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.

James Allen Wyatt

Though retired, Allen is a licensed mathematics teacher in North Carolina and is National Board Certified in mathematics teaching. Having admired surveyors and their profession since introductory trig in high school, he has studied and developed Lambert conformal conic projections informally since 2010.

View all posts by James Allen Wyatt

19 comments on “Raising A State Plane Projection

  1. A State Plane Coordinate System only exists at one level.

    You can do anything mathematically you want with it but it no longer is a State Plane Coordinate System.

    A State Plane Coordinate System is designed to encompass a large area. A local coordinate system is designed for a much smaller area and can thus be more precise. Precise being used very loosely.

    Paul in PA

  2. Spot on, Paul. This approach creates a different projection that shares its central parallel, central meridian, false northing, and false easting with the State Plane projection. What it does not share is the State Plane scale factor on the central parallel, so it is a unique projection with a unique geodetic definition.

  3. I think we are in at least the 4th decade of state plane being shifted to coordinates to use on the "surface". The SOP for the BIG users is to use a simple multiplication of the coordinates by a user defined scale factor and create an error budget to constrain just how "far" this surface coordinate system will allowed to be used.

    In the NAD27 days the SMALLER users of state plane would often use ground distances around a point. That point was usually a prominent NAD27 monument and control (other NAD27 monuments) was expanded around it by applying a user defined scale factor and then checked into with traverses that would often use ground distances.

    For larger projects state plane distances and coordinates would be calculated and then if needed adjusted by a scale factor, normally around a NAD27 monument holding those coordinates.

    However, GPS has seemed to introduce some surveyors to the state plane system that many have used for decades sans GPS and the GPSers seem to have no reference to the hands on use of the system and are uncomfortable with how it's been used for so long.

    And I am quite grateful that so many surveyors are uncomfortable with using the system as DOT does, so much work for me because of that.

  4. That's an interesting observation, Moe. When I first learned of State Plane and saw how grid and ground distances were related through combined factors, I thought that the massive computing power available today would make the system easier to use and a more popular choice. However, as you and so many others have pointed out, that is not the case.

    Properly used, it's hard to beat.

  5. That's an interesting observation, Moe. When I first learned of State Plane and saw how grid and ground distances were related through combined factors, I thought that the massive computing power available today would make the system easier to use and a more popular choice. However, as you and so many others have pointed out, that is not the case.

    Properly used, it's hard to beat.

    "Properly used…" Very eloquently said!

  6. That's an interesting observation, Moe. When I first learned of State Plane and saw how grid and ground distances were related through combined factors, I thought that the massive computing power available today would make the system easier to use and a more popular choice. However, as you and so many others have pointed out, that is not the case.

    Properly used, it's hard to beat.

    State plane was designed for LARGE uses, quad maps and such, not really suitable for subdivisions, road projects, ect.

    And speaking of massive computing power, why not get your coordinates near the surface as best you can instead of adjusting everything else.

    Consider a road project with a combined scale factor of .9995. What is the radius of a 1 degree highway curve in state plane for that project? Why bother with all that?

  7. I wouldn't attempt to recommend a projection system to a professional surveyor. As I recall Loyal's post, though, he was asked to create a similar raised plane.

    My purpose was to share a method that could be used by anyone who wanted to "…bother with all that," not to recommend that they do it. As to why someone would want to, that is determined solely by their own personal professional judgement.

  8. I wouldn't attempt to recommend a projection system to a professional surveyor. As I recall Loyal's post, though, he was asked to create a similar raised plane.

    My purpose was to share a method that could be used by anyone who wanted to "…bother with all that," not to recommend that they do it. As to why someone would want to, that is determined solely by their own personal professional judgement.

    Personally (and professionally) I see little value in these pseudo-LDP projections, but there is a time and a place for just about everything. Coincidently, I got a call just yesterday morning requesting help generating a Utah-Central-Zone-pseudo-LDP for a friend (PLS).

    I understand why different folks want to deal with modern Geodetic Coordinates (and projections) in different ways, and so long as they publish the Datum/Realization/Projection Parameters (constants), OR "modification recipe," I'm a happy camper.

    Loyal

  9. …. I thought that the massive computing power available today would make the system easier to use and a more popular choice. However, as you and so many others have pointed out, that is not the case..

    Of course massive computing power has made it easier to use and a more popular choice, but the problem I run into is many surveyors just using what the magic box kicks out, without the skills or willingness to check that it makes sense.  I also try to get guys to go actually measure a ground distance between a couple of points to make sure the actual resulting distances make sense.

    Whenever I check someone else's work I get out my calculator and check their math against the "metadata" they publish and see if I get their same results.  (No I don't redo a whole network of points but just some spot-checks).  The trouble with just calibrating in to their work is that you might just be duplicating their errors.  I've seen that happen.

    If there is not enough "metadata", I consider that an issue as well.

  10. Is there coursework anywhere that teaches the fundamentals of developing coordinate systems or does everyone learn whatever they can whenever they can?

  11. As a math teacher you should have been exposed to Topology. for others, Topology is the mathematical study of the geometric properties that are preserved through deformations, twistings, and stretchings of objects. I took a topology course as a Civil Engineering undergraduate at Lehigh University. Besides the basic and advanced Geometry in High school and Geodesy at NJIT. The short course in SPCs is included in almost every surveying Text. Satellite Geodesy, the true heart of GPS being a separate course. It would also be prudent to be well grounded in Physics, not just classical vectors but optical and quantum physics.

    Paul in PA

  12. Some of the fundamental constants for Lambert a plane can be developed using algebra, geometry, maybe a little trig, and calculus. When you start to convert angular latitudes and longitudes on the ellipse to linear measures on a plane, however, some higher math is needed. Oscar Adams did it with a complex plane, but other authors use vectors. James Stem "codified" the results so that Lambert planes could be developed algorithmically using his equations without wading through the theoretical development. Of course, he did the same for Transverse Mercator planes, which are much more complicated, as well.

    Fortunately, just as you don't need to know how to build a car in order to drive one, you don't need to know how a Lambert plane is developed in order to use it. But some understanding goes a long way if you want to create your own Lambert planes.

  13. Some of the fundamental constants for Lambert a plane can be developed using algebra, geometry, maybe a little trig, and calculus. When you start to convert angular latitudes and longitudes on the ellipse to linear measures on a plane, however, some higher math is needed. Oscar Adams did it with a complex plane, but other authors use vectors. James Stem "codified" the results so that Lambert planes could be developed algorithmically using his equations without wading through the theoretical development. Of course, he did the same for Transverse Mercator planes, which are much more complicated, as well.

    Fortunately, just as you don't need to know how to build a car in order to drive one, you don't need to know how a Lambert plane is developed in order to use it. But some understanding goes a long way if you want to create your own Lambert planes.

    Not to be too much of a stickler for detail, but I take issue with the continued usage of the generic term "plane" when talking about projected surfaces (except of course a "Tangent Plane" and such).

    I believe that [at least] some of the confusion many surveyors have with these discussions, is the use of "plane" when we are really talking about a "developed surface," than can be treated as a plane for coordinate purposes, but in fact is a portion of a mathematical SURFACE unique to the particular projection. An Elliptic Cylinder for Transverse Mercator (or Hotine), or a Cone for a Lambert Conic.

    Just a pet peeve of mine, carry on.

    BTW, I agree 100% with your analogy concerning cars!

    Loyal

  14. Some of the fundamental constants for Lambert a plane can be developed using algebra, geometry, maybe a little trig, and calculus. When you start to convert angular latitudes and longitudes on the ellipse to linear measures on a plane, however, some higher math is needed. Oscar Adams did it with a complex plane, but other authors use vectors. James Stem "codified" the results so that Lambert planes could be developed algorithmically using his equations without wading through the theoretical development. Of course, he did the same for Transverse Mercator planes, which are much more complicated, as well.

    Fortunately, just as you don't need to know how to build a car in order to drive one, you don't need to know how a Lambert plane is developed in order to use it. But some understanding goes a long way if you want to create your own Lambert planes.

    Math Teacher I would like to set the record straight (because I like to give credit where credit is due); It was Mr. T. Vincenty  who " compiled or developed"

    the math for the NOAA Manual NOS NGS 5, State Plane Coordinate System of 1983. Please see said manual page iv and read the first 3 lines under

    "ACKOWLEDGMENT". Also look at page 62 "BIBLIOGRAPHY" and you will see listed 3 papers published by Mr. T. Vincenty.

    years 1985,and 2 in 1986 dealing with State Plane Coordinate System which were incorporated into NOS NGS 5 which came out in 1989.

    JOHN NOLTON

  15. Fortunately, just as you don't need to know how to build a car in order to drive one, you don't need to know how a Lambert plane is developed in order to use it. But some understanding goes a long way if you want to create your own Lambert planes.

    Good analogy….to further it a bit, you need to know enough about cars to drive them, put in gas, air in tires, how the mirrors adjust, and the more you know the better chances you can figure out what's wrong when it breaks down…..With using a projection, you need to understand enough to drive it through your project….:D

  16. Absolutely, Tom. I think that designing a coordinate system is really the easy part, especially in today's world of boundless software. The hard part is what you and your guys do. But knowing whether an anomaly is likely to be coordinate system-related or measurement-related does make it easier to resolve.

     

    As @Mighty Moe pointed out, knowing the system's error characteristics throughout the project is a key to success. When you look at the documentation for the Iowa system and other similar systems, you see the amount of attention paid to that issue for intensively-used wide-area systems.

Leave a Reply