[ Table of contents ]

7        Reference datums, embeddings, and object reference models

7.1.      Introduction

This International Standard specifies reference datums as geometric primitives in position-space that are used to model aspects of object-space through a process termed reference datum binding. A reference datum binding is an identification of a reference datum in position-space with a corresponding constructed entity in object-space (see 7.2). Reference datums for celestial bodies of interest are specified in Annex D.

A normal embedding is a distance-preserving function from position-space to object-space. A normal embedding establishes a position-space model of object-space. The image of a bound reference datum under a normal embedding may or may not coincide with the constructed entity of the reference datum binding. If they coincide, the reference datum binding and the normal embedding are said to be compatible (see 7.3).

A set of bound reference datums can be selected so as to be compatible with only one normal embedding. In this way, a set of bound reference datums with properly constrained relationships can specify a unique normal embedding. Such a constrained set of bound reference datums is called an object reference model. Object reference models that use the same set of reference datum primitives and similar binding constraints are abstracted in the notion of an object reference model template. Object reference model templates provide a uniform method of object reference model specification (see 7.4).

Object reference models for celestial objects of interest are specified in Annex E. For these celestial objects, one object reference model is designated as the reference model for the object. The transformation from each object reference model to the reference model for the object is termed the reference transformation. Time-independent reference transformations are also specified.

Object-specific rules to bind reference datums in a way that is compatible with the binding constraints of an object reference model template are defined in 7.5. These object-specific binding rules are used to provide a uniform method of specifying object reference models for specific dynamically-related celestial bodies.

7.2.      Reference datums

7.2.1        Introduction

A reference datum (RD) is a geometric primitive in position-space that is used to model an aspect of object-space through a process termed RD binding. In this International Standard, the reference datum concept is defined for 1D, 2D, and 3D position-spaces. In the 2D and 3D cases, this International Standard specifies a small set of reference datums for use in its own specifications. This set is not intended to be exhaustive. Users of this International Standard may specify additional reference datums by registration in accordance with Clause 13.

7.2.2        Reference datums

In this International Standard, an RD geometric primitive is expressed in terms of analytic geometry in position-space. RDs are designed to correspond to constructed entities of similar geometric type in an object-space through a process called RD binding (see 7.2.5). These geometric types are limited to a point, a directed curve, or an oriented surface. The analytic form of the position-space representation and its corresponding object-space geometric representation are described by category and position-space dimension in Table 7.1. An RD of a given category is specified by the parameters and/or the analytic expression of its position-space representation.

Table 7.1 — RD categories

 

This International Standard specifies 2D and 3D RDs by RD category in Table 7.4 through Table 7.8. The specification elements of those tables are defined in Table 7.2. 3D RDs based on ellipsoids are described in 7.2.3 and 7.2.4 and specified in Annex D with specification elements defined in Table 7.9. Table 7.3 is a directory of RD specification tables or, in the case of 3D RDs based on ellipsoids, RD specification directories.

Table 7.2 — RD specification elements

Element	Definition
RD label	The label for the RD (see 13.2.2).
RD code	The code for the RD (see 13.2.3).
Description	A description of the RD including any common name for the concept.
Position-space representation	The analytic formulation of the RD in position-space

 

Table 7.3 — RD specification directory

 

Table 7.4 — 2D RDs of category point

RD label	RD code	Description	Position-space representation
ORIGIN_2D	1	Origin in 2D	(0,0)
X_UNIT_POINT_2D	2	x-axis unit point in 2D	(1,0)
Y_UNIT_POINT_2D	3	y-axis unit point in 2D	(0,1)

 

Table 7.5 — 3D RDs of category point

RD label	RD code	Description	Position-space representation
ORIGIN_3D	4	Origin in 3D	(0,0,0)
X_UNIT_POINT_3D	5	x-axis unit point in 3D	(1,0,0)
Y_UNIT_POINT_3D	6	y-axis unit point in 3D	(0,1,0)
Z_UNIT_POINT_3D	7	z-axis unit point in 3D	(0,0,1)

 

Table 7.6 — 2D RDs of category directed curve

RD label	RD code	Description	Position-space representation
X_AXIS_2D	8	x-axis in 2D
Y_AXIS_2D	9	y-axis in 2D

 

Table 7.7 — 3D RDs of category directed curve

RD label	RD code	Description	Position-space representation
X_AXIS_3D	10	x-axis in 3D
Y_AXIS_3D	11	y-axis in 3D
Z_AXIS_3D	12	z-axis in 3D

 

Table 7.8 — 3D RDs of category oriented surface

RD label	RD code	Description	Position-space representation
XY_PLANE_3D	13	xy-plane
XZ_PLANE_3D	14	xz-plane
YZ_PLANE_3D	15	yz-plane

 

7.2.3        Ellipsoidal RDs

The RDs specified in this International Standard include RDs based on oblate ellipsoids, prolate ellipsoids, and tri-axial ellipsoids. These RDs are 3D and of category oriented surface. These RDs are specified based upon certain geometrically-defined parameters. The position-space representations of oblate and prolate ellipsoid RDs are expressed in the form:

When  an RD of this form is an oblate ellipsoid RD with major semi-axis a and minor semi-axis b as illustrated in Figure 7.1.

Spheres shall be considered as a special case of oblate ellipsoids. If  an oblate ellipsoid RD may be called a sphere RD. In this case, the value  is the radius of the sphere RD.

NOTE          In general usage, spheres are a limiting case of oblate, prolate, and tri-axial ellipsoids. To remove ambiguity, in this International Standard spheres are a special case of oblate ellipsoids only.

When  an RD of this form is a prolate ellipsoid RD with major semi-axis b and minor semi-axis a, as illustrated in Figure 7.1.

Instead of specifying the parameters of an oblate ellipsoid RD as the major semi-axis a and the minor semi-axis b, it is both equivalent and sometimes convenient to use the major semi-axis a and the flattening f as defined in Equation (2). The minor semi-axis b may be expressed in terms of the major semi-axis a and the flattening f as in Equation (3). The flattening of a sphere RD is zero ( f = 0).

The position-space representation of a tri-axial ellipsoid RD is expressed in the form:

The semi-axes a, b, and c shall be positive non-zero and.

Figure 7.1 — Oblate and prolate ellipsoids

 

7.2.4        RDs associated with physical objects

In the case of ellipsoid RDs intended for modelling physical objects of interest, published parameter values for these RDs are used. The specification of these RDs includes the published ellipsoid parameters and the identification of the associated physical object. The specification elements for physical object RDs are defined in Figure 7.9.

Table 7.9 — Physical object RD specification elements

 

The RDs associated with physical objects are specified in Annex D. Figure 7.10 is a directory of these RDs organized by type of ellipsoid. The semi-axis and radius parameters are unitless in position-space, but are bound to metre lengths when the RD is identified with the corresponding physical object-space constructed entity.

Table 7.10 — Physical RD specification table locations

 

Additional RDs associated with physical objects may be specified by registration in accordance with Clause 13.

7.2.5        RD binding

An RD is bound when the RD in position-space is identified with a corresponding constructed entity in object-space. In this context, a "constructed entity" is defined to mean an intrinsic, artificial, measured, or conceptual entity in object-space that is uniquely identifiable within the user's application domain. The term "corresponding" in this context means that each RD is bound to a constructed entity of the same geometric object type. That is, position-space points are bound to identified points in object-space, position-space directed lines to constructed lines or line segments in object-space, position-space directed curves to constructed curves or curve segments in object-space, position-space oriented planes to constructed planes or partial planes in object-space, and position-space oriented surfaces to constructed surfaces or partial surfaces in object-space.

When a curve or surface RD is bound, the radii of curvature on the corresponding constructed entity in object-space shall correspond to the radii of curvature in position-space. In this International Standard, in the case of physical objects, one unit in position-space corresponds to one metre in object-space. In the case of abstract objects, one unit in position-space corresponds to the designated length scale unit in the abstract object-space. In particular, the semi-axes of an ellipsoid RD shall correspond to the semi-axes of the constructed ellipsoid to which it is bound.

If the constructed entity of an RD binding is fixed in position with respect to object-space, then the RD binding shall be called an object-fixed RD binding. This definition assumes that the position of the constructed entity does not change in time by an amount significant for the accuracy and time scale of an application.

EXAMPLE 1         For points on the surface of the Earth, tectonic plate movements are insignificant for many applications.

EXAMPLE 2         An RD X_AXIS_3D is bound to the line segment from the centre of the Earth to the centre of the Sun. This RD binding is not an object-fixed RD binding with respect to the spatial object Earth.

Figure 7.2 illustrates two distinct bindings of a point RD. On the left, it is bound to a specific point in the abstract object-space of a CAD/CAM model. On the right, it is bound to a point in physical object-space that is on an object that has been manufactured from that CAD model.

Figure 7.2 — An RD bound to an abstract object and to a real object

7.3.      Normal embeddings of position-space into object-space

7.3.1        Normal embeddings

An embedding is a position-space model of object-space formed by a one-to-one function of positions in position-space to points in object-space. A normal embedding is an embedding that satisfies the following distance-preserving property:

A function E from position-space to object-space is distance-preserving if for any two positions p and q in position-space, the measured distance in object-space from E(p) to E(q) in metres is equal to the Euclidean distance d(p, q).

NOTE 1       As a consequence of the normal distance-preserving property, a normal embedding is also a continuous function, that preserves angles, area, and other geometric properties.

 

Figure 7.3 — A right-handed normal embedding19

In position-space, the point E(0) is called the origin of the normal embedding E, and the point E(e₁) is the x-axis unit point of the normal embedding E. If the dimension of position-space is 2D or 3D, the point E(e₂) is the y-axis unit point of the normal embedding E. If the dimension of position-space is 3D, the point E(e₃) is the z-axis unit point of the normal embedding E.

A normal embedding of a 3D position-space is right-handed if the directed triangle formed by the three points, x-axis unit point, y-axis unit point, and z-axis unit point, in that sequence, has a clockwise orientation when viewed from the origin of the embedding. Otherwise, the embedding is left-handed. A right-handed normal embedding is illustrated in Figure 7.3. All 3D normal embeddings in this International Standard shall be right-handed.

7.3.2        Specification of 3D similarity transformations

A 3D object-space may have many normal embeddings of 3D position-space. Given two 3D normal embeddings E₁ and E₂ into the same object-space, one embedding can be expressed in terms of the other normal embedding. Given a position  in position-space, the normal embedding E₂ associates to it a unique point p in object-space. The normal embedding E₁ uniquely associates some position  to the same point p. This association of to may be expressed as a similarity transformation from E₂ to E₁ (see Figure 4.2). A similarity transformation is defined as a transformation on position-space that performs a translation, rotation, and/or scaling operation.

In general, E₂(0) may be displaced with respect to E₁(0) and the axes of the E₂ normal embedding may also be rotated and/or differently scaled with respect to the axes of the E₁ normal embedding (see Figure 7.4). If E₁ associates the position  to E₂(0), the similarity transformation from E₂ to E₁ may be specified in the form of the seven-parameter transformation:

where:

and where:

 

Figure 7.4 — 3D normal embedding relationships

The scale adjustment is needed to account for differing length scales in abstract object-space. In the case of physical object-space, small non-zero values of  may be required to adjust for spatial distortions in empirically estimated data. This is addressed in 7.4.5.

The convention of viewing the rotations with respect to E₁ is the position vector rotation convention. The coordinate frame rotation convention views rotations with respect to E₂ instead of E₁. The rotations  in Equation (5) are position vector rotations. If coordinate frame rotations are used, the rotations reverse sign (see Figure 7.5).

Note 1       Sign reversal does not affect cosine terms in the equation. Only the sine terms reverse sign.

Figure 7.5 — Rotation between E₁ and E₂ in two conventions

NOTE 2       A small rotation approximation of the seven-parameter transformation is described in Annex B.

The seven-parameter embedding specification of E₂ with respect to E₁ is defined by the seven-parameter values  in the position vector rotation convention (as in Equation (5)).

NOTE 3       In the cases that  the formula for the transformation from  to  reduces to a translation operation.

7.3.3        Specification of 2D similarity transformations

Given two 2D normal embeddings E₁ and E₂ into the same 2D object-space, one embedding can be expressed in terms of the other normal embedding. Given a position  in position-space, the normal embedding E₂ associates to it a unique point p in object-space. As in the 3D case, this association may be expressed as a similarity transformation.

If E1 associates  to E₂(0), is the scale factor, and w is the relative rotation, then the 2D similarity transformation from E2 to E1 may be specified in the form of the four parameter transformation:

with parameters:

7.4.      Object reference model

7.4.1        Introduction

A set of bound RDs can be selected so as to be compatible with only one normal embedding. In this way, a set of bound RDs with properly constrained relationships can specify a unique normal embedding. Such a constrained set of bound RDs is called an object reference model. Some object reference models use a set of RDs that model application-specific geometric aspects of the object-space. Of particular interest are object reference models that include an oriented surface RD that models a surface significant to the object (see 7.4.2).

A relationship between two or more bound RDs needed to ensure compatibility with a normal embedding is termed a binding constraint (see 7.4.3). Object reference models that use the same set of RD primitives and the same binding constraints are abstracted in the notion of an object reference model template. Object reference model templates provide a uniform method of object reference model specification. If the bound RDs of an object reference model are compliant with the RD set and binding constraints of a particular object reference model template, then the object reference model is said to realize that template (see 7.4.4).

A set of standardized object reference model templates are defined in this International Standard (see 7.4.5). Realizations of these templates are specified in Annex E.

7.4.2        ORM

A normal embedding and an RD binding are compatible if the normal embedding image of the RD primitive is coincident with the points (and direction or orientation, as applicable) of the constructed entity of the RD binding.

EXAMPLE 1         The constructed point in object-space to which RD ORIGIN_3D is bound is the origin of a normal embedding if, and only if, that normal embedding is compatible with the RD binding.

EXAMPLE 2         The directed line constructed in object-space to which RD X_AXIS_3D is bound is the locus of the x-axis image under a compatible normal embedding, and similarly for other axis RDs.

An object reference model (ORM) for a spatial object is a set of bound RDs for which there exists exactly one normal embedding that is compatible with each bound RD in the set. In the 3D case, this unique embedding shall also be right-handed.

An ORM is object-fixed if each of its RD bindings are object-fixed, otherwise it is called object-dynamic. The object-fixed definition assumes that the object itself is not changing in time by an amount significant for the accuracy and time scale of an application. The normal embedding determined by an ORM is, correspondingly, either an object-fixed embedding or an object-dynamic embedding.

EXAMPLE 3         The Sun and the gas giants Jupiter, Saturn, Uranus, and Neptune are not rigid. The ORM specified for the Sun uses RD bindings defined in part by ephemeris and is thus object-dynamic. In the case of the ORMs specified for the gas giants, the object for binding is the magnetic field of the planet, thus these ORMs are object-fixed.

An ORM is often selected to contain an RD of category oriented surface that corresponds to a physical or conceptual surface significant to the modelled spatial object. An RD is chosen and its position with respect to the object is bound so that that the RD instance is a “best fit” to the object in some application-specific sense. In particular, if the RD surface is “fitted” to a specific part of the object surface, the ORM is called a local model. If the RD is selected to best fit the entire surface, the ORM is called a global model.

An ORM may also contain an RD for the purpose of providing a CS binding parameter (see 8.3.2.2). In particular the radius of a sphere RD or the semi-axis values of an oblate ellipsoid RD may be used for this purpose.

An Earth reference model (ERM) is an ORM for which the spatial object is the Earth.

EXAMPLE 4         If the object is a planet, an ORM containing an oblate ellipsoid RD is usually selected to model all or part of the general shape of the planet.

7.4.3        Binding constraint

In an ORM, the existence of a unique and compatible normal embedding depends on establishing certain geometric relationships, called binding constraints, when the RDs are bound.

A binding constraint is a relationship in object-space between the constructed entities of two or more bound RDs, or a size relationship between a reference datum primitive and its corresponding constructed entity. Binding constraint relationships used in this International Standard include:

a)       containment of a point in a curve or a surface;

b)       containment of a curve in a surface;

c)       coincidence of a line with an axis of symmetry of a surface;

d)       in the 3D case, the right-handedness of sets of directed lines or oriented planes; or

e)       distance measurement in object-space between points.

In this International Standard every abstract object-space has an associated length scale. The term “(scaled) metres” in a binding constraint definition shall mean “metres” in the case of physical objects and shall mean “length scaled to metres with respect to the length scale of object-space” in the case of abstract objects.

7.4.4        ORM template

ORMs that have the same set of RD primitives and that have the same binding constraints are abstracted in the notion of an ORM template. An ORM template specifies how certain sets of RD primitives may be bound to ensure that the resulting set of bound RDs forms an ORM. An ORM template can be used to conveniently specify multiple ORMs.

An ORM template (ORMT) is a set of RDs and binding constraints such that, for a given object-space, whenever the RDs in the set are bound in compliance with the set of binding constraints, then that bound set of RDs forms an ORM.

An ORM is a realization of an ORMT if

1)       the RDs of the ORM match the RD set of the ORMT, and

2)       the RD bindings of the ORM are compliant with the binding constraints of the ORMT.

This International Standard specifies a set of ORMTs for 2D and 3D position-space in Table 7.14. The specification elements are defined in Table 7.11. Additional ORMTs may be registered in accordance with Clause 13. Table 7.12 is a directory of ORMT specification tables.

Table 7.11 — ORMT specification elements

 

Table 7.12 — ORMT specification directory

 

Table 7.13 — 2D ORMT specifications

 

Table 7.14 — 3D ORMT specifications

 

The specification of an ORMT does not determine a normal embedding. A normal embedding is determined when an ORMT is realized as an ORM.

The methods and techniques of binding the RDs of an ORMT realization draw from disciplines ranging from geometry to astronomy, surveying, geophysics, and satellite geodesy. In general, there are many ways to realize an ORMT for a spatial object. Techniques and methodologies for binding RD components are outside of the scope of this International Standard. The ORM concept is designed to be general enough to encompass these many application domains. As an illustration of the generality of this concept, the following Example outlines a method used in geodesy to define the RD bindings of an ORMT OBLATE_ELLIPSOID realization.

EXAMPLE            The North American Datum 1927 may be specified as a realization of the ORMT OBLATE_ELLIPSOID (see Table 7.14). The oblate ellipsoid RD component is RD CLARKE 1866 in Table D.2. The binding of this RD and the other two RD components in the Earth object-space may be defined as follows:

a)       the direction of the RD Z_AXIS_3D is identified as the north direction of the Earth’s rotational axis,

b)       a position (latitude 39º 13’ 26,686” N) on the oblate ellipsoid RD is identified to a position in the Earth object-space (Meades Ranch, Kansas, US),

c)       the direction of the surface normal to the ellipsoid at the identified point is specified ( χ = -1,32 ξ = 1,93), and

d)       the direction of the positive xz-plane normal is indirectly determined by specifying the longitude of Meades Ranch (98º 32’ 30,506” W).

Items a) through c) determine a unique oblate ellipsoid surface in the object-space of the Earth. The equatorial plane of the oblate ellipsoid is (by compatibility with the oblate ellipsoid surface generating function) the xy-plane, and its intersection with the oblate ellipsoid axis of rotation determines the origin point and the z-axis. Specification (d) together with the origin and the xy-plane determine the x-axis. Since there is one, and only one, oriented yz-plane that is both perpendicular to the xz-plane and right-handed compatible, the right-handed normal embedding is uniquely determined (see Figure 7.6).

This Example is based on a published specification of the North American Datum 1927. However, the methodology used to select the point in b) and determine the surface normal direction c) at that point was a complex process involving a mathematical best fit of the Clarke 1866 ellipsoid to a network of geodetic survey control points spanning the continental United States.

Figure 7.6 — Oblate ellipsoid ORMT binding

7.4.5        Standardized ORMs

The ORMs specified in this International Standard, are ORMT realizations. Standardized ORMs shall include:

a)       a specification of the spatial object and optionally a region in object-space,

b)       a specification of the ORM template,

c)       a specification of the ellipsoid RD (if any), and

d)       the binding year if the spatial object is a physical object.

A standardized ORM does not include a specification of the binding of its RD components. The binding specification is not directly needed for the scope of this International Standard. An ORM indirectly designates a unique normal embedding. A specification of an ORM normal embedding is important when two or more ORMs have been specified for the same spatial object. To inter-convert between spatial relationships with respect to two different normal embeddings, one normal embedding shall be expressed in terms of the other normal embedding by means of a similarity transformation, or both shall be expressed in terms of a third (reference) normal embedding by means of a similarity transformation for each ORM with respect to a third (reference) ORM.

If two or more object-fixed ORMs for the same object are specified (or registered), one of the ORMs shall be designated as the reference ORM for that object.

A reference transformation (RT) for an ORM is a similarity transformation from the ORM normal embedding to the normal embedding of the reference ORM for that object, ORM_R. The reference transformation for an ORM, ORM_S, shall be denoted by H_SR (see Table 10.1).

For 3D ORMs, a reference transformation shall be specified by the seven parameters of the corresponding seven-parameter embedding specification. For 2D ORMs, a reference transformation shall be specified by the four parameters of the corresponding four-parameter embedding specification.

Standardized object-fixed ORMs shall specify at least one reference transformation.

Some ERMs that are specified in Table E.5 are based on local geodetic datums. Historic local geodetic datums may exhibit distortions with respect to the reference ERM. In these cases, an RT specified by the seven-parameter transformation of the ERM is an approximation based on empirical measurements. In some of these cases, similarity transformation parameters for approximations based on different sets of measurements and/or sub-regions appear as multiple RT entries in Table E.6. In those cases, the RT labels shall share a common prefix derived from the name of the related local geodetic datum.

NOTE 1       The scale adjustment component of the seven-parameter specification is needed to account for differing length scales in abstract object-space. Theoretically, for two normal embeddings of a physical object-space. In practice, when the embeddings are indirectly determined by the RD bindings of an ORM, the determination of the parameters is an approximation that is the result of a mathematical best fit and the seventh parameter () is used to provide an additional degree of freedom to achieve the desired accuracy in the region of interest. In the case of the object Earth, published values for  are typically one part per million in magnitude (10^-6) or smaller.

A directory of reference ORMs is provided in Table E.1. The reference ORM for the Earth is ORM WGS_1984. This ORM is an Earth-fixed global model. The actual binding definition of ORM WGS_1984 (see [83502S] ) is a realization of ORMT BI_AXIS_ORIGIN_3D. However, for purposes of this International Standard, the RD WGS_1984 is associated with this ORM, so that its ORM specification is a realization of ORMT OBLATE_ELLIPSOID.

The elements of an ORM specification are defined in Table 7.15. Standardized ORMs are specified in Annex E. Additional ORMs may be registered in accordance with Clause 13.

Table 7.15 — ORM specification elements

 

For each object-fixed ORM there shall be specified one or more RTs that transform the ORM to the reference ORM of the ORM spatial object. The elements of an RT specification are defined in Table 7.16. Standardized RTs are specified in Annex E. Additional RTs may be registered in accordance with  Clause 13. The standard ORM for an abstract space of a given dimension specifies only an identity RT. Graphical applications use ad hoc transformations to scale, rotate, and translate one abstract space with respect to another as needed in an application. The API (Clause 11) provides support for non-standardized RTs.

Table 7.16 — Reference transformation specification elements

 

If an ORM is the reference ORM of a spatial object, then it shall have an RT with the RT label containing the string “IDENTITY”, the RT parameters of that RT shall be , and there shall be no error estimate. These reference ORM identity RTs are labelled and coded to provide uniform treatment of all object-fixed ORMs in the API (see Clause 11)

If a 3D ORM is not the reference ORM of a spatial object, and the RT parameters for it are   within the limits of precision in measurement and associated errors, then the RT label shall contain the string “IDENTITY_BY_MEASUREMENT”.

If a 3D ORM is not the reference ORM of a spatial object, and the RT parameters for it are   by intent, then the RT label shall contain the string “IDENTITY_BY_DEFAULT”.

NOTE 2       In the case of Earth-fixed ERM bindings axis rotations are either zero, or are very small with the following exceptions:

a)       Cases for which the xz-plane does not contain Greenwich, UK have relatively large  rotations.

b)       The Earth-fixed approximations of celestiomagnetic ERMs have large  rotations (see “GEOMAGNETIC” entries in Table 7.32).

NOTE 3       A geodetic datum that is global or geocentric is used as a basis for establishing a geocentric SRF (see 8.5.2). If geocentric coordinates are associated to a 3D linear embedding, then such a datum is conceptually equivalent to an ERM. The WGS 84 global datum (see [83502T]) is conceptually equivalent to the ORM WGS_1984 in Table E.5.

7.5.      Object binding rules for ORMT BI_AXIS_ORIGIN_3D realizations

7.5.1        Object binding rule set

ORMs for planets, satellites, and other celestial bodies include object-dynamic ORMs based on ORMT BI_AXIS_ORIGIN_3D. Many of these object-dynamic ORMs share common rules for the binding of ORMT BI_AXIS_ORIGIN_3D components based on physical characteristics and spatial arraignments of the applicable celestial bodies. To facilitate uniform specification of these ORMs, the concept of an object binding rule set for an ORMT BI_AXIS_ORIGIN realization is defined.

 An object binding rule set (OBRS) for ORMT BI_AXIS_ORIGIN_3D shall be comprised of:

a)       an object binding rule set name,

b)       a label and code,

c)       object restrictions that delineate the object-spaces for which the object binding rules apply, and

d)       a set of object binding rules for the RD components of ORMT BI_AXIS_ORIGIN_3D,

where an object binding rule for an ORMT is an object-space specific restriction for the binding of a single RD in the RD set of the ORMT. The set of object binding rules (d) shall comply with the binding constraints of the ORMT.

The specification elements for an OBRS for ORMT BI_AXIS_ORIGIN_3D are defined in Table 7.17.

Table 7.17 — OBRS for ORMT BI_AXIS_ORIGIN_3D specification elements

 

This International Standard provides a collection of OBRS specifications as identified in Table 7.18. Additional OBRS specifications may be registered in accordance with Clause 13.

Table 7.18 — OBRS for ORMT BI_AXIS_ORIGIN_3D specification directory

 

An OBRS name may be used to describe an ORM that that is compliant with the named OBRS. Thus a "celestiomagnetic ORM" denotes an ORM realization of ORMT BI_AXIS_ORIGIN_3D for an object-space that satisfies the celestiomagnetic OBRS object restrictions and whose RD bindings comply with the celestiomagnetic OBRS object binding rule set.

Several OBRS specifications in Table 7.18 are described using one or more of the following terms: rotational northwards, vernal equinox, inertial direction, quasi-inertial direction, first point of Aries, and Aries true of date. These terms are defined as follows:

The term rotational northwards, as used with respect to a rotating object, shall mean in a direction making an acute angle with respect to the direction from the centre of the object to its rotational north pole.

The ecliptic plane and the plane containing the Sun and parallel to the equatorial plane of a planet intersect in a line. An equinox is one of the two points of intersection between this line and the orbit of the planet. The vernal equinox is the equinox at which the direction from the planet to the Sun begins to ascend to the northern side of the equatorial plane.

An inertial direction is a direction with respect to the universe that is time invariant. A quasi-inertial direction is a direction that changes relatively slowly with respect to the universe.

At any given epoch as defined in 6.2.1, the direction of the Sun at the vernal equinox is fixed with respect to the Milky Way and other (detectible) galaxies. In the case of the Earth, the first point of Aries is the direction from the vernal equinox to the Sun. The current first point of Aries is termed Aries true of date.

NOTE          The effects of precession and nutation on the spin axis of a planet cause this direction to change over time. The change in direction of the first point of Aries is very slow (approximately one full rotation in 26 000 years).

EXAMPLE            The first point of Aries at a given epoch is an inertial direction, and Aries true of date is a quasi-inertial direction.

7.5.2        Equatorial inertial

The equatorial inertial OBRS is specified in Table 7.19. See also Figure 7.7.

Table 7.19 — Equatorial inertial OBRS

Element	Value
OBRS label	equatorial_inertial
OBRS code	1
OBRS name	equatorial inertial
Object restrictions	A planet in the solar system for which the ecliptic plane is distinct from the equatorial plane.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the planet. 2)       The RD X_AXIS_3D points in the direction of the Sun when the planet is at its vernal equinox. 3)       The RD Z_AXIS_3D is parallel to the rotational axis and points in the direction of rotational northwards.
References	[SEID]

 

The axis directions are quasi-inertial, but vary with respect to any object-fixed ORM for the planet.

In the case of ORM EARTH_INERTIAL_J2000r0, the International Earth Rotation Service (see [IERS]) specifies a very precise transformation to the ORM WGS_1984 reference embedding with a matrix whose coefficients represent the effects of polar motion, the Earth's rotation, nutation and precession20.

Under the assumption that the ORM z-axis and ORM WGS_1984 z-axis are parallel, the transformation simplifies to a single rotation matrix. This simplified form is described in Annex B.

Figure 7.7 — Equatorial inertial OBRS

Three equatorial inertial ERMs labelled ORM EARTH_INERTIAL_ARIES_1950, ORM EARTH_INERTIAL_J2000r0 and ORM EARTH_INERTIAL_ARIES_TRUE_OF_DATE, are specified in Table E.5. They are based on three determinations of the first point of Aries and the rotational axis. The first two have epoch-fixed inertial directions. ORM EARTH_INERTIAL_ARIES_TRUE_OF_DATE has a quasi-inertial x-axis direction that varies as a function of time.

Table 7.20 is a directory of the Earth and other celestial object equatorial inertial ORMs.

Table 7.20 — Equatorial inertial ORM directory

 

7.5.3        Solar ecliptic

The solar ecliptic OBRS is specified in Table 7.21. See [BHAV, 3.2.5] and Figure 7.8.

Table 7.21 — Solar ecliptic OBRS

Element	Value
OBRS label	solar_ecliptic
OBRS code	2
OBRS name	solar ecliptic
Object restrictions	A planet in the solar system.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the planet. 2)       The RD X_AXIS_3D is in the ecliptic plane and points in the direction of the Sun. 3)       The RD Z_AXIS_3D is perpendicular to the ecliptic plane and points northward.
References	[HAPG]

 

Figure 7.8 — Solar ecliptic ORM binding

NOTE          The x-axis slowly rotates once per orbit of the planet around the Sun. The y-axis also lies in the ecliptic plane.

In the case of the Earth, the transformation from a solar ecliptic ORM embedding to the ORM WGS_1984 reference embedding is described in Annex B.

Table 7.22 is a directory of the Earth and other planet object solar ecliptic ORMs.

Table 7.22 — Solar ecliptic ORM directory

 

7.5.4        Solar equatorial

A solar equatorial OBRS is specified in Table 7.23. See also Figure 7.9.

Table 7.23 — Solar equatorial OBRS

Element	Value
OBRS label	solar_eQUATORIAL
OBRS code	3
OBRS name	solar equatorial
Object restrictions	A planet in the solar system for which the ecliptic plane is distinct from the equatorial plane.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the planet. 2)       The RD X_AXIS_3D is in the ecliptic plane and points in the direction of the Sun. 3)       The RD Z_AXIS_3D is perpendicular to the RD X_AXIS_3D in the plane determined by the RD X_AXIS_3D and the rotational axis of the Sun and points northward.
References	[CRUS]

 

Figure 7.9 — Solar equatorial ORM binding

NOTE          The xz-plane of a solar equatorial ORM contains the rotational axis of the Sun. The x-axis slowly rotates once per orbit of the object around the Sun. The y-axis is parallel to the solar equatorial plane and points towards dusk [BHAV, 3.2.6].

Table 7.24 is a directory of the Earth and other planet solar equatorial ORMs.

Table 7.24 — Solar equatorial ORM directory

 

7.5.5        Heliocentric Aries ecliptic

The heliocentric Aries ecliptic OBRS is specified for a planet in Table 7.25. See [HAPG] and Figure 7.10.

Table 7.25 — Heliocentric Aries ecliptic OBRS

Element	Value
OBRS label	heliocentric_ARIES_ecliptic
OBRS code	4
OBRS name	heliocentric Aries ecliptic
Object restrictions	The Sun.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the Sun. 2)       The RD Z_AXIS_3D is perpendicular to the ecliptic plane of the Earth and points in the direction of rotational northwards. 3)       The RD X_AXIS_3D is in the ecliptic plane of the Earth and points towards the first point of Aries.
References	[HAPG]

 

Figure 7.10 — Heliocentric Aries ecliptic ORM binding

See Figure 7.10. Table 7.26 is a directory of the Earth and other planet solar equatorial ORMs. The heliocentric Aries ecliptic axis directions are inertial for ORM HELIO_ARIES_ECLIPTIC_J2000r0 and quasi-inertial ORM HELIO_ARIES_ECLIPTIC_TRUE_OF_DATE.

Table 7.26 — Heliocentric Aries ecliptic ORM directory

 

7.5.6        Heliocentric planet ecliptic

The heliocentric planet ecliptic OBRS is specified for a planet in Table 7.27. See [HAPG] and Figure 7.11.

Table 7.27 — Heliocentric planet ecliptic OBRS

Element	Value
OBRS label	heliocentric_planet_ecliptic
OBRS code	5
OBRS name	heliocentric planet ecliptic
Object restrictions	The Sun.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the Sun. 2)       The RD Z_AXIS_3D is perpendicular to the ecliptic plane for a specified planet and points in the direction of rotational northwards. 3)       The RD X_AXIS_3D is in the ecliptic plane of the specified planet and points towards the planet.
References	[HAPG]

 

Figure 7.11 — Heliocentric planet ecliptic OBRS

Table 7.28 is a directory of the Earth and other planet solar equatorial ORMs.

Table 7.28 — Heliocentric planet ecliptic ORM directory

ORM label	Published name
HELIO_EARTH_ECLIPTIC	Heliocentric Earth ecliptic

 

7.5.7        Heliocentric planet equatorial

The heliocentric planet equatorial OBRS is specified for a planet in Table 7.29. See also Figure 7.12.

Table 7.29 — Heliocentric planet equatorial OBRS

Element	Value
OBRS label	heliocentric_planet_equatorial
OBRS code	6
OBRS name	heliocentric planet equatorial
Object restrictions	The Sun.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the Sun. 2)       The RD Z_AXIS_3D is perpendicular to the solar equatorial plane of the specified planet and points in the direction of rotational northwards. 3)       The RD X_AXIS_3D is perpendicular to the RD Z_AXIS_3D in the plane determined by the RD Z_AXIS_3D and the planet centre and points towards the specified planet.
References	[HAPG]

 

Figure 7.12 — Heliocentric planet equatorial OBRS

Table 7.30 is a directory of the Earth and other planet solar equatorial ORMs.

Table 7.30 — Heliocentric planet equatorial ORM directory

ORM label	Published name
HELIO_EARTH_EQUATORIAL	Heliocentric Earth equatorial

 

7.5.8        Celestiomagnetic

The celestiomagnetic OBRS is specified in Table 7.31. See [BHAV, 3.3.1] and Figure 7.13.

Table 7.31 — Celestiomagnetic OBRS

Element	Value
OBRS label	celestiomagnetic
OBRS code	7
OBRS name	celestiomagnetic
Object restrictions	A planet or rotating satellite in a solar system with a magnetic dipole axis distinct from its rotational axis.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the planet. 2)       The RD Z_AXIS_3D is parallel to the magnetic dipole axis and points towards magnetic north. 3)       The RD X_AXIS_3D is contained in the plane through the origin that is parallel to the dipole and rotational axes, perpendicular to the RD Z_AXIS_3D and pointing away from the dipole northward direction.
References	[CRUS]

 

NOTE          The rotational south pole is contained in the x-positive xz-half-plane unless the planet has retrograde rotation. This binding is not applicable to Saturn whose magnetic and rotational poles are not distinguished.

Figure 7.13 — Celestiomagnetic ORM binding

In the case of the Earth, this dynamic ERM is approximated as an Earth-fixed ERM for a five-year epoch. The other celestial objects that have observed magnetic dipoles have object-fixed ORM approximations for the corresponding dynamic ORMs.

Table 7.32 is a directory of the Earth and other object-fixed celestiomagnetic ORM approximations.

Table 7.32 — Celestiomagnetic ORM directory

ORM label	Published name
GANYMEDE_MAGNETIC_2000	Ganymede magnetic
GEOMAGNETIC_1945	Geomagnetic 1945
GEOMAGNETIC_1950	Geomagnetic 1950
GEOMAGNETIC_1955	Geomagnetic 1955
GEOMAGNETIC_1960	Geomagnetic 1960
GEOMAGNETIC_1965	Geomagnetic 1965
GEOMAGNETIC_1970	Geomagnetic 1970
GEOMAGNETIC_1975	Geomagnetic 1975
GEOMAGNETIC_1980	Geomagnetic 1980
GEOMAGNETIC_1985	Geomagnetic 1985
GEOMAGNETIC_1990	Geomagnetic 1990
GEOMAGNETIC_1995	Geomagnetic 1995
GEOMAGNETIC_2000	Geomagnetic 2000

 

7.5.9        Solar magnetic ecliptic

The solar magnetic ecliptic OBRS is specified in Table 7.33. See [BHAV, 3.3.4] and Figure 7.14.

Table 7.33 — Solar magnetic ecliptic OBRS

Element	Value
OBRS label	solar_magnetic_ecliptic
OBRS code	8
OBRS name	solar magnetic ecliptic
Object restrictions	A planet in the solar system with a magnetic dipole.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the planet. 2)       The RD X_AXIS_3D is in the ecliptic plane of the planet pointing in the direction of the Sun. 3)       The RD Z_AXIS_3D is perpendicular to the RD X_AXIS_3D and points in the direction of rotational northwards in the plane determined by the x-axis and the planetary magnetic dipole axis.
References	[CRUS]

 

 

Figure 7.14 — Solar magnetic ecliptic ORM binding

Table 7.34 is a directory of the Earth and other planet solar magnetic ecliptic ORMs.

Table 7.34 — Solar magnetic ecliptic ORM directory

ORM label	Published name
EARTH_SOLAR_MAGNETOSPHERIC	Solar magnetospheric
JUPITER_SOLAR_MAG_ECLIPTIC	Jupiter solar magnetic ecliptic

 

7.5.10    Solar magnetic dipole

The solar magnetic dipole OBRS is specified in Table 7.35. See [BHAV, 3.3.5] and Figure 7.15.

Table 7.35 — Solar magnetic dipole OBRS

Element	Value
OBRS label	solar_magnetic_dipole
OBRS code	9
OBRS name	solar magnetic dipole
Object restrictions	A planet in the solar system with a magnetic dipole.
Object binding rules	1)       The RD ORIGIN_3D is the mass-centre of the planet. 2)       The RD Z_AXIS_3D is parallel to the planetary magnetic dipole axis and points towards magnetic north. 3)       The RD X_AXIS_3D is perpendicular to the RD Z_AXIS_3D and pointing towards the Sun in the plane determined by the Sun and the RD Z_AXIS_3D.
References	[CRUS], [BHAV]

 

Figure 7.15 — Solar magnetic dipole ORM binding

Table 7.36 is a directory of the Earth and other celestial object solar magnetic dipole ORMs.

Table 7.36 — Solar magnetic dipole ORM directory

ORM label	Published name
EARTH_SOLAR_MAG_DIPOLE	Solar magnetic dipole
JUPITER_SOLAR_MAG_DIPOLE	Jupiter solar magnetic dipole

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