Open Geospatial Consortium 
Submission Date: 20200121 
Approval Date: 20200824 
Publication Date: 20210226 
External identifier of this OGC® document: http://www.opengis.net/doc/BP/CDBSRF/1.2 
Internal reference number of this OGC® document: 16011r5 
Version: 1.2 
Category: OGC® Best Practice 
Editor: Carl Reed 
Volume 8: OGC CDB Spatial Reference System Guidance (Best Practice) 
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Copyright © 2021 Open Geospatial Consortium 
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Document type: OGC® Best Practice 
Document subtype: 
Document stage: Approved 
Document language: English 
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 1. Scope
 2. Conformance
 3. References
 4. Terms and Definitions
 5. Conventions
 6. CDB Approach
 7. The Three Systems
 Annex A: Revision History
i. Abstract
Volume 8 of the CDB standard defines the conceptual model and the methodologies that allow the description, and transformation or conversion, of geometric properties within a set of spatial reference frames supported by the CDB standard. The CDB Spatial Reference Model (SRM) supports an unambiguous specification of the positions, directions, and distances associated with spatial information. This document also defines algorithms for precise transformation of positions, directions and distances among different spatial reference frames.
In previous versions of the CDB standard, this CDB volume was Appendix K in CDB Version 3.2 as submitted to the OGC.
ii. Keywords
The following are keywords to be used by search engines and document catalogues.
ogcdoc, OGC document, cdb, spatial reference model, srm, coordinate systems, crs, srs
iii. Preface
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. The Open Geospatial Consortium shall not be held responsible for identifying any or all such patent rights.
Recipients of this document are requested to submit, with their comments, notification of any relevant patent claims or other intellectual property rights of which they may be aware that might be infringed by any implementation of the standard set forth in this document, and to provide supporting documentation.
iv. Submitting organizations
The following organizations submitted this Document to the Open Geospatial Consortium (OGC):
Organization name(s)

CAE Inc.

Carl Reed, OGC Individual Member

Envitia, Ltd

Glen Johnson, OGC Individual Member

KaDSci, LLC

Laval University

Open Site Plan

University of Calgary

UK Met Office
The OGC CDB standard is based on and derived from an industry developed and maintained specification, which has been approved and published as OGC Document 15003: OGC Common DataBase Volume 1 Main Body. An extensive listing of contributors to the legacy industryled CDB specification is at Chapter 11, pp 475476 in that OGC Best Practices Document (https://portal.opengeospatial.org/files/?artifact_id=61935 ).
v. Submitters
All questions regarding this submission should be directed to the editor or the submitters:
Name 
Affiliation 
Carl Reed 
Carl Reed & Associates 
David Graham 
CAE Inc. 
vi. Document Organization
For ease of editing and review, the standard has been separated into 16 Volumes, one being a schema repository.

Volume 0: OGC CDB Companion Primer for the CDB standard (Best Practice).

Volume 1: OGC CDB Core Standard: Model and Physical Data Store Structure. The main body (core) of the CDB standard (Normative).

Volume 2: OGC CDB Core Model and Physical Structure Annexes (Best Practice).

Volume 3: OGC CDB Terms and Definitions (Normative).

Volume 4: OGC CDB Rules for Encoding CDB Vector Data using Shapefiles (Best Practice).

Volume 5: OGC CDB Radar Cross Section (RCS) Models (Best Practice).

Volume 6: OGC CDB Rules for Encoding CDB Models using OpenFlight (Best Practice).

Volume 7: OGC CDB Data Model Guidance (Best Practice).

Volume 8: OGC CDB Spatial Reference System Guidance (Best Practice).

Volume 9: OGC CDB Schema Package: http://schemas.opengis.net/cdb/ provides the normative schemas for key features types required in the synthetic modeling environment. Essentially, these schemas are designed to enable semantic interoperability within the simulation context (Normative).

Volume 10: OGC CDB Implementation Guidance (Best Practice).

Volume 11: OGC CDB Core Standard Conceptual Model (Normative).

Volume 12: OGC CDB Navaids Attribution and Navaids Attribution Enumeration Values (Best Practice).

Volume 13: OGC CDB Rules for Encoding CDB Vector Data using GeoPackage (Normative, Optional Extension).

Volume 14: OGC CDB Guidance on Conversion of CDB Shapefiles into CDB GeoPackages (Best Practice).

Volume 15: OGC CDB Optional MultiSpectral Imagery Extension (Normative).
1. Scope
The handling of spatial data requires a good deal of rigor to accurately describe the position of points in space. Furthermore, it requires the ability to define directions and distances. Generally, this is accomplished through the use of coordinate systems.
A coordinate system (also called a spatial reference system) is a means of assigning coordinates to a location and establishing relationships between sets of such coordinates. Specifying a coordinate reference system enables the interpretation of a set of coordinates as a representation of a position in a real world space.
In the CDB standard, these terms can be used interchangeably. However, for consistency with previous versions of the CDB standard, the term spatial reference system (SRS) is used.
There is often the requirement to represent position in several different spatial reference systems. Each spatial reference system provides a particular way of defining positions within its domain. This level of abstraction also permits spatial reference systems to be decomposed into a series of, or even a hierarchy of reference, each relative to another reference system. This mechanism permits objects to be independently defined (positioned, oriented, and scaled) with respect to a local SRS and then be later incorporated into other reference systems. The reference systems can be abstract mathematical constructs or they can be bound to real world objects (e.g., a flap located defined in a flap reference system which in turn is bound to an aircraft wing reference system).
3. References
The following normative documents contain provisions that, through reference in this text, constitute provisions of this document. For dated references, subsequent amendments to, or revisions of, any of these publications do not apply. For undated references, the latest edition of the normative document referred to applies.
ISO: ISO 19111:2007, Geographic information ― Spatial referencing by coordinates, 2007
4. Terms and Definitions
This document uses the terms defined in Subclause 5.3 of [OGC 06121r8], which is based on the ISO/IEC Directives, Part 2, Rules for the structure and drafting of International Standards. In particular, the word “shall” (not “must”) is the verb form used to indicate a requirement to be strictly followed to conform to this standard. Any terms and definitions for this CDB document may be found in Volume 3: CDB Terms and Definitions (Normative).
6. CDB Approach
One of the primary objectives of the CDB standard is to provide the means to represent the entire earth. As such, a CDB compliant data store must handle spatial data with a good deal of rigor to accurately describe the position of points in space and must do so at the level of fidelity commensurate with the precision that is now possible in modern simulators.
The size, content, fidelity and precision of synthetic environments now warrant a different approach, an approach that entirely avoids the “problem with projections” and other approximations used in the past. To this end, the CDB standard mandates the use a geographic coordinate system (GCS). A GCS uses a threedimensional spherical surface to define locations on the earth. Since no projections are involved, full geometric coherence is assured without compromise and all four key spatial properties can be achieved simultaneously:

Preservation of distance

Preservation of direction

Preservation of area

Preservation of shape
6.1. CDB Spatial Reference Systems
The CDB standard is based on a surface geodetic coordinate system, i.e., points on the earth surface are specified as geographic latitude longitude and elevation coordinates. More specifically, geodetic coordinates (sometimes called geographic coordinates) are angular coordinates (longitude and latitude), closely related to spherical polar coordinates, and are defined relative to a particular Earth geodetic datum. For the CDB, this is the WGS 84 datum. The WGS 84 datum surface is an oblate spheroid (ellipsoid) with major (equatorial) radius a = 6378137 m at the equator and flattening f = 1/298.257223563 ^{[1]}. The polar semiminor axis b then equals a times (1−f), or 6356752.3142 m.
The CDB data store embeds modeled point features (e.g., the representation of 3D objects, moving and/or static) within a Cartesian Coordinate System ^{[2]}. Its use is generally constrained to objects that are small in comparison to the earth. As shown below, a modeled point feature can be referenced anywhere on the earth by providing the model’s orientation (the AO1 attribute specified in Chapter 5 of Volume 1: OGC CDB Core Standard: Model and Physical Database Structure) and the model’s origin using a set of latitude/longitude/elevation coordinates. Note that the model’s zaxis implicitly points upward with respect to the earth surface.
Figure 1: Cartesian Model positioned to WGS84 Coordinates
The earth shape is described by the WGS84 reference ellipsoid. The CDB standard also defines three related set of Spatial Reference Systems (and associated coordinate systems) for use in conjunction with the surface geodetic coordinate system; they are:

Earthcentered Cartesian (Geocentric)

Generic Cartesian

Local Vertical (LVCS)
7. The Three Systems
The following sections detail the normative aspects of the three Spatial Reference Systems (SRS) supported in the CDB model. We start with a general discussion of the Geographical Coordinate System before addressing the Spatial Reference Systems.
7.1. Geographic Coordinate System (Geodetic)
A geographic coordinate system (also called the geodetic coordinate system) is one in which the coordinates are expressed as latitude, longitude, and altitude relative to the reference ellipsoid. Geographical latitude φ and longitude λ are the angles of the normal on the reference ellipsoid along the point to the equator and zero meridian. The angles are normally given as degrees, minutes and seconds. Altitude is the distance above and normal to the reference ellipsoid in meters. The WGS 84 ellipsoid represents the actual geoid within an accuracy of 100 meters. The prime meridian and the equator are the reference planes used to define latitude and longitude.
In other terms, the geographic latitude – there are many other defined latitudes – of a point is the angle between the equatorial plane and a line normal to the reference ellipsoid’s surface. The geographic longitude of a point is the angle between a reference plane, Greenwich, and a plane passing through the point, both planes being perpendicular to the equatorial plane. The geographic height at a point is the distance from the reference ellipsoid to the point in a direction normal to the reference ellipsoid.
Table 1: Geographic Coordinate System (Geodetic)
Field 
Specification 
Properties 
Orthogonal. 
CS parameters and constraints 
a: major semiaxis length b: minor semiaxis length Constraints: a > b: (oblate ellipsoid) 
Coordinate components 
λ : longitude in radians, and φ : geodetic latitude in radians. 
Domain of the generating function or mapping equations 
. 
Domain of the inverse of the generating function or mapping equations 
. 
Notes 

7.1.1. EarthCentered SRS (aka Rectangular Geocentric SRS)
The earthcentered SRS defines a threedimensional Euclidian space with respect to the geometric center of the reference ellipsoid, the center of the earth. The reference datum of the Earthcentered SRS is based on the WGS84 ellipsoid reference model. In this SRS, the zaxis is pointing at the North Pole, the xaxis is pointing at the intersection of the equator and the Greenwich meridian, the prime meridian, and the yaxis is pointing at the intersection of the equator and 90^{o} east longitude. The associated coordinate system is called the World Coordinate System and its units are meters. The world coordinate system is used to specify the 3D position of objects with respect to the earthcentered SRS. This coordinate system is used as an intermediate system to convert geodetic coordinates to LVCS and vice versa
7.1.2. General Cartesian SRS
The Cartesian spatial reference frame defines a threedimensional Euclidian space with respect to an arbitrary origin. The reference datum specifying the origin and the orientation of the SRS is arbitrary, i.e., the reference datum can be specified within a geocentric SRS, a LVCS SRS or any other SRS. The SRS is righthanded and orthonormal. In this SRS system, the zaxis is pointing up and both the xaxis and yaxis lie in the horizontal plane. The associated coordinate system is called the General Cartesian Coordinate System; coordinates are specified in meters. This coordinate system is used for the representation of 3D objects, moving and/or static. Its use is generally constrained to objects that are small in comparison to the earth ^{[3]}.
7.1.3. Local Vertical SRS
The Local Vertical SRS (LVCS) spatial reference frame defines a threedimensional Euclidian space. It is a SRF similar to the Geocentric SRF except that the origin of the SRF is translated and rotated to a point on the surface of the WGS84 ellipsoid. At that point, the xy plane is tangent to the surface of the earth and the zaxis is normal to the ellipsoid. The associated coordinate system is called the local vertical coordinate system; the coordinates are specified in meters. In this coordinate system, the zaxis is pointing up, the yaxis is pointing north and the xaxis is pointing east. Its use is generally constrained to a surface that is small in comparison to the earth ^{[4]}.
7.2. Geodetic to Geocentric transformation ^{[1]}
The following equations are used to transform geodetic information to geocentric information according to the following:

If <ϕ, λ, h> represents the geodetic coordinates to be transformed, where ϕ is the latitude, λ is the longitude, and h is the WGS84 height above the reference ellipsoid; and

If <x, y, z> represents the geocentric coordinates; then using the WGS84 ellipsoid equatorial radius, a, of 6,378,137.0m and the WGS84 ellipsoid polar radius, b, of 6,356,752.314245m, the flattening f, the eccentricity e and the radius of curvature as a function of latitude N(ϕ) are given by equation eq. A1:
\[f = \frac{a  b}{a}\] \[e^2 = 2f  f^2\] \[N(φ) = \frac{a}{\sqrt[]{1  e^2 \sin^2 φ }}\] 
(eq. A‑1) 
From these equations, we define the transformation of each geodetic coordinate as:
\[x = (N(φ)+h) \cos φ \cos λ \] \[y = (N(φ)+h) \cos φ \sin λ \] \[z = (N(φ)(1e^2)+h) \sin φ\] 
(eq. A‑2) 
7.3. Geocentric to Geodetic Transformation
Geocentric coordinates cannot be transformed to the geodetic coordinate system directly. Instead, a successive approximation approach is used to compute the new coordinates. The following describes the algorithm to convert geocentric coordinates <x, y, z> to geodetic coordinates <ϕ, λ, h>, where ϕ is the latitude, λ is the longitude, and h is the WGS84 height above the reference ellipsoid. First, using the WGS84 ellipsoid equatorial radius, a = 6,378,137.0 m and the WGS84 ellipsoid polar radius, b = 6,356,752.314245 m, the flattening f and the eccentricity e of the ellipsoid are given by equation A3:
\[f = \frac{a  b}{a}\] \[e^2 = 2f  f^2\] 
(eq. A‑3) 
We first compute the longitude λ with equation A4:
\[λ = \tan ^{  1} \bigg(\frac{y}{x}\bigg)\] 
(eq. A‑4) 
We then compute a first approximation of the latitude assuming a spherical earth model with equation A5:
\[φ = \tan ^{  1} \bigg(\frac{z}{\sqrt[]{x^2 + y^2}}\bigg)\] 
(eq. A‑5) 
Then, we iteratively compute the radius of curvature as a function of latitude N(ϕ) and, as a result we iteratively converge to a new, more accurate latitude ϕ’ with equation A6:
\[N(φ) = \frac{a}{\sqrt[]{ 1  e^2 \sin^2 φ }}\] 
(eq. A‑6) 
For each iteration, ϕ is replaced with ϕ’, until the difference between the two values is less than a preset allowable error. The resulting latitude error will be less than ε. Finally, we compute the height above the reference ellipsoid h with equation A7
\[ h = \frac{\sqrt[]{x^2 + y^2}}{\cos φ}  N(φ) \] 
(eq. A‑7) 
7.4. Geodetic to LVCS Coordinate Transformation
The transformation of a geodetic coordinate into an LVCS coordinate is decomposed into two parts:

Apply a coordinate transformation to each coordinate of an object from the geodetic coordinate system to the rectangular geocentric coordinate system.

Then apply a second transformation to go from the geocentric coordinate system to LVCS.
The first transformation, from geodetic to rectangular geocentric is described in section K.4. The transformation is applied to the origin of the object. The result of this transformation is the origin x_{0} of the object in the geocentric coordinate system. Then for each coordinate x of the object, we apply the geodetic to geocentric transformation to coordinate x and we then compute the translation vector t between x and x_{0} in the geocentric coordinate system with equation A8
\[t = x  x_0\] 
(eq. A‑8) 
The second transformation, from geocentric to LVCS is presented here as an algorithm to transform all coordinates of an object from the geodetic coordinate system to LVCS. The transformation from geodetic to LVCS first requires the assembly of a 3x3 rotation matrix M with equation A9:
\[ M = \begin{bmatrix} \sin λ_0 & \cos λ_0 & 0 \\ \sin φ_0 \cos λ_0 & \sin φ_0 \sin λ_0 & \cos φ_0 \\ \cos φ_0 \cos λ_0 & \cos φ_0 \sin λ_0 & \sin φ_0 \end{bmatrix} \] 
(eq. A‑9) 
Where: ϕ_{0} and λ_{0} = the latitude and longitude of the origin of the object.
Finally, the rotation matrix M is applied to the translation vector t to obtain each coordinate x_{L} in the local vertical coordinate system with equation A10:
\[x_L = Mt\] 
(eq. A‑10) 
7.5. Angular Displacements to Linear Displacement
For WGS84, which is an elliptical representation of the earth, the transformation from angular displacements to equivalent linear displacements in a tangential plane is slightly different than that for a spherical earth.
For WGS84 we get…
\[\delta X = \rho _t \cos (lat) \delta lon \]
\[\delta Y = \rho _m \delta lat \]
… as opposed to for a spherical earth
\[\delta X = \rho \cos (lat) \delta lon \]
\[\delta Y = \rho \delta lat \]
where…
δX, δY are the linear displacements along the x and y axes.
ρ_{m}, ρ_{t} are the meridional and transverse radiuses of curvature.
ρ is the radius of the spherical earth.
δlat, δlon are small displacements at location lat/lon
we have…
\[ \rho _m = \frac{a(1  e^2)}{\big[1  e^2 \sin^2 (lat)\big] ^{3/2}} \]
\[ \rho _t = \frac{a}{\sqrt[]{1  e^2 sin^2 (lat)}} \]
\[ e^2 = 1  \frac{b ^2}{a ^2} \]
\[ f = \frac{a  b}{a} \Rightarrow b = a (1  f) \]
where…
e^{2} is the square of the eccentricity
a,b are the semimajor and the minor axes of the earth
f is the flattening
This section describes the transformations required to go toandfrom the DIS/HLA and the CDB moving model coordinate systems.
7.6. 3D Model Coordinate System
The CDB 3D model coordinate system conventions are presented earlier in the OGC CDB Rules for Encoding Data using OpenFlight Best Practice.
Figure 2: CDB 3D Model Coordinate System
The DIS coordinate system is used on a HLA network and is represented on the following figure.
Figure 3: DIS Entity ^{[5]} Coordinate System
The two coordinate systems differ in the axis conventions (Z is up in the CDB while Z is down in DIS). Furthermore, the position of the origin also differs; DIS requires that the origin of its coordinate system be located at the center of the entity’s bounding box excluding its articulated and attached parts ^{[6]}. The CDB standard uses a different convention.
The transformation from the CDB coordinate system to the DIS coordinate system involves one translation followed by two rotations. The translation represents the offset to the DIS origin as defined in chapter 6. Assume that P_{0} represents the coordinate of the DIS origin.
\[ P_0 = (x_0, y_0, z_0) \] 
(eq. A‑11) 
The two rotations are relatively simple. First, rotate 180° about the Xaxis. This rotation will position the Zaxis in its correct position. Equation A12 represents this rotation.
\[ M_x = \begin{Bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{Bmatrix} \] 
(eq. A‑12) 
Second, rotate 90° about this new Zaxis. This last rotation completes the transformation and is represented by equation A13.
\[ M_z = \begin{Bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{Bmatrix} \] 
(eq. A‑13) 
Now, if we combine equations A11, A12 and A13, we can transform a point P expressed in the CDB coordinate system into point P’ in the DIS coordinate system. Equation A14 presents the complete transformation.
\[ P' = M_z M_x (P  P_0) \] 
(eq. A‑14) 
The combined matrix gives equation A15 and the resulting individual terms are presented in A16.
\[ M_zx = \begin{Bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{Bmatrix} \] 
(eq. A‑15) 
\[ x = y  y_0 \] \[ y = x  x_0 \] \[ z' = z_0  z \] 
(eq. A‑16) 
If a single transformation matrix M is preferred then Matrix M_{zx} and point P_{0} are combined to obtain the set of equations A17.
\[ P' = MP \] \[where\] \[ M = \begin{Bmatrix} 0 & 1 & 0 & y_0 \\ 1 & 0 & 0 & x_0 \\ 0 & 0 & 1 & z_0 \\ 0 & 0 & 0 & 1 \end{Bmatrix} \] \[and…\] \[ P = \begin{Bmatrix} x \\ y \\ z \\ 1 \end{Bmatrix} \] 
(eq. A‑17) 
To convert from the DIS coordinate system back to the CDB coordinate system, the inverse transformation is applied. Knowing that unscaled rotation matrices (the upper 3 x 3 portion of M) have the property that their inverse is their transpose, we obtain the set of equations A18.
\[ P = M^{1} P' \] \[where\] \[ M^{1} = \begin{Bmatrix} 0 & 1 & 0 & x_0 \\ 1 & 0 & 0 & y_0 \\ 0 & 0 & 1 & z_0 \\ 0 & 0 & 0 & 1 \end{Bmatrix} \] 
(eq. A‑18) 
Annex A: Revision History
Date  Release  Editor  Primary clauses modified  Description 

20160205 
1.0 
C. Reed 
Various 
Prepare as version 1. 
20160803 
1.0 
C. Reed 
Various 
Modify based on 20160308 SWG discussions 
20160622 
1.0r2 
C.Reed 
Various 
Correct ambiguity WRT how altitude is defined. 
20160710 
R3 
C. Reed 
Various 
Final edits for publication 
20161121 
1.0 
C. Reed 
Ready for publication 

20171228 
1.1 
C. Reed 
Minimal 
Update for version 1.1. Very minor edits. 
20191216 
1.2 
C. Reed 
Various 
Changes for version 1.2 