[GRASSLIST:2574] Re: Multiple UTM zones
Quantitative Decisions
whuber at quantdec.com
Mon Oct 8 17:03:43 EDT 2001
At 11:48 AM 10/8/01 -0700, Rich Shepard wrote:
>On Mon, 8 Oct 2001, Glynn Clements wrote:
> > UTM isn't really "a projection"; it's a family of 60 projections.
...
> I won't argue semantics with you, but I always understood that the
>Universal Transverse Mercator was a projection of the 3D Earth onto a flat
>surface.
You are both right because you each use "projection" in a slightly
different sense. The problem with UTM is that the central meridian of each
zone is different. Therefore each zone is projected with the same _kind_
of projection but the 60 zones comprise 60 distinct projections.
UTM works by wrapping a cylinder around the earth "sideways"; that is, so
that the cylinder contacts the earth along a meridian rather than along the
equator. It projects points from the center of the earth onto the
cylindrical sheet, then distorts the sheet along the long axis of the
cylinder. (Specifically, for a spherical earth the distortion is 1/cos(x)
where x is about 1/111 times the shortest distance in kilometers from the
projection's meridian; a tiny adjustment is made for an ellipsoidal
earth.) The scale distortion increases as one moves east-west away from
the zone's meridian. UTM is designed to keep the maximum distortion within
limits throughout the entire six degree zone (three degrees on either side
of the meridian). To do this, all distances are shrunk slightly (by
0.040%) along the meridian. Scales increase to 1:1 about 1.62 degrees to
either side of the meridian, and are too large by 0.0966% at the zone
boundaries.
Officially, UTM zones extend up to 3.5 degrees from their meridians to
allow for one degree of overlap. At 3.5 degrees the scale distortion has
exceeded 0.14% (at the equator). Beyond there, scale distortions increase
ever more rapidly, as one would expect of a Mercator projection:
Distance (km) from meridian Scale distortion
----------------------------------- -------------------
445 0.20%
557 0.34%
668 0.51%
891 0.94%
1113 1.50%
1670 3.47%
2226 6.34%
3340 15.34%
5009 40.96%
(Only one scale distortion is needed because UTM is approximately conformal
throughout. The distances, which look strange, are nice multiples of one
conformal degree of 111,319.5 meters.)
> > Actually, GRASS doesn't really deal with projections. Most programs
> > just operate upon abstract 2D coordinates, treating the region as if
> > it were flat.
>
> Excellent point. But, aren't distances and directions dependent upon the
>projection used?
Yes. That is why an answer to your original question depends on your
accuracy requirements.
> My concern -- not based on a current project -- is when a project site
>spans the boundary of two UTM zones.
>
> > a) convert the coordinates using m.proj, or
>
> As long as this is an acceptable activity I'll use it.
A lot of people do this. However, if you are in a situation where your
study area spans two or more UTM zones, then either (a) you don't care
about scale distortions (and comparable direction distortions) of the size
tabulated above, or (b) you shouldn't be using UTM.
To help you decide, you can use the expression given above to estimate the
distortions you might experience: the actual scale will be approximately
0.9996/cos(x) (to account for the 0.040% shrinkage at the meridian). For
instance, if your site spans zones 11 and 12 at 40 degrees north latitude
and you elect to use the projection for zone 11 only, then the farthest
part of zone 12 from zone 11's meridian will be nine degrees to the
east. At this latitude one degree of longitude equals cos(40) degrees of
longitude at the equator, so x = (9 degrees * cos(40) * 111 km/degree) /
111 km = 6.89 degrees. The value 0.9996/cos(6.89) = 1.0069 indicates map
distances will be up to 0.69% too large: about seven times the maximum
scale distortion seen within the correct UTM zone. Area distortions will
be about twice as large, up to 1.38%.
--Bill Huber
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
William Huber, Ph.D.
Acting Editor
Directions Magazine
http://www.directionsmag.com
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