Figure 1. The Illinois Coordinate System
In the last article, I showed how to calculate state plane coordinates in a state that uses the Lambert conformal projection. In this article, I will calculate the state plane coordinates for a geodetic control point in a state that uses the transverse Mercator projection.

The problem is:
Calculate the state plane coordinates for station King whose NAD 27 coordinates are

latitude N40Â° 43' 37.302"
longitude W88Â° 41' 35.208"

The station is located in the State of Illinois, state plane zone Illinois East.

Figure 1 shows the map from the U.S. Coast and Geodetic Survey manual for the state of Illinois, also reproduced in Rayner and Schmidt1. Illinois uses the transverse Mercator projection with two zones, east and west. Each zone has its own axis for y, although both axes passing through the east and west zones are given an x-value of 500,000'. Both zones use theÂ sameÂ x-axis, which is located well below the southern limit of the state and has a value of zero feet.Â TheÂ central meridian of the East Zone is 88Â°20' west longitude; along this line the scale of the projection is one part in 40,000 parts too small. The lines of exact scale are parallelÂ toÂ theÂ central meridian and situated approximately 28 miles east and west. Of course, to the east and west of these lines, the scale is too large. The parallelÂ ofÂ latitude 36Â°40' defines the x-axis; the origin of coordinates for the east zone isÂ aÂ pointÂ onÂ theÂ 36Â°40'Â parallelÂ situated 500,000' west of longitude 88Â°20'.
Let's perform the calculations. Unlike the Lambert projection, there isn't a sketch that shows the geometric relations between latitude, longitude and x,y. The equations necessary to perform these calculations are as follows:

x = x' + 500,000Â Â Â  (1)
x' = H Dl" +/- a bÂ Â Â  (2)
y = yo + V ("/100)2 +/- cÂ Â Â  (3)

Where x' is the distance, the point is either east or west of the central meridian; yo, H, V and a are quantities based on the geodetic latitude; b and c are based on Dl" (the difference in longitude of the point from the longitude of the central meridian, in seconds-of-arc).

Tables are needed to get the values for H, V, a, b, yo and c. Fortunately, all values can be found in two tables, which are given in the publication for the state of Illinois; but for this article, Tables 1 and 2 (on page 18) from Rayner and Schmidt are abstracts of the original tables that cover the values needed to solve our problem.

Repeating the problem:

Given:Â Â Â
Station King
latitude N40Â° 43' 37.302"
longitude W88Â° 41' 35.208"
State - Illinois, East Zone
Central Meridian - W88Â° 20' 00

Solution:
1) Solve for Dl. Since we are in the western hemisphere, all values of longitude are minus.
Dl" = longitude - central meridian longitude.
Dl = -88Â° 41' 35.208" - (-88Â° 20' 00")
Dl = -0Â° 21' 35.208" = -1,295.208 seconds-of-arc

2) CalculateÂ Â Â  (Dl"/100)2
Â Â Â  (Dl"/100)2 = 167.756

Table 1. Values of H and V - Illinois East Zone

 Values of H and V - Illinois East Zone Lat. Y0 (feet) Â³Y0 per second H Â³H per second V Â³V per second a 40Â°Â Â Â  35' 40Â°Â Â Â  36' 40Â°Â Â Â  37' 40Â°Â Â Â  38' 40Â°Â Â Â  39' 40Â°Â Â Â  40' 40Â°Â Â Â  41' 40Â°Â Â Â  42' 40Â°Â Â Â  43' 40Â°Â Â Â  44' 40Â°Â Â Â  45' 40Â°Â Â Â  46' 40Â°Â Â Â  47' 40Â°Â Â Â  48' 40Â°Â Â Â  49' 40Â°Â Â Â  50' 40Â°Â Â Â  51' 40Â°Â Â Â  52' 40Â°Â Â Â  53' 40Â°Â Â Â  54' 1,426,385.98 1,432,457.79 1,438,529.61 1,444,601.45 1,450,673.31 1,456,745.19 1,462,817.08 1,468,888.99 1,474,960.92 1,481,032.87 1,487,104.84 1,493,176.82 1,499,248.82 1,505,320.84 1,511,392.88 1,517,464.93 1,523,537.01 1,529,609.10 1,535,681.20 1,541,753.33 101.19683 101.19700 101.19733 101.19767 101.19800 101.19817 101.19850 101.19883 101.19917 101.19950 101.19967 101.20000 101.20033 101.20067 101.20083 101.20133 101.20150 101.20167 101.20217 101.20233 77.158010 77.138853 77.119688 77.100517 77.081340 77.062156 77.042965 77.023768 77.004565 76.985354 76.966138 76.946914 76.927685 76.908448 76.889205 76.869956 76.850700 76.831437 76.812168 76.792893 319.28 319.42 319.52 319.62 319.73 319.85 319.95 320.05 320.18 320.27 320.40 320.48 320.62 320.72 320.82 320.93 321.05 321.15 321.25 321.38 1.216989 1.217100 1.217211 1.217321 1.217431 1.217540 1.217649 1.217757 1.217865 1.217973 1.218080 1.218187 1.218293 1.218399 1.218505 1.218610 1.218715 1.218819 1.218923 1.219027 1.85 1.85 1.83 1.83 1.82 1.82 1.80 1.80 1.80 1.78 1.78 1.77 1.77 1.77 1.75 1.75 1.73 1.73 1.73 1.72 - 0.509 - 0.507 - 0.505 - 0.503 - 0.501 - 0.499 - 0.497 - 0.495 - 0.493 - 0.491 - 0.489 - 0.487 - 0.485 - 0.483 - 0.481 - 0.479 - 0.477 - 0.475 - 0.473 - 0.471

Table 2. Values of b and c - Illinois Zones

 Values of b and c - Illinois Zones DlÂ¿ b Db c 0 Â Â Â  100 Â Â Â  200 Â Â Â  300 Â Â Â  400 Â Â Â  500 Â Â Â  600 Â Â Â  700 Â Â Â  800 Â Â Â  900 Â Â Â  1,000 Â Â Â  1,100 Â Â Â  1,200 Â Â Â  1,300 Â Â Â  1,400 Â Â Â  1,500 Â Â Â  1,600 Â Â Â  1,700 Â Â Â  1,800 Â Â Â  1,900 Â Â Â  2,000 0.000 Â Â Â  +0.212 Â Â Â  +0.424 Â Â Â  +0.634 Â Â Â  +0.842 Â Â Â  +1.049 Â Â Â  +1.252 Â Â Â  +1.453 Â Â Â  +1.649 Â Â Â  +1.841 Â Â Â  +2.028 Â Â Â  +2.209 Â Â Â  +2.384 Â Â Â  +2.553 Â Â Â  +2.715 Â Â Â  +2.868 Â Â Â  +3.014 Â Â Â  +3.151 Â Â Â  +3.279 Â Â Â  +3.397 Â Â Â  +3.504 +0.212 Â Â Â  +0.212 Â Â Â  +0.210 Â Â Â  +0.208 Â Â Â  +0.207 Â Â Â  +0.203 Â Â Â  +0.201 Â Â Â  +0.196 Â Â Â  +0.192 Â Â Â  +0.187 Â Â Â  +0.181 Â Â Â  +0.175 Â Â Â  +0.169 Â Â Â  +0.162 Â Â Â  +0.153 Â Â Â  +0.146 Â Â Â  +0.137 Â Â Â  +0.128 Â Â Â  +0.118 Â Â Â  +0.107 Â Â Â  +0.097 0.000 Â Â Â  0.000 Â Â Â  -0.001 Â Â Â  -0.002 Â Â Â  -0.003 Â Â Â  -0.005 Â Â Â  -0.007 Â Â Â  -0.010 Â Â Â  -0.014 Â Â Â  -0.018 Â Â Â  -0.022 Â Â Â  -0.027 Â Â Â  -0.032 Â Â Â  -0.038 Â Â Â  -0.043 Â Â Â  -0.049 Â Â Â  -0.055 Â Â Â  -0.061 Â Â Â  -0.067 Â Â Â  -0.073 Â Â Â  -0.079

Table 3. Values of Scale Factors - Illinois East Zone

 Values of Scale Factors - Illinois East Zone xÂ¿ (feet) Scale in Units of 7th Place of Logs Scale Expressed as a Ratio 0 Â Â Â  5,000 Â Â Â  10,000 Â Â Â  15,000 Â Â Â  20,000 Â Â Â  25,000 Â Â Â  30,000 Â Â Â  35,000 Â Â Â  40,000 Â Â Â  45,000 Â Â Â  50,000 Â Â Â  55,000 Â Â Â  60,000 Â Â Â  65,000 Â Â Â  70,000 Â Â Â  75,000 Â Â Â  80,000 Â Â Â  85,000 Â Â Â  90,000 Â Â Â  95,000 Â Â Â  100,000 Â Â Â  105,000 Â Â Â  110,000 Â Â Â  115,000 Â Â Â  120,000 Â Â Â  125,000 Â Â Â  130,000 Â Â Â  135,000 Â Â Â  140,000 Â Â Â  145,000 -108.6 Â Â Â  -108.5 Â Â Â  -108.1 Â Â Â  -107.5 Â Â Â  -106.6 Â Â Â  -105.5 Â Â Â  -104.1 Â Â Â  -102.5 Â Â Â  -100.7 Â Â Â  -98.5 Â Â Â  -96.2 Â Â Â  -93.6 Â Â Â  -90.7 Â Â Â  -87.6 Â Â Â  -84.3 Â Â Â  -80.7 Â Â Â  -76.8 Â Â Â  -72.7 Â Â Â  -68.4 Â Â Â  -63.8 Â Â Â  -58.9 Â Â Â  -53.9 Â Â Â  -48.5 Â Â Â  -42.9 Â Â Â  -37.1 Â Â Â  -31.0 Â Â Â  -24.7 Â Â Â  -18.1 Â Â Â  -11.3 Â Â Â  -4.2 0.9999750 0.9999750 0.9999751 0.9999752 0.9999755 0.9999757 0.9999760 0.9999764 0.9999768 0.9999773 0.9999778 0.9999784 0.9999791 0.9999798 0.9999806 0.9999814 0.9999823 0.9999833 0.9999843 0.9999853 0.9999864 0.9999876 0.9999888 0.9999901 0.9999915 0.9999929 0.9999943 0.9999958 0.9999974 0.9999990

3) From Table 1, interpolate for H, V, yo and a.
The argument for all values is the latitude of the point, 40Â°43'37.202". From Table 1 we must interpolate between 40Â° 43' and 40Â° 44' by the amount 37.202"/60" = 0.6217.

By doing this we get the following:Â Â Â

H = 76.992654
V = 1.217932
a = -0.492
yo = 1,478,725.73.

4) From Table 2, interpolate for b and c.
Â Â Â  b = +2.545
Â Â Â  c = -0.04

The argument for both values is Dl in seconds-of-arc.Â Â Â Â

5) Solve for H Dl and a b, which are needed to solve equation (2), given above:
Â Â Â  x' = HDl" +/- a b,
Â Â Â  HDl" = -99,721.50
Â Â Â  a b = -1.25
Â Â Â  x' = HDl +/- a b = -99,720.25

Note
The "sign convention" is as follows: when a b is negative, decrease HDl" numerically. If a b is positive, increase HDl" numerically. Since Dl" is negative because the station is west of the central meridian, x' is also negative.

6) Solve equation (3),
Â Â Â  y = yo + V ( Dl/100)2 +/- c
Â Â Â  y = 1,478,930.01 ft.

7) Solve for the scale factor.
The argument for scale factor is x'. Table 3 gives the scale factor for different values of x'. In our problem,
x' = -99,720.25. The "minus" sign is not needed for this calculation.

Solution
Scale factor = 0.9999863

As you can see, the calculations on the transverse Mercator grid are more complicated than calculations on the Lambert grid. However, at most, two conversions are needed for traversing a small area; after that all calculations are made using plane trigonometry. That"s what we will discuss in the next column. We are getting close to the end of this series, at most two more.