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

There are no normative provisions in this document.

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 semi-minor 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 3-D 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 z-axis implicitly points upward with respect to the earth surface.

Figure 1: Cartesian Model positioned to WGS-84 Coordinates

The earth shape is described by the WGS-84 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:

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)

The following equations are used to transform geodetic information to geocentric information according to the following:

From these equations, we define the transformation of each geodetic coordinate as:

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 A-3:

We first compute the longitude λ with equation A-4:

We then compute a first approximation of the latitude assuming a spherical earth model with equation 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 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 A-7

The transformation of a geodetic coordinate into an LVCS coordinate is decomposed into two parts:

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 x₀ 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 x₀ in the geocentric coordinate system with equation 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 A-9:

Where: ϕ₀ and λ₀ = 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 A-10:

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…

e² is the square of the eccentricity

a,b are the semi-major and the minor axes of the earth

f is the flattening

This section describes the transformations required to go to-and-from the DIS/HLA and the CDB moving model coordinate systems.

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 P₀ represents the coordinate of the DIS origin.

The two rotations are relatively simple. First, rotate 180° about the X-axis. This rotation will position the Z-axis in its correct position. Equation A-12 represents this rotation.

Second, rotate -90° about this new Z-axis. This last rotation completes the transformation and is represented by equation A-13.

Now, if we combine equations A-11, A-12 and A-13, we can transform a point P expressed in the CDB coordinate system into point P’ in the DIS coordinate system. Equation A-14 presents the complete transformation.

The combined matrix gives equation A-15 and the resulting individual terms are presented in A-16.

If a single transformation matrix M is preferred then Matrix M_zx and point P₀ are combined to obtain the set of equations 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 A-18.