Ask Norm - August 1998

Ask Norm - August 1998

Moving a UTM Zone

Some time ago, a major client in California had an interesting problem. Northern California resides entirely in UTM Zone 10, while southern California resides almost entirely in UTM Zone 11. Which Zone should be used to map the entire state? There are two answers which are worth exploring.

Solution One

You could choose either zone. Widths of the UTM zones were selected to meet a specific criteria. Nothing terrible happens when you cross the zone boundary. The mathemagics of the Transverse Mercator projection will still work; well sort of. Using either UTM zone will produce perfectly normal looking maps; but there are two problems which the Casual Cartographer should be aware of.

Problem One

First, the geographic data which is not within the UTM zone being used will not meet the design criteria of the UTM system. The UTM system was designed to enable mapping of the entire globe without having the scale distortion introduced by the projection exceed one part in 2,500. That is, the UTM system is designed such that the grid scale factor will never be less than 0.9996, nor greater than 1.0004. Using UTM Zone 10 to map geography which actually resides in Zone 11 simply causes the representation of reality of features in UTM Zone 11 to be distorted beyond the design limits. For example, if we mapped the entire state of California using Zone 10, the scale distortion of features in the Parker Dam region (the eastern most point in California) would be 1.008, about twenty times the design limit of the UTM system.

Problem Two

Second, the Transverse Mercator projection has a serious accuracy problem with points more than a several degrees away from its central meridian; points will not invert precisely. That is, if you convert a geographic coordinate (i.e. a lat/long) which is, say, 10 degrees away from the central meridian to X and Y, and then convert the X and Y back to geographic, the results will not be exactly the same as what you started with. This effect increases with distance from the central meridian (and also with the latitude).

Solution Two

cautm10.gif (7735 bytes)

A better alternative to the original problem is to create a new UTM zone, say UTM Zone 10˝. This can be done by simply copying the definition of UTM Zone 10 to a new coordinate system definition and changing the central meridian by 3 degrees. In the case of UTM Zone 10˝, we would start with the definition of UTM Zone 10, and change the central meridian from 123° west to 120° west longitude. The central meridian of our new system will now represent a median throughout the region to be mapped and, therefore, reduce the extremes of distortion experienced. In fact, the range of grid scale in this new map is now between 0.9996 and 1.0033. Thus, the maximum value of scale distortion is cut in half.

However, now that we have gone to the trouble of defining a new coordinate system, we can do even better. Recognize that the region being mapped extends from 114° west to 124° west. The range of this region is 10° and the center of the range is 119° west longitude. A minimum amount of distortion can be achieved by using 119° as our central meridian, and adjusting the scale reduction factor.

In choosing a scale reduction factor, a good rule of thumb for us Casual Cartographers is to set the Scale Reduction factor (for the Transverse Mercator Projection) to the cosine of ź of the zone width. For example, the normal width of a UTM zone is 6 degrees. One quarter of 6 degrees is 1˝ degrees. The cosine of 1˝ degrees is 0.999657325. Our zone width is 10° . One quarter of 10 is 2˝, and the cosine of 2˝° is 0.999048222.

So if we choose a central meridian of 119° west, and a scale reduction factor of 0.999, the grid scale distortion of our map ranges from 0.999 to 1.00163. The maximum grid scale deviation from one is now 0.00163, or about seven times better than if we just used UTM Zone 10 for the entire map, and three times better than using UTM Zone 10˝.

In writing this little article, I used our new Tralaine product’s coordinate test feature. It now calculates the grid scale factor and the convergence angle for each point converted. These two values are the most used measures of map projection distortion.

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