State Plane Coordinates vs. Surface Coordinates, Part 5.
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
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, 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 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 Rb = 19,471,398.75 feet
Find: State plane coordinates x and y, plus the scale factor.
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
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
Projection for Minnesota - North Zone
94Â°Â Â Â 21'
-0Â Â Â 55Â Â Â 35.4885
94Â°Â Â Â 41'
-1Â Â Â 10Â Â Â 24.9521
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"
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
47Â°Â Â Â 31'
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
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,
Station Blackduck Tank in Minnesota
Latitude N47Â° 43' 50.270'
Longitude W94Â° 32' 58.240'
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
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.