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State Plane Coordinates vs. Surface Coordinates, Part 6.

July 7, 2000
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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 N40° 43' 37.302"
longitude W88° 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 the same x-axis, which is located well below the southern limit of the state and has a value of zero feet. The central meridian of the East Zone is 88°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 parallel to the 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 parallel of latitude 36°40' defines the x-axis; the origin of coordinates for the east zone is a point on the 36°40' parallel situated 500,000' west of longitude 88°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' + 500,000    (1)
x' = H Dl" +/- a b    (2)
y = yo + V ("/100)2 +/- c    (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:   
Station King
latitude N40° 43' 37.302"
longitude W88° 41' 35.208"
State - Illinois, East Zone
Central Meridian - W88° 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 = -88° 41' 35.208" - (-88° 20' 00")
Dl = -0° 21' 35.208" = -1,295.208 seconds-of-arc

2) Calculate    (Dl"/100)2
    (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)

³Y0 per
second

H

³H per
second

V

³V per
second

a

40°    35'
40°    36'
40°    37'
40°    38'
40°    39'
40°    40'
40°    41'
40°    42'
40°    43'
40°    44'
40°    45'
40°    46'
40°    47'
40°    48'
40°    49'
40°    50'
40°    51'
40°    52'
40°    53'
40°    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

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

Table 3. Values of Scale Factors - Illinois East Zone

Values of Scale Factors -
Illinois East Zone

x¿ (feet) Scale in
Units of 7th Place of
Logs
Scale
Expressed
as a Ratio

0
    5,000
    10,000
    15,000
    20,000
    25,000
    30,000
    35,000
    40,000
    45,000
    50,000
    55,000
    60,000
    65,000
    70,000
    75,000
    80,000
    85,000
    90,000
    95,000
    100,000
    105,000
    110,000
    115,000
    120,000
    125,000
    130,000
    135,000
    140,000
    145,000

-108.6
    -108.5
    -108.1
    -107.5
    -106.6
    -105.5
    -104.1
    -102.5
    -100.7
    -98.5
    -96.2
    -93.6
    -90.7
    -87.6
    -84.3
    -80.7
    -76.8
    -72.7
    -68.4
    -63.8
    -58.9
    -53.9
    -48.5
    -42.9
    -37.1
    -31.0
    -24.7
    -18.1
    -11.3
    -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, 40°43'37.202". From Table 1 we must interpolate between 40° 43' and 40° 44' by the amount 37.202"/60" = 0.6217.

By doing this we get the following:   

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

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

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

5) Solve for H Dl and a b, which are needed to solve equation (2), given above:
    x' = HDl" +/- a b,
    HDl" = -99,721.50
    a b = -1.25
    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),
    y = yo + V ( Dl/100)2 +/- c
    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.

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