
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.

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'.
R47-43Â Â =Â 19,027,633.05 feet
              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 |
|||
Long. 94°   21' |
q -0Â Â Â 55Â Â Â 35.4885 |
Long 94°   41' |
q -1Â Â Â 10Â Â Â 24.9521 |
2. From Table 1, interpolate for q at longitude W94° 32' 58.240".   Â
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 |
y' |
Tabular |
Scale in |
Scale |
47°   31' |
19,100,580.81 |
370,817.94 |
101.31550 |
-355.2 |
0.9999182 |
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.