[ Table of contents ]

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:

                   

Figure 8.1 — A spatial embedding of a surface CS

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,

d)       (optionally) k^th 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

 

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:

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, and

b)       all augmented map projections.

As a further restriction, some CSs are based on spheres only. CS OBLIQUE_MERCATOR_SPHERICAL has this restriction.

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 specification of an SRF may optionally assign SRF-specific names to the k^th coordinate-components. The name assignment shall reflect the common use in the intended application domain.

EXAMPLE            For a spherical CS, the assignment of SRF-specific names to the k^th 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).

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:

k^th coordinate-component belongs to a non-empty interval of real numbers.

An extended valid-region specification is a finite (or empty) list of coordinate-component constraints of the form:

k^th coordinate-component belongs to an interval of real numbers, where  has been specified and.

Angular coordinate-component intervals shall be evaluated modulo 2p to represent an interval of the unit circle. Thus,

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 satisfy, then:

                                                               

EXAMPLE 2         The SRF is based on a transverse Mercator map projection (see SRFT TRANSVERSE_MERCATOR).
Valid-region specification:                         0 ≤ u ≤ 10 000 000,          0 ≤ v ≤ 500 000
Extended valid-region specification:        -100 < u,                            -100 < v
In this example, and are closed bounded intervals, and and are open semi-bounded intervals.

EXAMPLE 3         The SRF is based on a transverse Mercator map projection (see SRFT TRANSVERSE_MERCATOR).
Valid-region specification:                         -78º ≤ λ < -72º,          0º ≤ θ < 84º
Extended valid-region specification:        -78,5º ≤ λ < -71,5º
In this example, and are left-closed, right-open bounded intervals, as is . is not specified. This indicates that there are no constraints for latitude (except for the CS domain definition) in the extended valid-region specification.

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 CS on each coordinate-component surface (see 5.5.2). An SRF specification may optionally identify the 3^rd 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.

The CS GEODETIC and the CS PLANETODETIC 3^rd coordinate-components (h: ellipsoidal height), and the 3^rd 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.

An SRF realized from the SRF template LOCAL_TANGENT_SPACE_EUCLIDEAN specification (see 8.5.6) or the SRF template LOCAL_TANGENT_SPACE_CYLINDRICAL specification (see 8.5.8), the 3^rd coordinate-component, height, is specified as the vertical coordinate-component. In these cases, the zero-value vertical coordinate-component surface is a plane that is tangent to the oblate ellipsoid (or sphere) RD of the ORM. SRF templates are defined in 8.5.

The zero-value 3^rd 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 q, depression/elevation angle, is specified as a vertical coordinate. The zero-value vertical coordinate-component surface is a plane that is tangent to the oblate ellipsoid (or sphere) RD of the ORM.

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

An 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

 

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.

Table 8.3 — SRFT directory

 

8.5.2        Celestiocentric SRFT

Celestiocentric SRFs shall be derived from the SRFT specified in Table 8.4.

Table 8.4 — Celestiocentric SRFT

 

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

 

8.5.4        Celestiodetic SRFT

Celestiodetic SRFs shall be derived from the SRFT specified in Table 8.6.

Table 8.6 — Celestiodetic SRFT

 

8.5.5        Planetodetic SRFT

Planetodetic SRFs shall be derived from the SRFT specified in Table 8.7.

Table 8.7 — Planetodetic SRFT