3.1.1.  Discussion on keeping the Coordinate Systems simple

An ISO 19111 coordinate system is only a set of axes at no particular location, together with geometry formulas that are implied by the coordinate system class (not to be confused with the coordinate system unions, which are not covered in this section). For example, the use of a CartesianCS implies that angles, distances, surfaces, volumes, etc. are computed by formulas derived from Euclidean geometry. The use of a CylindricalCS or SphericalCS implies different sets of formulas for computing geometric properties. But all CoordinateSystem classes defined by ISO 19111 imply relatively simple and well-known formulas. This simplicity is necessary for performing tasks such as computing intersections or solving integral equations for computing surfaces. The most complex coordinate system class defined by ISO 19111 is the EllipsoidalCS. In that latter case, solving the integral equations requires series expansions (Clause 2.19), as done in map projections or geodesic (Clause 2.9) calculations. Extensions to ISO 19111 such as MinkowskiCS should not be much more complex.

Figure 2 — ISO 19111 Coordinate Systems

The simplicity mandated by the approach defined in the above paragraph may seem restrictive, but is an unavoidable step applied even with complex topographies. Complexity is usually managed by splitting a surface into small cells. Inside each cell, the “simple” mathematics of a coordinate system is assumed to be a good approximation. For example, the figure below zooms in on a single cell of a complex surface. If a software package computes the shortest path (the geodesic) between two points inside that cell by applying linear interpolations between (x₁, y₁, z₁) and (x₂, y₂, z₂) coordinate tuples, then that software is implicitly using a Cartesian coordinate system. The cells size (grid resolution) is chosen in such a way that approximating the real surface by a flat surface inside the cell area is considered an acceptable error.

Figure 3 — Surface with local Cartesian CS in each cell

In the above example, the shortest path inside a cell can be computed using the global (Clause 2.10) Cartesian coordinate system for the whole grid (gray axes), or a Cartesian coordinate system which is local (Clause 2.13) to the cell (red axes). In the example, both coordinate systems produce the same results, but the local CS is simpler because it can be reduced to a two-dimensional coordinate system (i.e., the perpendicular axis can be omitted) if all points lie on the surface. Coordinate transformations between those two systems are not discussed in this section, as these transformations require Coordinate Reference Systems (CRS) to provide context for the Coordinate Systems (CS).

The local coordinate system does not need to be of the same class as the global one. For example, the local CS could be an EllipsoidalCS if that is a better approximation of the real geometry inside the area of that cell, as illustrated in the figure below. In this case, doing geodesic calculations in the local CS produces different results than the cases illustrated in Figure 3.

Figure 4 — Surface with local ellipsoidal CS in each cell

In the above examples, calculations must be limited to a single cell. If the feature of interest spans more than one cell, then calculations must be performed separately in each cell and the results combined in some way. ISO 19111 does not specify how multi-cell calculations should be done.

An application can choose to use the same CoordinateSystem (CS) for all cells, even if axis directions vary slightly between different cells. Using a common CS means that all cells use the same geometric formulas with axes having the same units of measurement and the same orientations relative to the cell. The precise orientations of the axes are encapsulated in the Reference Frame, which is not part of a Coordinate System (it is part of a Coordinate Reference System). Using a uniform Coordinate System makes combining the results of adjacent cells easier, for example, by stating that all cells use a CartesianCS with measurements in meters. Therefore, volumes or areas computed in adjacent cells can be summed without unit conversions. This is true even for EllipsoidalCS using ellipsoids of different sizes and shapes. In the ISO 19111 model, the shape of the ellipsoid is not a property of the Coordinate System. Rather, it is a property of the Geodetic Reference Frame which, together with the EllipsoidalCS, forms a Geodetic Coordinate Reference System.

3.1.2.  Discussion on CRS domain of validity versus CS simplicity

An ISO 19111 Coordinate Reference System (CRS) is associated with exactly one Coordinate System (CS), except Compound CRS, which is omitted for the following discussion. A CRS is also associated, through subclasses of the datum class, with information about the CRS origin and “absolute” axis orientations (e.g., relative to stars). Consequently each cell in a grid such as Figure 3 or Figure 4 has its own local CRS because each cell has a different reference frame for different origin and “absolute” axis directions.

Coordinates can be transformed from the global CRS to the local CRS of a particular cell before computing geometric properties in a given cell. Note that in this scenario, the complexity of the irregular surface appears in the process of transforming coordinates from the global CRS to a local CRS. This complexity does not appear in any coordinate system before or after the transformation. In ISO 19111 modeling, the complexity is in CoordinateOperation, not in CoordinateSystem.

Figure 5 — ISO 19111 Operations and Coordinate Reference Systems

Given the restriction that each ISO 19111 CRS (excluding CompoundCRS) shall be associated with exactly one CS, and given that a CS should be mathematically simple, the CRS in Figure 3 and Figure 4 can be based on one of the following choices.

1. The global three-dimensional Cartesian coordinate system, in which case the complex surface plays no role. That surface is a feature inside the CS domain, but does not define the CS itself and is not involved in any calculation between pairs of (x, y, z) coordinate tuples.

2. A global two-dimensional coordinate system resulting from a curve-fitting of the complex surface. The result can be the surface of a plane, cylinder, sphere, ellipsoid, or something else defined outside ISO 19111. That approximation can be very good in some cases (e.g., Earth surface approximated by an ellipsoid). This option is nevertheless not discussed further in this Engineering Report.

3. A local two-dimensional coordinate system on the complex surface, i.e., the CS of one of the cells. That coordinate system may be Cartesian or ellipsoidal (among others) as illustrated in the two above examples, depending on the desired compromise between accuracy and mathematical simplicity.

Each alternative has pros and cons. Option #1 allows the referencing of any point anywhere in the domain covered by the global CRS. However, it does not allow accurate geodesic calculations (e.g., shortest path on the surface) between pairs of points. Conversely, option #3 is limited to a small area, but allows much more accurate geodesic calculations in that area.

Note also that complex numerical models, such as those based on fluid mechanic (meteorological and oceanographical models) or based on the equations of Einstein’s General Relativity (Annex E.2), do not change the fundamental approach. Those numerical models still need to split a large space into a grid (Clause 2.11) of cells small enough to allow the use of “simple” mathematics inside each cell with acceptable errors. The grids may be more sophisticated than grids of equilateral cells, but they are still grids with cells delimited by analytic functions.

The above discussion implies that if geometric properties or geodesics paths are required with high accuracy, then the associated calculations may require a CRS restricted to a local area. However, the area can be made larger by a choice of coordinate system class, or by a choice of kind of acceptable errors. This is partially illustrated for one-dimensional coordinate systems by the figure below. However, the discussion should be thought of as if the coordinate systems were two- or three-dimensional. The left side shows the case when Cartesian coordinate systems are wanted. No CartesianCS can be a good fit for the whole mountain, so the calculations must be restricted to smaller areas. The calculations can only be done between pairs of points on the same side of the mountain. If a calculation needs to be done between two points that are on each side of the mountain, it needs to be split into two parts, computed independently on each side, then combined somehow. In the ISO 19111 model, those two computations are done in two different Coordinate Reference Systems. However, if the coordinate system is based on a quadratic curve instead of straight lines, as shown in the right side of the figure below, then it becomes possible to do computation on the whole mountain with a single CRS. Therfore, computation between points on different sides of the mountain is no longer a problem.

Figure 6 — More complex coordinate system covering larger area

In a two-dimensional coordinate system, the use of EllipsoidalCS instead of CartesianCS allows covering a larger surface with a continuous function in a way similar to Figure 6. In the case of the Earth, using EllipsoidalCS is often an approximation good enough for covering the whole planet with a single CRS, such as WGS84. However, it does not change the general principle that EllipsoidalCS is still an approximation. On some irregular bodies, that approximation may not be good enough to support the use of a single EllipsoidalCS for the whole body with the desired accuracy.

3.1.3.  Coordinate System versus Coordinate Transformation roles in accuracy

A coordinate system based on a cubic curve would be more accurate or would cover a yet larger area than the examples shown in Figure 6, but at the cost of more mathematical complexity. Because the equations for intersections, surfaces, etc. can quickly become unsolvable, that mathematical complexity cannot be increased indefinitely. For example, ellipsoidal equations are only of degree 2, and already very difficult to solve. To cover a larger area, an alternative to increasing mathematical complexity is to keep a simpler coordinate system, such as CartesianCS, but accept some errors. For example, it is possible to favor accuracy in surface calculations at the expanse of accuracy in angles, or conversely. Controlling those errors is the art of map projections.

The overall accuracy does not depend only on the accuracy of formulas in the chosen coordinate system. Overall accuracy also depends on the accuracy of the transformation used for expressing points in that coordinate system. For example, Figure 7 below illustrates the results of a numerical experiment done at a point located at 45°N on Earth. The following test was repeated at distances varying from 1 meter to 1000 kilometers around the above-cited point.

• A circle is drawn around the center point (0°E 45°N) with a radius equal to the distance to test. The circle is determined by computing the locations of 360 points (each point is separated by 1°) using geodesic formulas on an ellipsoid published by Karney (2013).

• Distances and azimuths (horizontal angles from north) are computed for paths from the center point to the above-cited 360 points using the following methods.

  • Using SphericalCS formulas (distances are the arc lengths):

    • On an authalic sphere (sphere of the same surface as the ellipsoid) using the geographic coordinates as-is; and

    • On the same authalic sphere, but with geographic coordinates converted from the geodetic latitudes on the flattened sphere (Earth) to the geocentric latitudes on the authalic sphere. Those latitudes differ by about 21 kilometers. This operation is identified as a “datum shift” in Figure 7.

  • Using CartesianCS formulas (distances computed by the Pythagorean formula):

    • After projection to “Mercator (variant B)” with standard parallel at 45°; and

    • After projection to “Lambert Conic Conformal (2SP)” with the two standard parallels at 45°.

• The above calculations are compared with the expected distances and azimuths, which are the values used for drawing the circle in the first step.

The Y axis below shows the relative error, as a percentage of the distance shown on the X axis. For example, an error of 0.25% for a distance of 10 kilometers is an error of 25 meters. The error bars correspond to one standard deviation.

Figure 7 — Relative errors of distances around 0°E 45°N computed with different methods

An observation is that naive application of spherical formulas on Earth, with WGS84 latitudes and longitudes used directly, is not very good compared to other approaches. The relative errors are constant at around 0.13% over the range of tested distances, and the standard deviation is almost as large. A more sophisticated calculation converting the latitude values before applying spherical formulas gives better results, with relative errors at around 0.06% and negligible standard deviations. However, and maybe counter-intuitively, the use of CartesianCS gives better results than SphericalCS over small distances. Two factors contribute to that accuracy:

• the use of a “standard parallel” projection parameter configures the map projection for optimal accuracy around the latitude used in this test, namely 45°N; and

• the Mercator and Lambert Conic Conformal map projection formulas take into account the flattened shape of Earth.

On distances small enough, the geometrical distortions caused by using CartesianCS are smaller than the errors caused by ignoring Earth flatness in spherical formulas or caused by using a global sphere radius which may not be optimal for the test area. (Note: a sphere with geocentric radius computed at 45°N was also tested, but the results are not shown because they were similar to the authalic sphere case). In this test, the use of “Mercator (variant B)” projection is more accurate than spherical formulas for distances up to 100 kilometers and the use of “Lambert Conic Conformal (2SP)” is still more accurate even at 1000 kilometers.

The results of using “Popular Visualization Pseudo Mercator” are not shown in this graph because the errors are around 41% and would not fit in the Y axis scale. Those errors are caused mostly by the scale factor not well suited to 45°N. Other factors contributing to the errors are the facts that this projection ignores Earth flatness and uses the equatorial radius instead of some globally better value such as the authalic radius (radius of the authalic sphere).

3.1.4.  Implications for MinkowskiCS

MinkowskiCS is a four-dimensional coordinate system developed for use in Special Relativity. It ignores the spacetime curvature caused by gravity according to General Relativity (Annex E.2). However, even in the context of General Relativity, the Minkowski coordinate system can be useful as an approximation good enough at short distances from the object of interest. Such an approximation is like the use of CartesianCS in map projections, which are flat approximations of a curved surface considered good enough inside a known region.

Because Cartesian and Minkowski coordinate systems are flat spacetime approximations, it is tempting to conclude that geodesic (Clause 2.9) calculations performed in those coordinate systems can only be less accurate than geodesic calculations performed in a coordinate system taking the curvatures in account, such as a spherical coordinate system. But as shown in Figure 7, this is not necessarily the case because the overall accuracy depends on two factors: the fitness of the coordinate system to the local geometry and the accuracy of the coordinate operation applied to get there (Clause 3.1.3).

The choice of a coordinate system impacts the complexity of a fundamental object of study in General Relativity. In the context of Einstein’s relativity, spacetime geometry is described by a metric tensor. The metric can be represented as a matrix and defines how distances, angles, and other geometry properties are computed. While those metrics are of particular importance in relativity, this concept can be used in classical geometry as well. The Euclidean metric used in CartesianCS is the simplest:

Let p be the coordinate tuple of a point in space or space-time, and p_i (or p_j) be one of the coordinates of that tuple. For a small displacement dp, the small distance ds is given by:

For the Euclidean metric, the development gives the familiar Pythagorean formula:

By comparison, the metric at the surface of a SphericalCS is as below, assuming a radius of 1, coordinates in (latitude, longitude) order and units in radians. The main thing to note is that this metric is not constant. The matrix varies at every position, because it depends on the latitude φ. Consequently, computing the geodesic distance s² on a non-short path requires resolving an integral equation (not done in this ER).

MinkowskiCS can be seen as CartesianCS generalized to the spatiotemporal domain. Its metric is shown below, using the (- + + +) sign convention. Coordinates are in (t, x, y, z) order with the time coordinate already multiplied by c, as required by the MinkowskiCS definition proposed in this ER:

In the same way as the metric on the surface of a SphericalCS may be considered of manageable complexity, there are some alternatives to MinkowskiCS which are also of manageable complexity. For example, assuming a system with spherical symmetry residing in a vacuum, then the Schwarzschild metric (Annex E.7) may provide a computationally feasible solution for ds² in Formula (2). This ER does not propose classes for modeling those alternatives yet, but it is expected that if MinkowskiCS is accepted, other classes could follow the path.

MinkowskiCS may not provide sufficient accuracy in some situations in the same way as CartesianCS on the Earth’s surface is not accurate when the area of interest becomes too large. The accuracy on Earth can be improved by “upgrading” CartesianCS to EllipsoidalCS. Likewise, the accuracy in a relativist context may be improved by “upgrading” MinkowskiCS to another relativist coordinate system using a more complex metric. However, those “upgrades” cannot be such that the geometry formulas become unsolvable. When the combination of a large region and high accuracy exceeds what can be solved, splitting the region in smaller cells as illustrated in Figure 3 become unavoidable.

This Engineering Report does not explore how to combine calculations done in adjacent cells. Doing so may require the participation of other OGC working groups such as Discrete Global Grid Systems (DGGS). However, those combinations would probably require coordinate transformations between cells. This is where the complexity of geoid surfaces or gravitational fields are involved. But this complexity is in the transformation between two CRSs, not in the CRSs themselves.

3.1.5.  Engineering CRS for spacecraft

The coordinate reference system of a spacecraft can be described by an EngineeringCRS. This type of CRS was discussed in OGC 22-038r2 and an example is provided with the DART.xml file. ISO 19111 restricts the set of CS classes that can be associated to an EngineeringCRS. The allowed CS classes are affine, Cartesian, cylindrical, linear, ordinal, polar, and spherical implying that MinkowskiCS is also accepted because it is a subtype of CartesianCS. However, this is not necessarily the case for other metrics such as Schwarzschild (Annex E.7). Relaxing this restriction on CS type is proposed in Clause 3.2.3.3.

3.2.1.  Additions to OGC Topic 2/ISO 19111

Most classes listed below are new and do not require modifications to the ISO 19111 Standard, as the classes can be specified as an extension in a separate document. An exception is CelestialBody which may need to be included in ISO 19111 if an association from Datum to CelestialBody is added.

3.2.2.  Additions to OGC WKT 2 / ISO 19162

This section proposes new keywords for the Well Known Text (WKT) 2 format. Those additions require changes in the definitions provided in Backus-Naur form (BNF). The following table shows a summary of new WKT keywords. Those new keywords should be added (except Minkowski) to the Well-known text representation of coordinate reference systems Standard (OGC 18-010r11) §6.6 — Reserved keywords, with “TBD” replaced by the actual section numbers where the keywords are defined.

Table 5 — Summary of new WKT keywords

[ { <wkt separator> <alias> } ] ...

<alias>         ::= <alias keyword> <left delimiter> <alias text> <right delimiter>
<alias keyword> ::= ALIAS
<alias text>    ::= <quoted Latin text>

<celestialBody>          ::= <celestial body keyword> <left delimiter>
                             <celestial body name> <right delimiter>
<celestial body keyword> ::= CELESTIALBODY
<celestial body name>    ::= <quoted Latin text>

<geodetic reference frame keyword> <left delimiter> <datum name> <wkt separator>
[ <wkt separator> <celestialBody> ] <ellipsoid>  [ <wkt separator> <datum anchor> ]
[ { <wkt separator> <identifier> } ]…  <right delimiter>
[ { <wkt separator> <prime meridian> } ]

3.2.3.  Changes to ISO 19111

This section describes proposed changes to ISO 19111 abstract model. If accepted, those changes would require the publication of an ISO 19111 amendment and/or an OGC corrigendum.

3.2.3.2.  Remove ObjectUsage for generalization

In the ISO 19111 model, most classes inherit a set of properties from the IdentifiedObject parent class. Those properties are name, alias, identifier, and remarks. But there is also one additional property, namely domain, which is inherited only by the Datum, CRS, and CoordinateOperation classes. ISO 19111:2019 does this restriction by defining those properties in an intermediate class named ObjectUsage. ISO 19111:2007 did this restriction by repeating properties in the three above-cited classes, in a way similar to the UML diagram provided in section Clause 3.1.

The following UML diagram shows the ISO 19111:2019 model. The ObjectUsage contains only a domain property and is inserted between the IdentifiedObject parent class and the Datum, CRS, and CoordinateOperation subclasses. This insertion was done for preserving the same restriction as in ISO 19111:2007, where only Datum, CRS, and CoordinateOperation could have a scope and domain of validity.

Figure 12 — Current ISO 19111:2019 model with object usage

Removing the restriction saying that only Datum, CRS, and CoordinateOperation can have a scope or a domain of validity is proposed. There is nothing inherently wrong in specifying a scope on most kinds of CRS-related objects. For example:

• A MinkowskiCS definition may want to specify “for use with special relativity” as the scope.

• An Ellipsoid definition may want to specify a limited geographic domain of validity if the celestial body has an irregular shape with different ellipsoidal approximations for different geographic areas.

• A PrimeMeridian definition may want to specify a limited temporal extent if it is determined by an ephemeral feature, for example of the surface of a gas planet.

• A CoordinateSystemAxis with direction such as “South along 60°E meridian” may want to specify ”for use at North pole” as the scope.

The restriction can be removed by deleting the ObjectUsage class (not to be confused with the ObjectDomain class, which is kept unchanged), and moving the domain attribute up to the IdentifiedObject parent class.