic ecliptic SRFT
Element Specification
SRFT label SOLAR_MAGNETIC_ECLIPTIC
SRFT code 13
solar magnetic ecliptic SRFT
A Euclidean 3D CS based planet centred SRF aligned with the direction to
Short name and description
the Sun and the plane determined by that direction and the magnetic dipole
of the planet.
Object type A planet in the solar system with a magnetic dipole.
Based on ORMT BI_AXIS_ORIGIN_3D and
ORM constraint
OBRS SOLAR_MAGNETIC_ECLIPTIC.
CS label
EUCLIDEAN_3D
CS coordinate-component
The same as the CS.
names and/or symbols
© ISO/IEC 2009 – All rights reserved
Element Specification
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
1) See 7.5.9.
Notes
2) In the case of planet Earth, this SRFT is also known as a geocentric solar
magnetospheric SRFT.
References [CRUS]
8.5.15 Solar magnetic dipole SRFT
Solar magnetic dipole SRFs shall be derived from the SRFT specified in Table 8.17.
Table 8.17 — Solar magnetic dipole SRFT
Element Specification
SRFT label SOLAR_MAGNETIC_DIPOLE
SRFT code 14
solar magnetic dipole SRFT
Short name and description A Euclidean 3D CS based planet centred SRF with the z-axis aligned with the
magnetic dipole and the xz-plane containing the Sun.
A planet in the solar system with a magnetic dipole axis distinct from its
Object type
rotational axis.
Based on ORMT BI_AXIS_ORIGIN_3D and
ORM constraint
OBRS SOLAR_MAGNETIC_DIPOLE.
CS label EUCLIDEAN_3D
CS coordinate-component
The same as the CS.
names and/or symbols
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
Notes See 7.5.10.
References [CRUS] , [BHAV]
8.5.16 Heliospheric Aries ecliptic SRFT
Heliospheric Aries ecliptic SRFs shall be derived from the SRFT specified in Table 8.18.
Table 8.18 — Heliospheric Aries ecliptic SRFT
Element Specification
SRFT label HELIOSPHERIC_ARIES_ECLIPTIC
© ISO/IEC 2009 – All rights reserved
Element Specification
SRFT code 15
Heliospheric Aries ecliptic SRFT
An equatorial spherical CS based Sun centred SRF with zero spherical
Short name and description
latitude aligned with the ecliptic plane and zero longitude aligned to the first
point of Aries.
Object type Sun.
Based on ORMT BI_AXIS_ORIGIN_3D and
ORM constraint
OBRS HELIOCENTRIC_ARIES_ECLIPTIC.
CS label
EQUATORIAL_SPHERICAL
CS coordinate-component
The same as the CS.
names and/or symbols
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
Notes See 7.5.5.
References [HAPG]
8.5.17 Heliospheric Earth ecliptic SRFT
Heliospheric Earth ecliptic SRFs shall be derived from the SRFT specified in Table 8.19.
Table 8.19 — Heliospheric Earth ecliptic SRFT
Element Specification
SRFT label HELIOSPHERIC_EARTH_ECLIPTIC
SRFT code
heliospheric Earth ecliptic SRFT
An equatorial spherical CS based Sun centred SRF with zero spherical
Short name and description
latitude aligned with the ecliptic plane and zero longitude aligned to the centre
of the Earth.
Object type Sun.
Based on ORMT BI_AXIS_ORIGIN_3D and
ORM constraint
OBRS HELIOCENTRIC_PLANET_ECLIPTIC.
CS label EQUATORIAL_SPHERICAL
CS coordinate-component
The same as the CS.
names and/or symbols
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
Notes See 7.5.6.
References [HAPG]
© ISO/IEC 2009 – All rights reserved
8.5.18 Heliospheric Earth equatorial SRFT
Heliospheric Earth equatorial SRFs shall be derived from the SRFT specified in Table 8.20.
Table 8.20 — Heliospheric Earth equatorial SRFT
Element Specification
SRFT label HELIOSPHERIC_EARTH_EQUATORIAL
SRFT code
heliospheric Earth equatorial SRFT
An equatorial spherical CS based Sun centred SRF with zero spherical
Short name and description
latitude aligned with the equator of the Sun and zero longitude aligned to the
centre of the Earth.
Object type Sun.
Based on ORMT BI_AXIS_ORIGIN_3D and
ORM constraint
OBRS HELIOCENTRIC_PLANET_EQUATORIAL with respect to Earth.
CS label EQUATORIAL_SPHERICAL
CS coordinate-component
The same as the CS.
names and/or symbols
Template parameters none
CS parameter binding rules none
Coordinate valid-region No additional restrictions.
Notes See 7.5.7.
References [HAPG]
8.5.19 Mercator SRFT
Mercator SRFs shall be derived from the SRFT specified in Table 8.21.
Table 8.21 — Mercator SRFT
Element Specification
SRFT label MERCATOR
SRFT code 18
Mercator SRFT.
Short name and description A Mercator and augmented Mercator map projection of the oblate or sphere RD
component of the ORM.
Object type
physical
Shall be derived from:
ORM constraint ORMT OBLATE_ELLIPSOID, OBLATE_ELLIPSOID_ORIGIN,
SPHERE, or SPHERE_ORIGIN.
CS label MERCATOR
© ISO/IEC 2009 – All rights reserved
Element Specification
CS coordinate-component Same as the CS.
names and/or symbols h: ellipsoidal height is the vertical coordinate-component.
λ : longitude of origin (-π < λ ≤ π)
origin origin
k : central scale (0< k ≤ 1)
0 0
Template parameters
u : false easting
F
v : false northing
F
CS parameters match RD values:
Oblate ellipsoid RD case -
2 2
Major semi-axis a, ε = 1−b a
CS parameter binding rules
( )
Sphere RD case -
Radius a, ε =0
Coordinate valid-region No additional restrictions.
1. The augmented Mercator CS induces the Mercator CS on the zero-value
vertical coordinate-component surface (which coincides with the RD
surface).
Notes
2. True scale (point distortion = 1) may be specified at a given latitude ϕ by
setting: k = 1a R ϕ cosϕ .
( ) ( ) ( )
0 N 1 1
References [SNYD]
8.5.20 Oblique Mercator spherical SRFT
Oblique Mercator spherical SRFs shall be derived from the SRFT specified in Table 8.22.
Table 8.22 — Oblique Mercator spherical SRFT
Element Specification
SRFT label OBLIQUE_MERCATOR_SPHERICAL
SRFT code 19
Oblique Mercator SRFT for a sphere ORM.
Short name and description An oblique Mercator and augmented oblique Mercator map projection of the
sphere RD component of the ORM.
Object type physical
ORM constraint Shall be derived from ORMT SPHERE or SPHERE_ORIGIN.
CS label OBLIQUE_MERCATOR_SPHERICAL
CS coordinate-component Same as the CS.
names and/or symbols h: ellipsoidal height is the vertical coordinate-component.
© ISO/IEC 2009 – All rights reserved
Element Specification
λ ,ϕ : first point on the central line
( )
1 1
λ ,ϕ : second point on central line
( )
2 2
k : central scale (0< k ≤ 1)
0 0
u : false easting
F
v : false northing
F
Template parameters
λ ,ϕ and λ ,ϕ are two distinct points on the shortest great circle
( ) ( )
1 1 2 2
arc on the sphere representing the desired central line, k is the
point distortion on the central line, and
π π π π
− <ϕ < , − <ϕ < , ϕ + ϕ > 0,
1 2 1 2
2 2 2 2
−π <λ ≤ π, −π <λ ≤ π, λ ≠λ , and λ −λ ≠ π.
1 2 1 2 1 2
The CS parameter R matches the RD value:
Radius R = r.
CS parameter binding rules
The values of λ ,ϕ ,λ ,ϕ ,k ,u , and v match
1 1 2 2 0 F F
the corresponding template parameters.
Coordinate valid-region No additional restrictions.
The augmented oblique Mercator CS induces the oblique Mercator CS on the
Notes zero-value vertical coordinate-component surface (which coincides with the RD
surface).
References [SNYD]
8.5.21 Transverse Mercator SRFT
Transverse Mercator SRFs shall be derived from the SRFT specified in Table 8.23.
Table 8.23 — Transverse Mercator SRFT
Element Specification
SRFT label TRANSVERSE_MERCATOR
SRFT code 20
Transverse Mercator SRFT
Short name and description A transverse Mercator and augmented transverse Mercator map projection of
the oblate or sphere RD component of the ORM.
Object type physical
Shall be derived from:
ORM constraint ORMT OBLATE_ELLIPSOID, OBLATE_ELLIPSOID_ORIGIN,
SPHERE, or SPHERE_ORIGIN.
CS label TRANSVERSE_MERCATOR
CS coordinate-component Same as the CS.
names and/or symbols h: ellipsoidal height is the vertical coordinate-component.
© ISO/IEC 2009 – All rights reserved
Element Specification
λ : longitude of origin (-π < λ ≤ π)
origin origin
ϕ : latitude of origin (−π 2<ϕ < π 2)
origin origin
Template parameters k : central scale (0< k ≤ 1)
0 0
u : false easting
F
v : false northing
F
CS parameters match RD values:
Oblate ellipsoid RD case -
2 2
Major semi-axis a, ε = 1−b a
CS parameter binding rules ( )

Sphere RD case -
Radius a, ε =0
Coordinate valid-region No additional restrictions.
The augmented transverse Mercator CS induces the transverse Mercator CS
Notes on the zero-value vertical coordinate-component surface (which coincides
with the RD surface).
References [SNYD]
8.5.22 Lambert conformal conic SRFT
Lambert conformal conic SRFs shall be derived from the SRFT specified in Table 8.24.
Table 8.24 — Lambert conformal conic SRFT
Element Specification
SRFT label LAMBERT_CONFORMAL_CONIC
SRFT code
Lambert conformal conic SRFT
Short name and description A Lambert conformal conic and augmented Lambert conformal conic map
projection of the oblate or sphere RD component
...

6 Temporal coordinate systems
6.1 Introduction
There is a requirement to identify time as well as location in environmental representation. Time is that
physical quantity perceived as the continued progress of existence measured by an observer as events which
are relatively ordered as “before” or “after” and which, at a given point in time, give rise to the notions of past,
present and future. Time and location are often used together by an application to describe when a given
condition exists or when an object was present at a given location.
This International Standard uses the concept of time in several ways. Dynamic systems are treated as
systems with a time parameter. These systems reduce to the case of a static relationship by fixing a value for
the time parameter. Object reference model bindings are often based on physical measurements of objects or
systems that change with time. Time is also used to identify the epoch for which these measurements are
applicable.
A temporal coordinate system is a Euclidean 1D CS (see Table 5.35) that assigns distinct coordinates to
distinct times so that larger coordinate values are assigned to later times. This International Standard uses
Coordinated Universal Time (UTC) (see 6.2.4) to provide a temporal coordinate system that enables a unique
temporal coordinate to be assigned to an event. In this International Standard, times and dates refer to UTC
unless explicitly indicated otherwise.
6.2 Temporal coordinate systems
6.2.1 Integrated and dynamic temporal coordinate systems
An integrated temporal coordinate system is a Euclidean 1D CS (see Table 5.35) based on a unit of duration
that is derived from a physical phenomenon. Fixing an origin (called the epoch) and then integrating
continuously by accumulating units of the duration specifies an integrated temporal coordinate system.
EXAMPLE  The wave length of certain atomic energy emissions determine a wave period which serves as the
physical duration that is accumulated to specify atomic clock time.
A dynamic temporal coordinate system is a Euclidean 1D CS (see Table 5.35) based on data derived from the
observation of a dynamic physical system, typically planetary motion. The specification of a dynamic temporal
coordinate system depends on the observed system being described by a mathematical model where time is
one of the parameters that unambiguously identifies the configuration of the system. The time measurement
can then be considered to be a measurement of position with units defined as a specified duration. Fixing an
origin by specifying the initial conditions of the physical system and then continuously accumulating units of
the duration specifies a dynamic temporal coordinate system.
A dynamic temporal coordinate system differs from an integrated temporal coordinate system in that the
former ties a mathematical model to the state of a physical system while the latter accumulates the duration of
a periodic phenomenon.
6.2.2 Universal time
Universal time (UT) is a general designation of a set of dynamic temporal coordinate systems based on the
rotation of the Earth. There are different forms of UT whose values may differ by a few hundredths of a
second:
a) Universal Time observed (UT0) is the mean solar time of the prime meridian obtained from direct
astronomical observation.
© ISO/IEC 2009 – All rights reserved

b) Universal Time polar motion corrected (UT1) is UT0 corrected for the effects of small movements of
the Earth relative to the axis of rotation (polar variation).
c) Universal Time Earth rotation corrected (UT2) is UT1 corrected for the effects of a small seasonal
fluctuation in the rate of rotation of the Earth.
Complete definitions of UT0, UT1, UT2, and the concepts involved in their definitions may be found in the
publications of the International Earth Rotation Service [IERS] that maintains these three temporal coordinate
systems.
6.2.3 International atomic
...

3 Terms, definitions, symbols, and abbreviated terms
3.1 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
NOTE  Symbols or abbreviations used as representations for the term are listed immediately following the term. After
the definition, where necessary, an additional clarifying note may be provided. Terms defined in the body of this document
are presented in italics at the point where they are defined. The Index provides a directory of those terms defined in the
body of this International Standard.
3.1.1
Earth gravitational model
spherical harmonic expansion of the gravitational field potential
NOTE  Rotational effects are not included in this model; gravity includes rotational effects.
3.1.2
ecliptic plane
plane defined by the orbit of a planet at a point in time
3.1.3
equatorial plane
plane through a designated centre of an object and normal to the rotational axis of the object
3.1.4
geodetic datum
datum describing the relationship of a coordinate system to the Earth
[ISO 19111]
NOTE  In most cases, the geodetic datum includes an ellipsoid definition.
3.1.5
north pole
that pole of rotation that lies on the north side of the invariable plane of the solar system
[RIIC]
NOTE 1 Some planets have retrograde rotation with respect to this definition.
NOTE 2 Map north (see Clause 5) may be unrelated to this direction.
3.1.6
spatial object
physical or virtual object to which spatial information applies
3.1.7
spatial operation
mathematical function that re-expresses coordinates, directions, and/or distances expressed in one spatial
reference frame in terms of a different spatial reference frame or a mathematical function for distance or other
geometric quantities within a single spatial reference frame
3.2 Notation, symbols and abbreviated terms
Table 3.1 lists the mathematical notation conventions used in this document.
© ISO/IEC 2009 – All rights reserved

Table 3.1 — Mathematical notation

Style Use Examples
lower case, bold, points, vectors x, p
italic
lower case, italic variables, scalars, a, b, f, x-axis
scalar-valued functions,
axes of a linear
coordinate system
upper case, bold, vector-valued functions, F, G, M
italic matrices
upper case, italic sets S, T
Upper case italic letter symbols are also used for scalar-valued functions that are customarily capitalized.
EXAMPLE  R in Table 5.6.
N
Table 3.2 lists the symbols used in this document.
Table 3.2 — Symbols
Symbol Definition
...

Contents Page
Foreword.xix

0    Introduction.xxi
0.1    Purpose .xxi
0.2    Design criteria.xxi

1 Scope . 1

2 Normative references. 3

3 Terms, definitions, symbols, and abbreviated terms . 5
3.1 Terms and definitions. 5
3.2 Notation, symbols and abbreviated terms. 5

4 Concepts . 9
4.1 Introduction . 9
4.2 Spatial objects and object-space . 10
4.3 Position-space and normal embeddings. 11
4.4 Reference datums. 13
4.5 Object reference models. 15
4.6 Coordinate systems . 17
4.6.1 Abstract coordinate systems. 17
4.6.2 Temporal coordinate systems. 20
4.6.3 Spatial coordinate systems . 20
4.7 Spatial reference frames. 22
4.8 Designated spatial surfaces and vertical offsets. 23
4.9 Spatial reference frame operations. 24
4.10 Application program interface . 25
4.11 SRM units. 25
4.12 Profiles . 25
4.13 Registration . 25

5 Abstract coordinate systems. 27
5.1 Introduction . 27
5.2 Preliminaries . 27
5.3 Abstract CS . 27
5.4 CS types. 29
5.5 Coordinate surfaces, induced surface CSs, and coordinate curves. 30
5.5.1 Introduction . 30
5.5.2 Coordinate-component surfaces and induced surface CSs . 30
5.5.3 Coordinate-component curves. 31
5.6 CS properties . 32
5.6.1 Linearity. 32
5.6.2 Orthogonality. 33
5.6.3 Linear CS properties: Cartesian, and orthonormal . 33
5.6.4 CS right-handedness and coordinate-component ordering. 33
5.7 CS localization . 34
5.8 Map projection coordinate systems . 35
5.8.1 Map projections. 35
5.8.2 Map projection as a surface CS. 36
5.8.3 Map projection geometry. 37
5.8.4 Relationship to projection functions . 40
5.8.5 Map projection CS common parameters . 43
5.8.6 Augmented map projections . 44
iii
© ISO/IEC 2009 – All rights reserved

5.9 CS specifications.45
5.9.1 Specification table elements and common functions and parameters.45
5.9.2 Euclidean 3D CS specification .49
5.9.3 Lococentric Euclidean 3D CS specification.50
5.9.4 Spherical CS specification.51
5.9.5 Lococentric spherical CS specification.53
5.9.6 Azimuthal spherical CS specification .54
5.9.7 Lococentric azimuthal spherical CS specification .56
5.9.8 Geodetic 3D CS specification.57
5.9.9 Planetodetic 3D specification .60
5.9.10 Cylindrical CS specification.62
5.9.11 Lococentric cylindrical CS specification .63
5.9.12 Mercator CS specification .64
5.9.13 Oblique Mercator spherical CS specification .67
5.9.14 Transverse Mercator CS specification .70
5.9.15 Lambert conformal conic CS specification .74
5.9.16 Polar stereographic CS specification .77
5.9.17 Equidistant cylindrical CS specification.80
5.9.18 Surface geodetic CS specification .82
5.9.19 Surface planetodetic CS specification.84
5.9.20 Lococentric surface Euclidean CS specification .85
5.9.21 Lococentric surface azimuthal CS specification.87
5.9.22 Lococentric surface polar CS specification .88
5.9.23 Euclidean 2D CS specification .90
5.9.24 Lococentric Euclidean 2D CS specification.91
5.9.25 Azimuthal CS specification.93
5.9.26 Lococentric azimuthal CS specification.94
5.9.27 Polar CS specification .95
5.9.28 Lococentric polar CS specification .96
5.9.29 Euclidean 1D CS specification .98

6 Temporal coordinate systems .101
6.1 Introduction.101
6.2 Temporal coordinate systems .101
6.2.1 Integrated and dynamic temporal coordinate systems .101
6.2.2 Universal time.101
6.2.3 International atomic time .102
6.2.4 Coordinated universal time.102
6.3 Specified temporal coordinate systems .102
6.4 Registered temporal coordinate systems.103

7 Reference datums, embeddings, and object reference models.105
7.1 Introduction.105
7.2 Reference datums.105
7.2.1 Introduction.105
7.2.2 Reference datums.105
7.2.3 Ellipsoidal RDs .108
7.2.4 RDs associated with physical objects .109
7.2.5 RD binding.111
7.3 Normal embeddings of position-space into object-space .111
7.3.1 Normal embeddings .111
7.3.2 Specification of 3D similarity transformations .112
7.3.3 Specification of 2D similarity transformations .114
7.4 Object reference model.114
7.4.1 Introduction.114
7.4.2 ORM .115
7.4.3 Binding constraint.116
7.4.4 ORM template .116
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7.4.5 Standardized ORMs. 125
7.5 Object binding rules for ORMT BI_AXIS_ORIGIN_3D realizations. 129
7.5.1 Object binding rule set . 129
7.5.2 Equatorial inertial . 130
7.5.3 Solar ecliptic. 132
7.5.4 Solar equatorial . 133
7.5.5 Heliocentric Aries ecliptic . 134
7.5.6 Heliocentric planet ecliptic. 135
7.5.7 Heliocentric planet equatorial. 136
7.5.8 Celestiomagnetic. 137
7.5.9 Solar magnetic ecliptic . 139
7.5.10 Solar magnetic dipole. 140

8 Spatial reference frames. 143
8.1 Introduction . 143
8.2 Spatial coordinate systems . 143
8.3 Spatial reference frame. 144
8.3.1 Specification. 144
8.3.2 SRF specification elements . 144
8.4 SRF induced surface spatial reference frame. 147
8.5 SRF templates . 147
8.5.1 Introduction . 147
8.5.2 Celestiocentric SRFT . 150
8.5.3 Local space rectangular 3D SRFT. 150
8.5.4 Celestiodetic SRFT . 151
8.5.5 Planetodetic SRFT. 152
8.5.6 Local tangent space Euclidean SRFT. 153
8.5.7 Local tangent space azimuthal spherical SRFT . 154
8.5.8 Local tangent space cylindrical SRFT. 155
8.5.9 Lococentric Euclidean 3D SRFT. 157
8.5.10 Celestiomagnetic SRFT . 158
8.5.11 Equatorial inertial SRFT. 158
8.5.12 Solar ecliptic SRFT . 159
8.5.13 Solar equatorial SRFT. 160
8.5.14 Solar magnetic ecliptic SRFT. 160
8.5.15 Solar magnetic dipole SRFT . 161
8.5.16 Heliospheric Aries ecliptic SRFT. 161
8.5.17 Heliospheric Earth ecliptic SRFT . 162
8.5.18 Heliospheric Earth equatorial SRFT . 163
8.5.19 Mercator SRFT. 163
8.5.20 Oblique Mercator spherical SRFT. 164
8.5.21 Transverse Mercator SRFT . 165
8.5.22 Lambert conformal conic SRFT . 166
8.5.23 Polar stereographic SRFT. 167
8.5.24 Equidistant cylindrical SRFT . 168
8.5.25 Local space rectangular 2D SRFT. 169
8.5.26 Local space azimuthal 2D SRFT . 170
8.5.27 Local space polar 2D SRFT. 170
8.6 Standardized SRFs. 171
8.6.1 Introduction . 171
8.6.2 British national grid. 173
8.6.3 UK ordnance survey GRS80 grid. 173
8.6.4 Delaware (US) state plane coordinate system . 173
8.6.5 Geocentric WGS 1984 . 174
8.6.6 Geodetic Australia 1984. 174
8.6.7 Geodetic WGS 1984 . 175
8.6.8 Geodetic north american 1983. 175
8.6.9 Irish grid . 175
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8.6.10 Irish transverse Mercator.176
8.6.11 Lambert-93 .176
8.6.12 Lambert II étendu (Lambert II wide) .177
8.6.13 Mars planetocentric .177
8.6.14 Mars planetographic.178
8.6.15 Maryland (US) state plane coordinate system .178
8.7 Standardized SRF sets .179
8.7.1 Introduction.179
8.7.2 Alabama (US) state plane coordinate system.181
8.7.3 GTRS global coordinate system (GCS) .182
8.7.4 Japan plane coordinate system .184
8.7.5 Lambert NTF .189
8.7.6 Universal polar stereographic.191
8.7.7 Universal transverse Mercator .192
8.7.8 Wisconsin (US) state plane coordinate system.193

9 Designated spatial surfaces and vertical offsets.195
9.1 Introduction.195
9.2 Designated spatial surface.195
9.3 Vertical offset surface.196
9.4 Geoidal separation .197
9.5 Vertical offset height and elevation .197
9.6 Use of vertical offset height in spatial referencing.198
9.7 Other vertical measurements .198
9.8 Geoidal and equipotential DSS specifications .199

10 SRF operations .203
10.1 Introduction.203
10.2 Notation and terminology .203
10.3 Operations on ORMs.204
10.3.1 Introduction.204
10.3.2 ORMs for a single object.204
10.3.3 Relating ORMs for different objects .206
10.4 Operations to change spatial coordinates between SRFs.206
10.4.1 Introduction.206
10.4.2 Change coordinate SRF operation.207
10.4.3 The matched normal embeddings case .209
10.4.4 Map projection SRF and celestiodetic SRF with matched normal embeddings case.209
10.4.5 Linear orthonormal 3D SRF to linear orthonormal 3D SRF cases.210
10.4.6 Changing abstract space linear SRF coordinates to a linear SRF in the space of another object
......................................................................................................................................................211
10.5 Spatial directions and change SRF operations on directions .212
10.5.1 Introduction.212
10.5.2 Specification of direction .213
10.5.3 Changing the reference coordinate of a direction .214
10.5.4 Changing the SRF representation of a direction.215
10.6 Euclidean distance .216
10.7 Geodesic distance and azimuth on an oblate ellipsoid .216
10.7.1 Introduction.216
10.7.2 Geodesic distance.216
10.7.3 Geodetic azimuth .217

11 Application program interface.219
11.1 Introduction.219
11.2 Non-object data types .220
11.2.1 Overview.220
11.2.2 Abbreviations.220
11.2.3 Numbers.221
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11.2.4 Logicals . 221
11.2.5 Object_Reference . 222
11.2.6 Enumerated data types. 222
11.2.7 Selection data types. 223
11.2.8 Array types . 226
11.2.9 Structured data types. 227
11.3 Object classes. 231
11.3.1 Introduction . 231
11.3.2 Class specification format . 232
11.3.3 LifeCycleObject . 233
11.3.4 Private objects. 234
11.3.5 Abstract classes . 236
11.3.6 SRF concrete subclasses of BaseSRF2D . 261
11.3.7 SRF concrete subclasses of BaseSRF3D . 263
11.3.8 SRF concrete subclasses of BaseSRFwithTangentPlaneSurface . 273
11.3.9 SRF concrete subclass of BaseSRFwithEllipsoidalHeight . 276
11.3.10 SRF concrete subclasses of BaseSRFMapProjection. 278
11.4 Standard SRFs. 286
11.5 SRF set classes . 286
11.6 Implementation support query functions. 287
11.7 Object inheritance hierarchy . 288
11.8 Method precedence for life cycle objects and examples . 292
11.9 Data storage structures. 293
11.9.1 Introduction . 293
11.9.2 SRFT_Parameters . 293
11.9.3 SRFS_Info. 294
11.9.4 SRF_Parameters_Info_Code. 294
11.9.5 SRF_Parameters_Info . 294
11.9.6 SRF_Reference_Surface_Info. 295
11.9.7 Coordinate structures. 295
11.9.8 Spatial_Coordinate_Code. 298
11.9.9 Coordinate. 299
11.9.10 RD_Code . 300
11.9.11 OBRS_Code . 300

12 Profiles . 301
12.1 Introduction . 301
12.2 Profile specification . 301
12.3 Default profile . 303

13 Registration . 305
13.1 Introduction . 305
13.2 Specification elements for SRM registered items . 306
13.2.1 Introduction . 306
13.2.2 Label. 306
13.2.3 Code. 307
13.2.4 Description . 308
13.2.5 References. 309
13.3 Guidelines for specific SRM concepts . 309
13.3.1 Guidelines for registration of abstract CSs . 309
13.3.2 Guidelines for registration of temporal CSs . 310
13.3.3 Guidelines for registration of RDs. 310
13.3.4 Guidelines for registration of ORMTs. 311
13.3.5 Guidelines for registration of ORMs. 311
13.3.6 Guidelines for registration of RTs . 312
13.3.7 Guidelines for registration of OBRSs. 312
13.3.8 Guidelines for registration of SRFTs. 313
13.3.9 Guidelines for registration of SRFs. 314
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13.3.10 Guidelines for registration of SRF sets and their members .314
13.3.11 Guidelines for registration of DSSs .316
13.3.12 Guidelines for registration of profiles.316

14 Conformance.319
14.1 Introduction.319
14.2 Functional implementation conformance .319
14.3 Conformance of exchange formats.320
14.4 Conformance of language bindings of the SRM API .320
14.5 Conformance of applications that use the SRM API.321
14.6 Conformance of specifications that reference this International Standard .321

Annex A Mathematical foundations .323
A.1 Introduction.323
n
A.2 R as a real vector space .323
n
A.3 The point set topology of R .324
n
A.4 Smooth functions on R .324
A.5 Functional composition.325
A.6 Smooth surfaces in R .326
A.6.1 Implicit definition.326
A.6.2 Ellipsoid surfaces .326
n
A.7 Smooth curves in R .327
A.7.1 Parametric definition.327
A.7.2 Implicit definition.329
A.7.3 Arc length and geodesic distance .330
A.8 Special functions .330
A.8.1 Double argument arctangent function .330
A.8.2 Jacobian elliptic functions.330
A.9 Projection function.331
A.9.1 Geometric projection functions into a developable surface .331
A.9.2 Planar projection functions.331
A.9.3 Cylindrical projection function.333
A.9.4 Conic projection function.333

Annex B Implementation notes .335
B.1 Introduction.335
B.2 General observations .335
B.2.1 Finite precision .335
B.2.2 Computational efficiency .
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