Issues with Extending a UTM Zone
Page 1
Issues with Extending a UTM Zone
Ryan Brazeal, M.Sc., P.Eng., P.Surv., PMP
Geomatics Technology Manager, Caltech Surveys Ltd.
July 2016
Introduction
The Universal Transverse Mercator (UTM) system is designed to provide a means of representing each
point on the Earth (except for areas surrounding the Poles) using a set of Cartesian (X,Y or E,N)
coordinates. The benefits of representing points on the Earth with Cartesian coordinates are for easy planar
mapping, easily deriving spatial information (e.g., distances, angles, areas, etc.) from points’ coordinates as
well as the easy ability to calculate the coordinates of a point of interest based on spatial information.
However, the trade-off for the easiness of these operations is often at the expense of the accuracy of the
solution. The UTM system was adopted by the U.S. Army in 1947 for large-scale military maps of the entire
world (Synder, 1987). Here in Canada, the UTM system has also been utilized for many national and
provincial mapping purposes. The Canada Centre for Mapping and Earth Observation (CCMEO) uses the
Universal Transverse Mercator Projection for its popular National Topographic System (NTS) series at
1:50,000 and 1:250,000 scales (NRCan, 2016). Information Services Corporation of Saskatchewan (ISC)
requires that surveys submitted must have two points georeferenced to the NAD83(CSRS) datum UTM
Extended Zone 13 in order to be approved for filing in the Land Surveys Directory (ISC, 2014). Before we
begin discussing the issues with extending a UTM zone, let us first clearly understand the ideas and
concepts behind a regular UTM zone.
The Regular UTM Zone
The regular northern UTM zone coordinate system definition (for all but a select few geographical areas on
the Earth) only allows geodetic points within a 6 degree longitudinal area, between 84 degrees north latitude
and the Equator, to be converted into projected points. Our discussion will be constrained to these northern
UTM zones and as such southern UTM zones will not be discussed here; an interested reader is referred to
Snyder, 1987. These 6 degree longitudinal areas are predefined, and hence cannot be any arbitrary 6 degree
longitudinal area. The 1st predefined area (called a zone) is between 180 degrees longitude and 174 degrees
west longitude with each successive zone progressing easterly. Figure 1 illustrates the extents of each UTM
zone (zone numbers are shown along the bottom axis in Figure 1).
Issues with Extending a UTM Zone
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Figure 1: UTM zones (source: Synder, 1987)
There are 6 defining parameters underpinning a regular northern UTM zone coordinate system, they are:
1) A transverse Mercator projection (i.e., a set of mathematical algorithms) is used to convert a
longitude, latitude coordinate pair (i.e., geodetic point) to an X, Y coordinate pair (i.e., projected/
Cartesian point). Metric units are traditionally used.
2) The Y axis of the Cartesian UTM coordinate system is aligned with the meridian at the midpoint of
the zone limits (e.g., for the 1st Zone it’s Y axis would be aligned with the meridian at 177 degrees
west longitude). The meridian that is used for this purpose is called the central meridian of the
UTM zone. Any geodetic point that has a longitude equal to the central meridian’s longitude would
have a projected X coordinate of 0m.
3) The X axis of the Cartesian UTM coordinate system is aligned with the equator. This is common
amongst all UTM zones. This parallel is called the latitude of origin of the UTM zone. Any
geodetic point that has a latitude equal to the latitude of origin will have a projected Y coordinate
equal to 0m.
4) In order to avoid experiencing negative X coordinate values within a UTM zone, an offset is
applied to the X Cartesian coordinates along the central meridian. An offset of +500,000m is
applied to all X coordinates. As a result, any geodetic point that has a longitude equal to the central
meridian’s longitude will now have a projected X coordinate equal to 500,000m. This concept is
commonly called the false easting of the UTM zone and is common amongst all UTM zones.
Issues with Extending a UTM Zone
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5) For northern UTM zones there is no chance of experiencing negative Y Cartesian coordinates and
as such no Y coordinate offset needs to be applied (i.e., offset of 0m). This concept is commonly
called the false northing of the UTM zone and is common amongst all northern UTM zones.
6) The final defining parameter is a bit harder to describe. One needs to have a firm understanding of
the geometric perspectives of map projections in order to clearly visualize this point (this is
outside the scope of this article). In short, the derived X and Y coordinates are scaled by a numeric
factor (commonly called the scale factor or grid scale factor) in order to minimize the scale
variations within a UTM zone (Snyder, 1987). For all UTM zones this scale factor is defined to be
0.9996 (a 1:2,500 scale reduction) and is applicable along the central meridian. This has the
implication that spatial quantities (e.g., distances) within a UTM zone will be either too small, too
large or truly correct, with respect to quantities measured on the reference ellipsoid. This depends
upon where within the UTM zone the quantities are being observed. One important point to
recognize is that this scale factor relates how points on the surface of a reference ellipsoid (i.e.,
geodetic points) relate to their derived projected points on the mapping/planar surface. Neither of
these surfaces are guaranteed to align with the topographical surface of the Earth where we
actually make measurements (commonly referred to as the ground surface). Extra considerations
must be taken into account in order to use UTM coordinates in deriving so-called ground
information.
In summary, there are 6 defining map projection parameters for a UTM zone. For northern UTM zones only
the central meridian parameter varies while the other 5 parameters remain constant. If the defining rules for
UTM zones are rigidly applied, the derived spatial information from a UTM defined map/dataset can be
guaranteed to have an accuracy of no worse than 1:1,000 with respect to the reference ellipsoid (Synder,
1987). However, in Saskatchewan if the UTM extent rules are rigidly applied, the derived spatial
information from a UTM dataset is guaranteed to have an accuracy of no worse than 1:2,500 with respect to
the reference ellipsoid (we will call this term the UTM accuracy). These rigid rules dictate that any point on
the Earth only belongs within a single UTM zone. Even two points positioned geographically close to one
another can each be within a different UTM zone. These two points are then not directly relatable to one
another as their coordinates are each in a different coordinate system. This by-product of the rules can be
undesirable and can present challenges in mapping, measurement, calculation and design tasks. A
commonly used ‘solution’ to this problem is to disregard the rule that states that only points within +/- 3
degrees of longitude with respect to the central meridian are allowed to be projected into the applicable
Issues with Extending a UTM Zone
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UTM zone. This ‘solution’ mathematically extends the UTM zone but in doing so it introduces greater scale
(and angular) variations as well as the potential to introduce more mathematic fragility in the derived UTM
X, Y coordinates. These aspects need to be closely examined in order to better understand the issues
surrounding extending a UTM zone.
Transverse Mercator Algorithms
The transverse Mercator projection mathematics require the use of calculus in order to convert a geodetic
point to a projected point, and vice versa. However, in order to create practical algorithms, certain calculus
operators (e.g., an integral) are numerically approximated using one of many different techniques. These
numerical approximations are a necessity but can introduce rounding type errors within a solution.
Commonly used transverse Mercator algorithms are based on Taylor series expansions and begin to grade
beyond the standard 6-degree zone-width of UTM (Stuifbergen, 2009). The transverse Mercator algorithms
shown in Figure 2 are somewhat cryptic in appearance but nicely illustrate - besides the typewriting skills of
someone in the early 70’s - how a series expansion (which technically contains an infinite number of terms)
is cut off after a certain number of terms. However, when a UTM zone is extended, the probability increases
that more terms should be included within these algorithms in order to achieve suitably accurate results.
This is a difficult concept to test, as it requires one to be aware of the exact algorithms that his/her software
is utilizing. Nonetheless, it is important to understand that different software may produce different results
for the projected coordinates of a point within an extended UTM zone.
Figure 2: Transverse Mercator Forward Mapping Algorithms
(source: Krakiwsky, 1973)
Testing Points
In order to understand the issues of scale variations within an extended northern UTM Zone 13, our
discussion will now focus on some selected testing points within Saskatchewan. In order to conserve space,
angular variations are not quantitatively discussed in this article. Figure 3 illustrates the three different
Issues with Extending a UTM Zone
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northern UTM zones (i.e., 12, 13 and 14) that are applicable in Saskatchewan. The testing points are also
shown in Figure 3 and labelled as A, B and C. These three points were selected because they reside within
Saskatchewan but are outside of the regular northern UTM Zone 13. These three points will represent the
extreme cases of the scale (and angular) variations experienced as a result of using an extended UTM Zone
13 within Saskatchewan.
Figure 3: Saskatchewan’s UTM zones
(source: Ryan Brazeal, 2016)
Examining testing point A (49o 00’ 00” N, 101o 21’ 43” W), we notice that it is just slightly outside the
extents of the regular UTM Zone 13 (approximately 47 km); but the scale factor has already worsened by
over 2 times what it was at the regular extent of UTM Zone 13. The scale factor at testing point A is
1.000470 (1:2,127 UTM accuracy) while the scale factor at the regular extent of UTM Zone 13, nearest to
testing point A, is 1.000191 (1:5,235 UTM accuracy). Testing point C (60o 00’ 00” N, 110o 00’ 00” W) is
located approximately 111 km outside the extents of the regular UTM Zone 13. The scale factor at this point
is 1.000552 (1:1,811 UTM accuracy) while the scale factor at the regular extent of UTM Zone 13, nearest to
testing point C, is 0.999943 (1:17,543 UTM accuracy). Testing point B (49o 00’ 00” N, 110o 00’ 00” W) is
located approximately 146 km outside the extents of the regular UTM Zone 13. The scale factor at this point
is 1.001243 (1:804 UTM accuracy). As noticed, testing point B experiences the worst UTM accuracy of any
point within Saskatchewan when using an extended UTM Zone 13. This means points located within the
southwest corner of the province will be effected the most when calculated/mapped within an extended
Issues with Extending a UTM Zone
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UTM Zone 13. Previously it was stated, that in Saskatchewan if the regular UTM zone extent rules are
rigidly applied, the derived spatial information from a UTM dataset is guaranteed to have a UTM accuracy
of no worse than 1:2,500. By now applying an extended UTM Zone 13 across the province, this statement
must be changed to state that the guaranteed UTM accuracy will be no worse than approximately 1:800.
Practical Numerical Problem
A practical question concerning the use of extended UTM Zone 13 coordinates was submitted to the
Practice Committee by SLSA members and was the main reason for the creation of this article. The question
asked was (and I am paraphrasing here), when calculating the position of the midpoint between the
NE-36-41-28-W3 and the NE-36-42-28-W3 when using the extended UTM Zone 13 versus the regular UTM
Zone 12, the position varies by ~12cm. Why is this?
Let us begin with a geographical analysis of the problem using mapping software. First, the NAD83
geodetic positions (i.e., latitude and longitude) of the two NE township corners in question were calculated
using NRCan’s GeoMath software (also known as MathTool), available here http://geoscan.nrcan.gc.ca/
starweb/geoscan/servlet.starweb?path=geoscan/fulle.web&search1=R=296440). Next, these geodetic points
were then converted into UTM Zone 12 and extended UTM Zone 13 projected coordinates using NRCan’s
TRX online tool, found here http://webapp.geod.nrcan.gc.ca/geod/tools-outils/trx.php. Table 1 displays the
respective coordinates.
Point
Latitude
Longitude
UTM Zone
12 Northing
UTM Zone
12 Easting
Ext. UTM Zone
13 Northing
Ext. UTM Zone
13 Easting
NE36-42-28-W3
52 40’ 03.0700”
o
-109 53’ 17.2000”
o
5835865.985
575189.980
5846505.849
169548.777
NE36-41-28-W3
52 34’ 49.2300”
o
-109 53’ 17.2000”
o
5826169.273
575339.536
5836817.974
168890.375
Table 1: Township corners’ NAD83 coordinates
Next, the coordinates of the midpoint between the NE township corners were calculated by simply finding
the mean coordinate value between township corners’ coordinates (i.e., add together and divide by 2). We
will call the midpoint coordinates that were calculated using this approach the average midpoint. The
following table displays these coordinates.
Issues with Extending a UTM Zone
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UTM Zone
12 Northing
UTM Zone
12 Easting
Ext. UTM Zone
13 Northing
Ext. UTM Zone
13 Easting
o
5835865.985
575189.980
5846505.849
169548.777
-109 53’ 17.2000”
o
5826169.273
575339.536
5836817.974
168890.375
∑/2
∑/2
∑/2
∑/2
∑/2
∑/2
52 37’ 26.1500”
-109 53’ 17.2000”
5831017.629
575264.758
5841661.912
169219.576
Point
Latitude
Longitude
NE36-42-28-W3
52 40’ 03.0700”
o
-109 53’ 17.2000”
NE36-41-28-W3
52 34’ 49.2300”
o
Average midpoint
o
o
Table 2: Average midpoint NAD83 coordinates
Lastly, taking all of these coordinates and reprojecting them into a single projected coordinate system (here
we used the Google Mercator system) yields the following map.
Figure 4: Map of the Practical Numerical Problem (source: Ryan Brazeal, 2016)
As is seen within Figure 4, the distance reported between the average midpoint of the line between the NE
township corners in UTM Zone 12 and extended UTM Zone 13 is 0.117m. This value agrees with the SLSA
members’ observation, but the question still remains as to what is causing this apparent problem?
In an attempt to best explain the problem at hand, I would like to begin by asking the question, what
constitutes something being called level? We say that two points within a small project area that have the
same elevation coordinates are level, but do we still say that an ocean’s surface is level when in fact we
know it is actually curved? The answer relies on how localized our view of the Earth’s surface is and how
we choose to represent it (i.e., our frame of reference).
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At the core of the problem at hand is a practitioner’s understanding/intuition as to what constitutes a straight
line. We like to utilize a planar surface/frame of reference to represent the spatial position of points within a
limited region on the Earth (e.g., a UTM zone). Within these frames of reference the concept of a straight
line is easy to understand and mathematically define. The definition of a straight line on a planar surface is
that it connects two points via the shortest possible path. However, when we expand our view of the Earth
and represent it by a reference ellipsoid, the concept of a straight line between two points now becomes
unclear. Rather than use the term, straight line, we now need to use the more general term of a geodesic line.
The definition of a geodesic line is that it connects two points via the shortest possible path along a surface.
Similar to the concept of levelness discussed above, a geodesic line also has curvature. This means that our
simple mathematical tools for straight lines in 2D planes are not guaranteed to yield accurate results when
trying to answer questions pertaining to geodesic lines.
The difference between the UTM Zone 12 and extended UTM Zone 13 positions for the midpoint of the
line, is because the UTM coordinates of the township corners were used to mathematically define a 2D
straight line between the corners. The simple add together and divide by two mathematics used to calculate
the average midpoints were in fact calculating the midpoints along these 2D straight lines. These UTM
derived straight lines closely approximated the true geodesic line (i.e., the meridian) between the township
corners, but not perfectly. The results of calculating the midpoint using the UTM Zone 12 coordinates were
much better than calculating it using the extended UTM Zone 13 coordinates. This is because the township
corners involved in this practical problem were near the testing point B we previously discussed. We know
that the largest amount of error for UTM derived spatial information within the entire province occurs near
testing point B.
We will discuss two approaches to solving for the midpoint position of the geodesic line connecting the
township corners within both UTM zones. The first approach, which is accurate and relatively easy,
involves solving the geodetic coordinates (i.e, latitude and longitude) of the geodesic midpoint and then
projecting this geodetic point into each of the respective UTM coordinate systems. Using the reference
ellipsoid to relate points of interest is a near perfect way to retain spatial information but it is an underutilized approach because Cartesian coordinates and planar mathematics cannot be used.
In order to solve the geodetic coordinates of the geodesic midpoint properly one cannot just simply calculate
the mean latitude and longitude coordinates between the township corners (as previously done and shown in
Issues with Extending a UTM Zone
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Table 2). This is because there is not a linear relationship between surface distance and angular change on a
reference ellipsoid. In general, what needs to be done is to first solve for the ellipsoidal distance between
points A and B along with the initial azimuth from point A to point B (solved via the indirect ellipsoid
problem). Then, the midpoint’s geodetic position can be computed by starting from point A and going in the
direction of the initial azimuth and at half the ellipsoidal distance calculated in the previous step (solved via
the direct ellipsoid problem). Both the indirect and direct ellipsoid problems can be solved using NRCan’s
INDIR online tool, found here http://webapp.geod.nrcan.gc.ca/geod/tools-outils/indir.php. 1 The table below
illustrates the correct midpoint coordinates for the geodesic line between the township corners. For our
practical problem, one will notice that there is negligible change in the geodetic coordinates for the
midpoint when compared to the coordinates shown in Table 2. This is because of the relatively small
ellipsoidal distance between the township corners.
Point
Latitude
Longitude
Correct midpoint
52 37’ 26.1506”
-109 53’ 17.2000”
o
o
UTM Zone
12 Northing
UTM Zone
12 Easting
Ext. UTM Zone
13 Northing
Ext. UTM Zone
13 Easting
5831017.630
575264.780
5841661.925
169219.482
Table 3: Correct midpoint NAD83 geodetic and UTM coordinates (using first approach)
If we were to take these two different sets of UTM coordinates and reproject them into a single projected
coordinate system (again we used the Google Mercator system) the following map would result.
Figure 5: Map of the Correct Midpoint (source: Ryan Brazeal, 2016)
1 To better understand this concept, use the INDIR online tool to calculate the ellipsoidal distance between the points (30o N, 105o W) and (45o N, 105o W) and
o
o
o
o
o
o
o
o
then between the points (45 N, 105 W) and (60 N, 105 W). You will notice that the point (45 N, 105 W) is not the midpoint between (30 N, 105 W) and (60
o
o
o
N, 105 W). The correct midpoint is (45 01’ 09.6” N, 105 W).
o
Issues with Extending a UTM Zone
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Notice that there is now only a single correct midpoint displayed. The distances between each of the average
midpoints and the correct midpoint are reported as being 0.022m to the UTM Zone 12 average midpoint and
0.096m to the extended UTM Zone 13 average midpoint. These values can be easily calculated by the
interested reader by solving the 2D distance between the average midpoint coordinates shown in table 2 and
the correct midpoint coordinates shown in table 3 using each respective set of UTM coordinates.
The second approach to solving for the midpoint position of the geodesic line connecting the township
corners, involves only using the UTM coordinates and the arc-to-chord (also known as the t-T) correction.
This approach is technically an approximation. However, it works well within the extents of a regular UTM
zone but quickly becomes less accurate when used within an extended UTM zone. In order to best
understand how this approach works, one needs to be able to visualize the geodesic line (i.e., a meridian in
our problem) in relation to the UTM derived mathematic straight lines. The following figure illustrates how
meridians and mathematic straight lines are represented within a transverse Mercator based coordinate
system.
Figure 6: Meridians, parallels and mathematic straight lines in a transverse
Mercator coordinate system (source: Stuifbergen, 2009)
As seen in Figure 6, the mathematic straight line (dashed pink) at the central meridian (0o E in this figure)
perfectly aligns with the projected meridian. As we move easterly, away from the central meridian, the
mathematic straight lines begin to increasingly misalign with the projected meridians; scale and angular
Issues with Extending a UTM Zone
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errors also increase. The projected meridians within a UTM coordinate system will always have a concave
curvature towards the central meridian (Deakin, 2006). This implies that spatial positions calculated based
on UTM coordinates will tend to have a longitudinal (east/west) shift towards the Zone’s central meridian in
comparison to their correct geodetic positions. Latitudinal (north/south) shifts can also result, but are
significantly smaller. These shifts can be seen in Figure 5, where the UTM Zone 12 average midpoint is
shifted to the west of the projected meridian (as the UTM Zone 12 central meridian is at 111o W) while the
extended UTM Zone 13 average midpoint is shifted to the east of the projected meridian (as the extended
UTM Zone 13 central meridian is at 105o W). The extended UTM Zone 13 average midpoint is also
noticeably shifted south.
In order to account for these shifts within the UTM derived spatial information, one needs to adjust the
position of the calculated average midpoint. The adjustment requires that the arc-to-chord angular
correction, the mathematic line distance (i.e., grid distance) and the mathematic line azimuth (i.e., grid
azimuth) to be determined. Figure 7 illustrates the arc-to-chord angular correction (t-T) when the geodesic
line of interest is a meridian.
Figure 7: Arc-to-chord angular correction (source: Ryan Brazeal, 2016)
Issues with Extending a UTM Zone
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In this specific case, the t-T angular correction could be computed by subtracting the grid declination angle
(𝛄), (which has not been discussed previously), from the grid azimuth angle (αG).
t -T = αG - 𝛄
A general equation for solving the arc-to-chord angular correction for any geodesic that only requires the
UTM coordinates of the end points of the mathematic straight line, along with the reference ellipsoid
parameters, is given in Deakin (2006) as,
t -T
,
the interested reader is referred to this reference for a full explanation of the terms in this equation and
further information on the subject.
The grid azimuth and grid distance (dG ) are simply solved via an inverse problem using the UTM projected
coordinates of the township corners. Next, the projected geodesic between the township corners is
approximated by a circular arc where the midpoint of the circular arc is assumed to align with the correct
midpoint of the geodesic. This allows for the distance of the perpendicular line (dP ), which connects the
average midpoint to the correct midpoint, to be calculated using the following equation.
The perpendicular line distances should agree with the distances reported by the mapping software as shown
in Figure 5. Next, the azimuth of the perpendicular line from the average midpoint to the correct midpoint
(αP) is calculated by adding or subtracting 90o from the grid azimuth (this depends upon what side of the
central meridian the geodesic is on). Lastly, the position of the correct midpoint can be calculated via
solving a direct problem starting from the average midpoint’s UTM position, moving in the direction of the
azimuth of the perpendicular line, and going the perpendicular line distance (dP ). The following table
displays the intermediate values and results of this second approach. It should be noted that the results for
the correct midpoint’s UTM coordinates are very close to the same values as previously calculated using the
first approach. Only the UTM Northing coordinates vary by a handful of millimetres.
Issues with Extending a UTM Zone
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Name
UTM Zone 12
Northing
UTM Zone 12
Easting
Ext. UTM Zone
13 Northing
Ext. UTM Zone
13 Easting
Average midpoint
5831017.629
575264.758
5841661.912
169219.576
t-T
—>
—>
-1.849”
—>
—>
8.112”
αG
—>
—>
-0 53’ 01.05”
o
—>
—>
3 53’ 16.53”
αP
—>
—>
89 06’ 58.95”
o
—>
—>
-86 06’ 43.47”
dG
—>
—>
9697.8653m
—>
—>
9710.2222m
dP
—>
—>
0.0217m
—>
—>
0.0955m
Correct midpoint
5831017.6293
575264.7797
5841661.9180
169219.4808
Value
Value
o
o
Table 4: Correct midpoint NAD83 UTM coordinates (using second approach)
In closing, this article brought to surface some important aspects for the SLSA membership to consider
when choosing to use UTM coordinates within their practices. Even though UTM coordinates have a
comfortable feel to them and allow us to easily calculate spatial information, there is a potentially large
accuracy trade-off as a result of this comfort and easiness. Using the reference ellipsoid and working
directly with geodetic coordinates can often be used to ‘easily’ tackle geospatial problems, but for some
practitioners this may require a new set of calculation tools.
Food For Thought
Why is today’s official map projection/projected coordinate system for the province of Saskatchewan one
that was defined nearly 70 years ago for the purpose of mapping the entire world? Is it believed, that using
UTM Zone 13 is the only way that allows geospatial information within Saskatchewan to be relatable to
other geospatial data? In the United States each state utilizes a projected coordinate system (or several of
them) that best work for that state’s mapping purposes. New Brunswick uses a special Double Stereographic
projection for their provincial mapping purposes. Why are we not defining a much better fitting and
applicable map projection/projected coordinate system for Saskatchewan? The possibility certainly exists to
create a moderate map projection for Saskatchewan that brings about substantial improvements in accuracy,
commonality in the geospatial data collected and improved standard operating procedures for practitioners.
Just a thought!
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References
Deakin, R.E., (2006). Traverse Computation on the UTM Projection for Surveys of Limited Extent, School of
Mathematical and Geospatial Sciences, RMIT University, March 2006.
ISC, (2014). Saskatchewan CAD File & Georeferencing Specifications, Information Services Corporation of
Saskatchewan, Plan Processing Service, October 4, 2014.
Krakiwsky, E.J., (1973). Conformal Map Projections in Geodesy, Department of Geodesy and Geomatics Engineering, University of New Brunswick, September 1973.
NRCan, (2016). The UTM Grid - Map Projections, Natural Resources Canada, Earth Sciences, Retrieved from:
http://www.nrcan.gc.ca/earth-sciences/geography/topographic-information/maps/9775 on July 10th, 2016.
Stuifbergen, N., (2009). Wide Zone Transverse Mercator Projection, Canadian Technical Report of Hydrography
and Ocean Science No. 262.
Snyder, J.P., (1987). Map Projections - A Working Manual, U.S. Geological Survey Professional Paper: 1395.