Distance measurements across UTM zones: use geographic or planar approaches?

I have a survey grid spread across 3 UTM zones (36N, 36S, 37S). I want to find the nearest (or shortest) distances of the centroids of these grids to roads and various points between.

It seems like there are too many compromises when using any kind of planar projection (read: with regards to preserving distance between any number of points on the map). Should one just forget about using projections in this case and go for goedesic, or ellipsoidal (read: Geographic) techniques?

Is there to anyone's knowledge a planar technique that will preserve distance between any number of points on the map? It does not seem like I can use an equidistant projection with exception of the gnomonic projection. Is this correct?

• What levels of accuracy do you need? (Using centroids as proxies for entire polygonal cells already suggests your accuracy requirements are low.) – whuber May 28 '13 at 15:20
• Because you still have not specified what level of accuracy you need, your last question is unanswerable. A thorough analysis of the errors made in using one UTM zone to make measurements in neighboring ones appears at gis.stackexchange.com/questions/31701/…. Whether the gnomonic projection is a better choice depends on latitude: at Equatorial latitudes it can be superior to UTM for this purpose, but at more extreme latitudes it will be inferior. Note that the gnomonic projection is not equidistant. – whuber Jul 11 '13 at 19:46
• @whuber the centroid issue is one that I cannot get around, nevertheless I need measurements to be <250 meters of known distance – XNSTT Jul 12 '13 at 7:08
• It's hard to understand what you might mean by "preserving the shortest route." A Gnomonic projection merely maps geodesics (on the sphere) to line segments (in the plane). To do this, it grievously distorts the distances. An equidistant projection relative to a base point O, which we may assume appears at the map's origin, has the property that the apparent distances from every mapped point P to the origin are equal to the actual spherical distances between P and O. A Gnomonic projection does not do this. – whuber Jul 12 '13 at 13:37
• Re the accuracy: you won't get that accuracy over long distances even when staying in your proper UTM zone! By design, UTM has a scale factor that is short by 400 parts per million along its central meridian. If you were to measure, say, a distance of 1000 Km north to south along that meridian in the projected coordinates, you would get 999.6 Km: 400 meters too short. Usually people assess accuracy as a fraction of the total distance, expecting the absolute error to increase with distance. (A 250 m error in measuring a football field would be awful!) – whuber Jul 12 '13 at 13:44

Here is a paper that may help in beginning to drive your selection of distance measures. Take note of table 1 (pg. 4), copied below. On geodesic distance modeling and spatial analysis (2004) - S. Banerjee

I would suggest that if you intend to use inter-UTM zone distance computations you should be using a geographic measure. Likewise, the spatial distribution of the points to roads within the UTM may be sufficient in the N/S extent to warrant the use of geographic distance measures.

The real question needs to start as: How accurate do my measures need to be? How many measures will I be making and is the added computational cost of a geographic measure inline with the required solution speed?

Edit for the comment: The answer goes back to your accuracy tolerance. If I needed to compute in planar space over a large distance (3 UTM zones at mid latitudes is sufficiently large) with a high level of accuracy I would likely use a sinusoidal projection. The distances computed using a gnomonic projection are only completely accurate 'from a single reference point' (ref. as above). Are you only measuring from a single point in each UTM zone? If so, use the gnomonic projection. Otherwise, think about computing chordal distance, using a sinusoidal projection, or accepting the accuracy issues.

Given the accuracy requirement without any constraint on potential distance measures you really should be using geodesic measurements. Additionally, the gnomonic projection is not azimuthal equidistant, it just happens to draw the great circle curves as straight lines. As an alternative to geodesic computation you could reproject your data centered on the origin point of your measurement into an azimuthal equidistant projection*.

Having done this for a project involving 20,000+ points and some buffering, it is not efficient to perform for extremely fast lookup. It is a one time, let it run for a minute or so operation.

• thanks - let's say that the required solution speed means that I don't have time for a geographic measurement solution. Will the gnomonic projection suffice? – XNSTT Jul 11 '13 at 18:45

Computing geodesic distances is comparable in speed to anything else you might do with your points. E.g., on my machine (2.66GHz 64-bit Intel) with C++ implementations:

• UTM <-> geographic conversions take about 1 us each way
• 2 geographic coords -> geodesic distance takes about 2.5 us

The conversion from UTM to gnomonic incurs you the cost a UTM to geographic conversion and even then (as whuber points out) the gnomonic isn't a useful projection for distance computations. Perhaps doing the honest-to-goodness distance computations won't be so bad? In 5 minutes you can do about 100 million distance calculations and you won't then have to worry about the accuracy.

Since nothing has been accepted yet, I'll take a shot.

Given the three UTM Zones you listed in your question, is the data contained within Kenya? Or within 4-6 degrees longitude? If so, it may be easiest to just reproject the data into a custom transverse Mercator projection by moving the central meridian a little. From there, you can calculate projected distances.

I'm not sure how or where this calculation is being used, but if that won't work, I'd suggest trying the Vincenty Formula for calculating distance along the ellipsoid. And given modern computers, not that expensive of a computation. For best results in Africa, your datum should be Clarke 1880, since that ellipsoid is the closest fit to the actual Earth for that area.

If that's too slow, there's always the Haversine formula or spherical law of cosines.