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Raising A State Plane Projection

Back in January, 2016, Loyal Olson described a consulting project that required raising a state plane so that ground distances were very close to grid distances and state plane grid azimuths were preserved. All of the posts in that thread are interesting and informative, so the entire thread is worth reading. Here? the link to Loyal? post:

https://rplstoday.com/community/threads/ldp-site.325120/#post-353522

The purpose of this post is to share the results of the research that Loyal? post prompted me to do. Let? explore the relationships and mathematics that make it possible to raise a Lambert state plane projection to provide ground distances and grid azimuths over a limited area.

The diagram below illustrates the situation in a two-dimensional equatorial view. The state plane is a secant plane, intersecting the GRS80 ellipsoid in two points. The single point that will provide the basis for raising the state plane lies above the state plane on the earth? surface. The idea is to raise the state plane, preserving its central parallel and central meridian, until it intersects this base point.

Diagram
Created with the FooPlot online grapher

In state plane language, the combined factor of the base point on the raised plane will equal 1. The raised plane will be a new projection surface, so the coordinates of all its points will differ from their state plane values. The convergence angle, however, will not change.

In the diagram, the Normal to the Central Parallel is an important element. In this two-dimensional drawing, this line is perpendicular to both the state plane and the ellipsoid and it is the only line that can be drawn perpendicular to both.

The scale factor on the central parallel is a key parameter for any state plane. It is the distance from the minor axis to the plane along this normal divided by the distance from the minor axis to the ellipsoid along this normal.

For the state plane, the scale factor on the central parallel will be less than 1 because the state plane is below the ellipsoid. The raised plane is above the ellipsoid, so this scale factor will be greater than 1 for the raised plane.

The plane can be viewed as moving up or down on its normal, its position determined by this scale factor. Thus, one way to raise the state plane to the desired level is to find the appropriate scale factor for its central parallel.

As it turns out, this is relatively easy to do, because scale factors on the state plane and the raised plane are proportional. That is, if the scale factor of a point on the raised plane is, say, 1.05 times its scale factor on the state plane, then the scale factors for all of the points on the raised plane are 1.05 times their corresponding scale factors on the state plane.

Here? the relationship, applied to a base point and the central parallel, in equation form:

equation1

Rearranging the proportion gives:

equation2

The scale factor of the base point on the raised plane is the only unknown on the left side of the equation. To find it, find the reciprocal of the base point? elevation factor. Here? the math:

equation3

where EF is Elevation Factor, SF is Scale Factor, and CF is Combined Factor.

Combining everything into one equation gives:

equation4

Note that the denominator is just the combined factor of the base point.

Here? an example. Raise the North Carolina state plane so that the combined factor for NGS mark DF7999 (Hampton) in Boone is equal to 1. The relevant state plane data are:

Base Point State Plane SF = 1.0000121972768

Base Point EF = 0.9998536878

Central Parallel State Plane SF = 0.999872591882

The scale factor for the central parallel for the raised plane is:

equation5

All of the numbers in the example contain at least 10 significant digits. This lessens the effect of rounding on the final result. The final result can be rounded as desired. For example, rounding the raised central parallel scale factor to 8 decimal places (4 significant digits) does not change the plane coordinates of DF7999, rounded to millimeters.

The new projection can be documented as follows:

Description of Hampton Raised Plane Projection

Projection name

Hampton Raised Plane

Ellipsoid

GRS 80

Projection type

Lambert Conformal Conic

Central parallel

North 35.2517586002 degrees

Central meridian

West 79 degrees

Scale factor at central parallel

1.00000670948967

False northing

166,589.9922 meters

False easting

609,601.2199 meters

The coordinates of DF7999 on the Hampton Raised Plane projection are different from its coordinates on the NC State Plane projection. Here? a comparison of DF7999 coordinates and other parameters on the two planes:

Point DF7999 on Hampton Raised Plane Compared to State Plane

Parameter

Hampton Raised Plane

NC State Plane (Data Sheet)

Northing

276,083.571 m

276,068.886 m

Easting

369,193.183 m

369,226.104 m

Convergence

?d 32m 34.7s

?d 32m 34.7s

Scale factor

1.0001463336

1.00001220

Elevation factor

0.9998536878

0.99985369

Combined factor

1.0000000000

0.99986589

Distortion

0.00 ppm

?34.11 ppm

Some key points are:

  • The coordinates of DF7999 on the Hampton Raised Plane are north and west of the coordinates on the state plane. Any Lambert conformal conic projection uses straight lines for meridians and arcs of concentric circles for parallels. Raising a Lambert plane lengthens the radii of the concentric circles. Consequently, coordinates of every point in the northwest quadrant on the raised plane will have coordinates that are northwest of those on the original plane. The situation is different for points in the other quadrants and for points on the central meridian, but those points are outside the intended area of the projection.
  • The elevation factor is the same for both projections. The elevation factor describes the height of a point above the ellipsoid, so it will not change unless the ellipsoid changes.
  • Convergence is the same for both projections. Convergence depends on the position of a point relative to the central meridian. Since the central meridian did not change, convergence did not change. Convergence is actually a consequence of replacing elliptical meridians on an ellipsoid with straight lines on a plane.

The purpose of the raised plane is to provide coordinates for points within a specific project area. The next table compares the Hampton Raised Plane coordinates for four nearby points to their NC State Plane coordinates.

Some Selected Points Near DF7999

Point

Hampton

Raised Plane Coordinates

(meters)

Distortion (ppm)

NC State Plane Coordinates

(meters)

Distortion (ppm)

FZ2928

?5.873

?59.990

Northing

274,135.037

274,120.613

Easting

366,969.304

367,001.845

FZ2946

10.582

?23.530

Northing

279,644.615

279,629.452

Easting

370,838.094

370,870.116

FZ2945

?.522

?37.640

Northing

274,497.962

274,483.490

Easting

371,018.698

371,050.696

FZ2023

?8.760

?72.880

Northing

278,550.109

278,535.093

Easting

368,090.074

368,122.465

The reduction in distortion on the Hampton Raised Plane is obvious. Also, since distortion is a function of the combined factor, the table provides all of the information needed to compare grid distances on the Hampton Raised Plane to ground distances using NC State Plane parameters.

Distortion provides a tangible idea of how unadjusted grid distances compare to ground distances, but combined factors are used to convert grid distances to ground distances. The following formula can be used to convert distortion to a combined factor:

equation6

The grid distance from FZ2928 to FZ2946 on the Hampton Raised Plane is 6732.235 meters. The ground distance, using the average combined factor and NC State Plane coordinates, is 6732.286 meters. Using the Hampton Raised Plane grid distance instead of the ground distance produces accuracy of better than 1:130,000.

The unadjusted distance on the Hampton Raised Plane that has the poorest accuracy is from FZ2928 to FZ2023. The distortion is high and has the same numeric sign at both points. This means that the distortions accumulate rather than offset. Relative accuracy can be calculated directly from distortion for this distance, and other shorter distances, like this:

equation7

The difference in the relative accuracies is striking. Points FZ2928 and FZ2023 have the greatest ellipsoid heights of the five points. The impressive accuracy of distances between points with offsetting distortions can be very misleading, so care and caution are advised when developing or using any ground coordinate system.

Modifying a Lambert state plane projection by raising it is an interesting alternative to other grid-to-ground methods. While raised state plane coordinates have values that are very similar to state plane coordinates, they are somewhat better differentiated than those produced by many other schemes. Also, the raised state plane is a geodetically sound projection that can be used directly in mapping and surveying software. And, grid distances are very close to ground distances.

Loyal introduced an intriguing concept that presented an entertaining (for me, anyway) research opportunity. Hopefully, many other people have benefited from his post (and his other posts, too) as much as I have.

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