Accuracy of planimetric geometry vs. ellipsoidal model

Question

I'm examining the two geospatial options for IBM's DB2 database. In particular I am interested in the accuracy between these two options and whether it's worth it (the more accurate representation isn't free).

The thing I am most concerned about is someone doing a query (e.g. looking for a point or other geometry within a polygon) and the database incorrectly returning an empty resultset. My use case is a catalogue indexed by several means (temporal, geospatial and XML) and entries can be anywhere on the globe.

IBM states (PDF link) that:

Spatial Extender treats the Earth as a flat map. It uses planimetric (flat-plane) geometry, which means that it approximates the round surface of the Earth by projecting it onto a flat plane. This projection causes distortions, which can vary across the extent of the data, but the distortions generally increase toward the edges of the projected region…Spatial Extender is best used for local and regional data sets that are well represented in projected coordinates, and for applications where location accuracy is not important.

The other option is:

Geodetic Extender treats the Earth as a globe. It uses a latitude and longitude coordinate system on an ellipsoidal Earth model. Geometric operations are precise, regardless of location…Geodetic Extender is best used for global data sets and applications that cover large areas on the Earth, where a single map projection cannot provide the accuracy required by the application.

If I was using something like WGS84 in the Spatial Extender geospatial implementation, can anyone give me a ballpark figure of the worst-case accuracy errors I might encounter when compared to using the Geodetic Extender? Is there a formula or process for working it out for anywhere on the globe for a given map projection?

Additional information: "When to use DB2 Geodetic Data Management Feature and when to use DB2 Spatial Extender".

Answer 1

Ultimately, because all extensive features are given by curves and those curves are approximated by line segments (when projected) or geodesic segments (when not projected), any errors are due to the fact that a line segment joining two projected points likely deviates (at least a little) from the projection of a geodesic between those two points. The amount of deviation obviously depends on the projection. It also depends on how the curve is approximated by a sequence of segments (its "digitized," vector version). This makes it impossible to give a general answer or formula.

The problem and a process to assess it is illustrated in the question and answers at Why is the 'straight line' path across continent so curved? . The question shows two points and a piecewise approximation to the projected geodesic between them. Obviously this approximation differs from the line segment connecting the points (in the map). Indeed, the problem was found by obtaining a sequence of points along that geodesic and projecting them, then comparing those points (vertices of the broken curve) to the straight line between the projected points. The error in this example is extreme. Errors can become practically infinite when features come close to or cross singularities of the projection: a feature that crosses one of the poles in a Mercator projection is a good example.

My reply to that question maps the same feature with a different projection. In the new projection, there is no error even when the original geodesic is not broken. Although this is not shown, one demonstration that the new projection is free of error for this geodesic would be to break the geodesic into a sequence of points and separately project them: they would lie along the same projected line. (The new projection makes errors for other geodesics, of course: no projection can render all geodesics free of some kind of metric distortion.)

Thus, a general process for evaluating error consists of making two maps of the features: the original map and another one in which all sufficiently long segments of the digitized features are first subdivided into smaller segments. Potential errors occur where corresponding features in those two maps disagree substantially.

In general, when features are digitized at a scale appropriate for the application and a suitable projection is shown, calculations made in projected coordinates are adequate. When features extend over large parts of the globe, it often becomes necessary to change projections (on the fly) to accommodate specific analytic needs, such as preserving area or relative angles. If you know that all analyses will be based on geodetic coordinates (that is, using spherical geometry) and you exploit that fact when creating the digitization (such as when portions of features fall along long geodesics), you will likely need the "Geodetic Extender" option. When features extend over large parts of the globe, there may be few or no projections that adequately render all the features with sufficient accuracy for the intended analyses and again the "GE" option is worthwhile. Good examples are climatological models and oceanographic research covering multiple ocean basins. In all other cases--including almost every project at a continent level or smaller--this option is probably not necessary.