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State Plane Coordinates vs. Surface Coordinates, Part 5.

July 7, 2000
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Figure 1. Minnesota State Plane Coordinate Zones. U.S. Coast and Geodetic Survey.
Up to this point in time, we have described the State Plane Coordinate Systems of 1927. Part 4 of this column had a long table defining the state zones and constants. Each state had its own publication, and all calculations were performed by hand1. In this article, I want to show how to generate NAD 27 state plane coordinates for a known geodetic position using hand computations. To simplify things, I'm going to "borrow" an example from an old surveying textbook.2

Here is the problem:
Calculate the State plane coordinates for station Blackduck Tank whose NAD 27 coordinates are

Latitude N47° 43' 50.270"
Longitude W94° 32' 58.240"

The station is located in the State of Minnesota, State plane zone Minnesota North.

Figure 2. Lambert Coordinates
Figure 1 shows the map from the U.S. Coast and Geodetic Survey manual from the State of Minnesota, also reproduced in Rayner and Schmidt. Minnesota uses the Lambert Conformal Conic Projection with three zones. If you were to look at the table in Part 4 of this series, you would find the constants for Minnesota North. Figure 2, from Rayner and Schmidt, is a sketch that shows graphically the Minnesota North constants and the basis of transforming the geodetic coordinates of point P into the state plane coordinates on the Lambert grid. The central meridian is W93° 06', which has a false x-coordinate of 2,000,000 feet.

The y=0 coordinate occurs at N46° 30', which is far enough south of the Minnesota North zone so that all y-coordinates will be positive. Given the latitude and longitude of point P, you will need to know the values of the angle, radius Rb and radius R in order to calculate x,y coordinates of point P. Remember, this is a conic projection; the point A represents the apex of the cone on which the area is projected, and the arc EP represents a portion of the parallel of latitude through point P.

Let's perform the calculations. Referring to Figure 2, the x and y coordinates of point P can be calculated using the following equations:

x = R sin q + C
y = Rb - R sin q.

As can be seen from Figure 2, C=2,000,000 feet. Although not shown, Rb=19,471,398.75 feet, a constant for Minnesota North.

Tables are needed to get R and q. These tables are given in the publication for the State of Minnesota, but for this article, Tables 1 and 2, from Rayner and Schmidt, are abstracts of the original tables that cover the values needed to solve our problem. Table 1 gives the values of q as a function of longitude, from longitude W94° 21' to longitude W95° 00'. Table 2 gives values of R, y' and scale factor as a function of latitude, from latitude N47° 31' to latitude N47° 50' (y' is not needed for our problem).

Repeating the problem:

Given: Station Blackduck Tank
            Latitude N47° 43' 50.270"
            Longitude W94° 32' 58.240"
            State - Minnesota, North Zone
            C = 2,000,000 feet
            Rb = 19,471,398.75 feet

Find: State plane coordinates x and y, plus the scale factor.

Solution:
1. From Table 2, interpolate to get R for latitude N47° 43' 50.270"

For latitude 47° 43',
R = 19,027,633.05 feet
For latitude 47° 44',
R = 19,021,553.99 feet
Difference = 6,079.06 feet

Interpolate for latitude 47° 43' 50.270"

6,079.06 x 50.270" = 5093.24
--------------------
60"

Since the value of R is decreasing from latitude 47° 44' to 47° 43', in order to get R at latitude 47° 43' 50.270" you subtract 5093.24 from the value of R at latitude 47° 43'.

R47-43   =  19,027,633.05 feet
                (-) 5,093.24 feet
R47-43-50.220  =  19,022,539.81 feet
-----------------------------------------


Table 1. Values of q - Minnesota North Zone

Lambert Projection for Minnesota - North Zone
1" of long. = 0" .7412196637 of q

Long.

94°    21'
        22
        23
        24
        25
    94°    26'
        27
        28
        29
        30
    94°    31'
        32
        33
        34
        35
    94°    36'
        37
        38
        39
        40

q

-0    55    35.4885
    -0    56    19.9617
    -0    57    04.4348
    -0    57    48.9080
    -0    58    33.3812
    -0    59    17.8543
    -1    00    02.3276
    -1    00    46.8007
    -1    01    31.2739
    -1    02    15.7471
    -1    03    00.2202
    -1    03    44.6935
    -1    04    29.1666
    -1    05    13.6398
    -1    05    58.1130
    -1    06    42.5862
    -1    07    27.0594
    -1    08    11.5325
    -1    08    56.0057
    -1    09    40.4789

Long

94°    41'
        42
        43
        44
        45
    94°    46'
        47
        48
        49
        50
    94°    51'
        52
        53
        54
        55
    94°    56'
        57
        58
        59
    95°    00'

q

-1    10    24.9521
    -1    11    09.4253
    -1    11    53.8984
    -1    12    38.3716
    -1    13    22.8448
    -1    14    07.3180
    -1    14    51.7912
    -1    15    36.2643
    -1    16    20.7375
    -1    17    05.2107
    -1    17    49.6839
    -1    18    34.1571
    -1    19    18.6302
    -1    20    03.1034
    -1    20    47.5766
    -1    21    32.0498
    -1    22    16.5230
    -1    23    00.9961
    -1    23    45.4693
    -1    24    29.9425


2. From Table 1, interpolate for q at longitude W94° 32' 58.240".    

For longitude W94° 32',
q = -1° 03' 44.6935"
For longitude W94° 33',
q = -1° 04' 29.1666"
Difference = -0° 00' 44.4731"

Interpolate for longitude
94° 32' 58.240"

44.4731" x 58.240" = 43.1686"
--------------------------------
60"

Since the value of q is increasing negatively from 94° 32' to 94° 33', algebraically add 43.1686" to the value at 94° 32'.

q94-32   =   -1° 03' 44.6935"
                 (-) 43.1686"
q94-32-58.240  =   -1° 04' 27.8621"

3. Solve the equation x = R sin q + C: x = 1,643,311.67 feet.

4. Solve the equation y = Rb - R cos q:
y = 452,203.34 feet.


Table 2. Values of R, y', and
Scale Factors - Minnesota North Zone

Lambert Projection for Minnesota - North Zone

Lat.

R
(ft.)

y'
y Value on
Central Meridian (ft)

Tabular
Difference
for 1" of Lat. (ft)

Scale in
Units of
7th Place
of Logs

Scale
Expressed
as a
Ratio

47°    31'
        32
        33
        34
        35
47°    36'
        37
        38
        39
        40
47°    41'
        42
        43
        44
        45
47°    46'
        47
        48
        49
        50

19,100,580.81
19,094,501.88
19,088,422.95
19,082,344.01
19,076,265.06
19,070,186.10
19,064,107.13
19,058,028.15
19,051,949.16
19,045,870.15
19,039,791.13
19,033,712.10
19,027,633.05
19,021,553.99
19,015,474.92
19,009,395.83
19,003,316.72
18,997,237.60
18,991,158.46
18,985,079.30

370,817.94
376,896.87
382,975.80
389,054.74
395,133.69
401,212.65
407,291.62
413,370.60
419,449.59
425,528.60
431,607.62
437,686.65
443,765.70
449,844.76
455,923.83
462,002.92
468,082.03
474,161.15
480,240.29
486,319.45

101.31550
101.31550
101.31567
101.31583
101.31600
101.31617
101.31633
101.31650
101.31683
101.31700
101.31717
101.31750
101.31767
101.31783
101.31817
101.31850
101.31867
101.31900
101.31933
101.31950

-355.2
-362.0
-368.4
-374.5
-380.2
-385.6
-390.6
-395.2
-399.4
-403.3
-406.8
-410.0
-412.8
-415.2
-417.3
-419.0
-420.3
-421.2
-421.8
-422.1

0.9999182
0.9999166
0.9999152
0.9999138
0.9999125
0.9999112
0.9999101
0.9999090
0.9999080
0.9999071
0.9999063
0.9999056
0.9999050
0.9999044
0.9999039
0.9999035
0.9999032
0.9999030
0.9999029
0.9999028


5. Solve for the scale factor:

From Table 2, last column,

Latitude N47° 43',
scale factor = 0.9999050
Latitude N47° 44',
scale factor = 0.9999044
Difference = 0.0000006

Interpolate for latitude 47° 43' 50.270"

0.0000006 x 50.270" = 0.0000005
60"
----

Since the scale factor is decreasing from 47° 43' to 47° 44', subtract 0.0000005 from the value at 47° 43':

Scale factor =
0.9999050 - 0.0000005 = 0.9999045.

That¿s it; in summary,

Given:
Station Blackduck Tank in Minnesota
Latitude N47° 43' 50.270'
Longitude W94° 32' 58.240'

Calculated:
Minnesota North Zone, NAD 27
x = 1,643,311.67 feet
y = 452,203.34 feet
scale factor = 0.9999045.

In order to traverse, a second geodetic control point is needed and the State plane coordinates must be calculated for that point. If the two geodetic control points are intervisible, inversing between the two state plane coordinates gives the "grid azimuth" (It's also possible to use a solar or star azimuth, more on that later). Then all distances measured on the surface must be scaled down to the grid and all traverse calculations made using plane trigonometry; we'll do that in the next article.

As you can see, the calculations on the Lambert grid are straightforward if you have the tables. In the next article I'll do a transformation onto the transverse Mercator grid, not as simple as on the Lambert grid, as you will see.

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