The width of each UTM longitude zone decreases poleward from the equator (see Figures 2.3
and2.7), so the range of UTM easting coordinates depends on the latitude. We can exploit this dependence to refine (narrow) the range approximation derived in SectionA.1.
Consider any fixed latitude, the corresponding parallel, and the coincident arc that extends between the two adjacent bounding meridians. Neglecting distortion, the range of UTM easting coordinates at that latitude is approximated by the length of the arc. Assuming the longitude zone is symmetric about its central meridian, we obtain the following approximation for the range: xcm− Sφ 2 ≤ x ≤ xcm+ Sφ 2 (A.3.1) where
φ is the latitude angle (|φ| ≤ π/2 rad), with positive and negative values corresponding to northern and southern latitudes, respectively; and
Sφ is the arc length as measured at the parallel of latitude φ.
Assuming that every parallel of latitude is circular, the length of an arc measured along a parallel is given by
Sφ= rφ∆λ rad−1 (A.3.2)
where rφis the radius of the circle coincident with the parallel of latitude φ. As a first approxi-
mation assume the earth is spherical; then from trigonometry the radius of the circle is related to the radius of the sphere (rsphere), and to the latitude angle, as
This result is valid for both positive and negative values of φ (i.e., both northern and southern latitudes), because the cosine function is symmetric about the angle φ = 0. Substituting the right-hand side of equation (A.3.3) for rφ in equation (A.3.2) gives
Sφ= rsphere cos φ ∆λ rad−1 (A.3.4)
The earth is not spherical, but is more closely approximated by an ellipsoid with its equa- torial radius (a) slightly larger than its polar radius (b). For example, for the WGS 84 datum a = 6378.137 km and b = 6356.752 km [NIMA, 2000]. One can see from equation (A.3.4) and expression (A.3.1) that using the equatorial radius to estimate rsphere will result in a greater es-
timate of the arc length, and hence a more conservative (wider) estimate for the range of easting coordinates, so that’s what we’ll do. Thus, the result obtained for the arc length depends in part on the particular datum used.
Example: Ranges of UTM Easting Coordinates, for Regular UTM Zones, at Extreme Latitudes
Problem: What are the approximate ranges of UTM easting coordinates at the most southerly and most northerly limits of regular (6◦-wide) UTM zones? Base your results on the WGS 84 geoid.
Solution: Use ∆λ = π/30 rad, rsphere= a = 6378.137 km (WGS 84), and xcm = 500 000 m.
Part 1: Most Northerly Limit
At the most northerly limit the latitude is 84◦N., so φ = 84◦ = 7 π/15 rad. Substituting these results for the corresponding variables in equation (A.3.4) gives Sφ= 69 816.33 m. Substituting the above values for the corresponding variables
in expression (A.3.1) yields
465 092 m ≤ x ≤ 534 908 m (WGS 84)
Part 2: Most Southerly Limit
At the most southerly limit the latitude is 80◦S., so φ = −80◦ = −4 π/9 rad. Substituting these results for the corresponding variables in equation (A.3.4) gives Sφ= 115 982.6 m. Substituting the above values for the corresponding variables
in expression (A.3.1) yields
442 009 m ≤ x ≤ 557 991 m (WGS 84)
As expected, the range of easting coordinates is wider for the most southerly limit than for the most northerly limit.
Example: Ranges of UTM Easting Coordinates for 12◦-wide Irregular UTM Zones
Problem: Estimate the maximum range of easting coordinates for those irregular UTM zones that are 12◦ wide (i.e., UTM zones 33X and 35X). Base your results on the WGS 84 geoid.
Solution: UTM zones in the northern hemisphere are widest at their most southerly limits. For UTM zones that lie within UTM latitude zone X, this corresponds to 72◦N. lat. Like all regular UTM zones, the 12◦-wide irregular zones 33X and 35X are symmet- ric about their respective central meridians. Therefore, the range expression (A.3.1) applies to zones 33X and 35X as well.
Use φ = 72◦= 2 π/5 rad, ∆λ = 12◦ = π/15 rad, rsphere= a = 6378.137 km (WGS 84),
and xcm = 500 000 m. Substituting these numerical values for the corresponding vari-
ables in equation (A.3.4) gives Sφ= 412 795.374 m. Substituting the above numerical
values for the corresponding variables in equation (A.3.1) yields
293 603 m ≤ x ≤ 706 397 m (WGS 84)
Thus, the maximum range is more than 3.5 times that of the regular UTM zones at the same latitude (see previous example).
Exercise: Low-Precision Approximation for Range of UTM Easting Coordinates at Specified Latitude
For quick error screening of coordinate data, an easy-to-remember, low-precision approxi- mation is useful. Several datums used in the U.S. are based on ellipsoids whose semimajor axes are approximately 6378 km (to four significant digits). These include NAD 27, NAD 83, and WGS 84. Use the expressions given in Section A.2 to show that if the semimajor axis a = 6378 km, then to three significant digits the range of values for UTM easting coordinates at latitude φ, for all regular UTM zones, is approximately
500 000 m −S 0 φ 2 ≤ x ≤ 500 000 m + Sφ0 2 where Sφ0 = (668 000 m) cos φ
Approximate Ranges for UTM
Northing Coordinates
Ignoring distortion, one can assume the UTM northing coordinate is measured parallel to the central meridian of the corresponding UTM longitude zone. Then the difference between the zone’s maximum and minimum northing coordinates is approximated by the length of the arc that extends from the zone’s most southerly parallel to its most northerly parallel, along the central meridian.
As a first approximation we assume the earth is spherical; then the meridian is circular so the arc length is given by
Smeridian= rmeridian∆φ rad−1 (B.0.1)
where rmeridian denotes the radius of the circle coincident with the meridian and ∆φ denotes the
positive angle (i.e., the latitude increment) subtended by the arc.
The earth is not spherical, but is more closely approximated by an ellipsoid with its equatorial radius (a) slightly larger than its polar radius (b). For example, for the WGS 84 datum a = 6378.137 km and b = 6356.752 km [NIMA, 2000]. Using the equatorial radius to estimate rmeridian
will result in a greater estimate of the arc length, and hence a more conservative (wider) estimate for the range of northing coordinates, so that’s what we’ll do. Thus, the result obtained for the arc length depends in part on the particular datum used.
B.1
Northern Hemisphere
Each UTM longitude zone in the northern hemisphere extends from the equator northward to the parallel of 84◦N. lat. The northing coordinate is assigned a value of y = 0 m at the equator and increases poleward, so it (approximately) satisfies the following:
0 m ≤ y ≤ Smeridian (B.1.1)
Substituting ∆φ = 84◦= 7 π/15 rad and rmeridian= a in equation (B.0.1) leads to
Smeridian=
7 π 15 a
Combining this result with equation (B.1.1) yields the following approximation for the maxi- mum possible range of northing coordinates within any UTM longitude zone in the northern
hemisphere:
0 m ≤ y ≤ 7 π
15 a (B.1.2)
For example, using the numerical value of the equatorial radius corresponding to the WGS 84 datum in expression (B.1.2) gives the following approximation:
0 m ≤ y ≤ 9 350 837 m (N. hemis., WGS 84)