State Plane Coordinates vs. Surface Coordinates, Part 5.

July 7, 2000

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 N47Â° 43' 50.270"
Longitude W94Â° 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 W93Â° 06', which has a false x-coordinate of 2,000,000 feet.

The y=0 coordinate occurs at N46Â° 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, R_b=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 W94Â° 21' to longitude W95Â° 00'. Table 2 gives values of R, y' and scale factor as a function of latitude, from latitude N47Â° 31' to latitude N47Â° 50' (y' is not needed for our problem).

Repeating the problem:

Given: Station Blackduck Tank
Â Â Â Â Â Â Â Â Â Â Â Latitude N47Â° 43' 50.270"
Â Â Â Â Â Â Â Â Â Â Â Longitude W94Â° 32' 58.240"
Â Â Â Â Â Â Â Â Â Â Â State - Minnesota, North Zone
Â Â Â Â Â Â Â Â Â Â Â C = 2,000,000 feet
Â Â Â Â Â Â Â Â Â Â Â R_b = 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 N47Â° 43' 50.270"

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

Interpolate for latitude 47Â° 43' 50.270"

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

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

R47-43Â Â =Â 19,027,633.05 feet
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (-) 5,093.24 feet
R47-43-50.220Â =Â 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.

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

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

Long

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

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

2. From Table 1, interpolate for q at longitude W94Â° 32' 58.240". Â Â Â

For longitude W94Â° 32',
q = -1Â° 03' 44.6935"
For longitude W94Â° 33',
q = -1Â° 04' 29.1666"
Difference = -0Â° 00' 44.4731"

Interpolate for longitude
94Â° 32' 58.240"

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

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

q94-32Â Â =Â Â -1Â° 03' 44.6935"
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (-) 43.1686"
q94-32-58.240Â =Â Â -1Â° 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
47Â°Â Â Â 31' Â Â Â Â Â Â Â 32 Â Â Â Â Â Â Â 33 Â Â Â Â Â Â Â 34 Â Â Â Â Â Â Â 35 47Â°Â Â Â 36' Â Â Â Â Â Â Â 37 Â Â Â Â Â Â Â 38 Â Â Â Â Â Â Â 39 Â Â Â Â Â Â Â 40 47Â°Â Â Â 41' Â Â Â Â Â Â Â 42 Â Â Â Â Â Â Â 43 Â Â Â Â Â Â Â 44 Â Â Â Â Â Â Â 45 47Â°Â Â Â 46' Â Â Â Â Â Â Â 47 Â Â Â Â Â Â Â 48 Â Â Â Â Â Â Â 49 Â Â Â Â Â Â Â 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 N47Â° 43',
scale factor = 0.9999050
Latitude N47Â° 44',
scale factor = 0.9999044
Difference = 0.0000006

Interpolate for latitude 47Â° 43' 50.270"

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

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

Scale factor =
0.9999050 - 0.0000005 = 0.9999045.

ThatÂ¿s it; in summary,

Given:
Station Blackduck Tank in Minnesota
Latitude N47Â° 43' 50.270'
Longitude W94Â° 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.