Here? a question for Lambert state plane coordinate system users. Over how big an area can a Lambert state plane coordinate system give usable results?
Before you answer, consider the following table. It shows the calculation of the length of an airport runway in Honolulu using North Carolina state plane coordinates.
These calculations work because of the fundamental characteristics of Lambert projections. While projecting an ellipsoid onto a plane requires heady mathematics, seeing why the North Carolina state plane coordinate system works in Hawaii does not. Here? how.
The Lambert cone
Every Lambert state plane coordinate system is a life-sized, full-scale map. Furthermore, each one maps the entire northern hemisphere. Two illustrations will help to visualize these ideas.
The illustration at the left is a modified version of a diagram that appears in the National Geodetic Survey publication NOAA Manual NOS NGS 5. The original diagram pictures the ellipsoid and the imaginary cone that Lambert conic conformal projections are based on. This modified version shows the radius of the base of the cone, the height of the cone, and the slant height of the cone for the North Carolina state plane coordinate system.
The slant height is the constant K listed in the table of constants in Manual 5 for the North Carolina state plane coordinate system. The height and radius are calculated from the slant height by observing that the angle between the height and the slant height has the same degree measure as the latitude of the projection? central parallel.
The slant height extends from a point above the pole to the equator and the base of the cone encircles the ellipsoid above the equator. Thus, the imaginary cone encompasses the entire northern hemisphere. Consequently, the resulting Lambert projection, the North Carolina state plane in this case, also covers the entire northern hemisphere.
Flattening a cone
The second illustration shows a conical paper drinking cup that has been flattened. The cup was cut along the dark line from its vertex to its base. The flattened cone is a Lambert projection. The remaining line on the flattened cone is the central meridian on the projection.
Note that the radius of the flattened cone is the slant height of the cone. This result creates the measure of convergence of the meridians on Lambert projections. The 360 degrees allotted to the equator by the base of the cone is less than 360 degrees on the flattened cone. As a result, a one-degree difference in longitude on the ellipsoid is less than one degree on a Lambert projection.
Meridians become straight lines that form angles smaller than those formed by corresponding meridians on the ellipsoid. The equator and all other parallels remain circular.
The green area
Cutting and flattening the conical drinking cup divided the surface of the cup along the meridian that is 180 degrees from the central meridian of the Lambert projection. The green area between the edges of the flattened cone is on the coordinate plane but not on the Lambert projection. Coordinates of points in this green area are on the same coordinate system as the Lambert projection, but they do not have unique corresponding latitude and longitude coordinates on the ellipsoid.
Convergence for the meridian where the green area is created, the edges of the flattened cone, can be found by substituting ?80 for the difference in longitude in the formula for convergence on page 28 in NGS Manual 5. For the North Carolina state plane projection, the formula is:
? = (?? – ?) sin(F?) = ?80 sin(35.2517586002) = ?03?53?26.32540?/p>
This diagram shows approximately the appearance of the North Carolina state plane projection. Note that parallels are concentric circles and meridians are straight lines (radii). The projection transforms the entire northern hemisphere of the GRS 80 ellipsoid into a sector of a circle. The radius of the circle is 13,178,320.6222 meters (K in the computed constants for North Carolina in Manual 5). The central angle of the sector is 207?46?52.65079? Remember; the actual projection is life-sized.
The grid lines form a conventional Cartesian coordinate system. While the Lambert projection occupies only a portion of the coordinate plane, the coordinate system it defines extends infinitely in every direction.
Lambert state plane projections for other states are similar in appearance, but with different dimensions. Radii for projections for states north of North Carolina are shorter, while states south of North Carolina have longer radii. Central angles are larger for more northern states and smaller for states further south.
Except at the central meridian, meridian lines on the projection are not directed due north to the pole. For meridian lines between the central meridian and the ? = ?0?marks, these lines are northeast or northwest. At the ? = ?0?marks, meridian lines run east or west on the projection. Past ? = ?0?degrees, meridian lines are the reflections of those below them.
Thus, the azimuths of meridian lines on the projection are not the same as azimuths of meridians on the ellipsoid. Consequently, a geodetic azimuth and its corresponding azimuth on the projection will not be equal.
Convergence and azimuths
Two numbers that stand out in the table for the Runway 4L calculations are the state plane azimuth (grid azimuth) and the geodetic azimuth. Note that the grid azimuth is southeast while the geodetic azimuth is northeast. Airport survey data confirms that the geodetic azimuth is correct. In addition, the measured runway is named 4L, indicating that the magnetic heading of the runway is about 40 degrees. Compass variation at the airport is roughly 10 degrees east, so the geodetic azimuth is about 50 degrees.
The Corpscon conversion to state plane coordinates also calculates convergence for the point. For runway 4L, convergence is -45?33?07.89240? Geodetic azimuth is the sum of grid azimuth and convergence minus, if significant, the arc-to-chord correction. For runway 4L, the sum of grid azimuth and convergence is
98?33?01.2050 + (-45?33?07.89240? = 52?59?53.3126?
The arc-to-chord correction, calculated by using the geographic coordinate formula given in NGS Manual 5, is -6.2706 seconds. The geodetic azimuth computed from the grid azimuth for runway 4L is
52?59?53.3126?- (-0?0?06.2706? = 52?59?59.5832?
Geodetic azimuth calculated from grid azimuth differs from the NGS Inverse calculation by only 0.0458 seconds. The arc-to-chord correction is needed because using the North Carolina state plane coordinate system in Hawaii creates a very, very wide Lambert projection.
So, how much is usable?
Any Lambert state plane projection can be used anywhere in the northern hemisphere for short lines like Runway 4L. Even far from the central meridian, only standard Lambert reduction procedures are needed, but very large adjustments are to be expected.
It may be tempting to use a local state plane projection to calculate a very long line, say from New York to Honolulu. The grid distance can be readily calculated, but calculating the average scale factor is problematic. The Simpson? Rule calculation given in Manual 5 is inadequate for lines millions of meters long.
State plane coordinate calculations beyond ? = ?0?are troublesome for Corpscon and the NGS Geodetic Tool Kit. Corpscon can do the forward calculation, but not the inverse. The NGS Toolkit output format limits the number of digits that can be displayed. Also, coordinates that lie in the empty space between the edges of the flattened cone and grid distances for lines that cross that space are not usable.
As a practical matter, then, any Lambert state plane projection provides usable results for points in the northern hemisphere whose convergence is within a range of ?0 degrees. These points have latitudes greater than zero and longitudes within ?0 degrees divided by the sine of the projection? central parallel of the central meridian.
Foldable NC State plane
The current North Carolina road map produced by NCDOT is the North Carolina state plane coordinate system at a scale of 1 : 825,000. At this scale, the slant height of the Lambert cone is 52.407 feet and the mapping radius of the central parallel is 35.923 feet.
The map shows a few latitudes and longitudes labeled and marked as faint red lines. Even with their long radii, the circular shape of the parallels can be seen clearly. However, the map does not include Honolulu.
The intent of this post is not to recommend the use of an east-coast state plane coordinate system in the middle of the Pacific Ocean. Rather the intent is to use an extreme example to reveal characteristics of Lambert projections that are subtle in normal state plane usage. Convergence adjustments, modest at the state level, are large and easy to see at the extreme. Scale factors on Lambert projections are near to one in normal state plane applications, but they are significant in the extreme. Still, standard state plane adjustments work even very far from the projection? origin.
The mathematics used to derive Lambert conformal conic projections is somewhat advanced. The formulas that result are workable, but many of them show no intuitive connection to an ellipsoid, a cone, or a plane. The geometry that does work between the parameters of a cone and an ellipsoid does not extend to calculating plane coordinates.
Today, even though we can easily work backward and forward among software solutions, the underlying mathematics, and the underlying concepts, understanding can still be difficult. The genius of the mathematicians, surveyors, and geodesists who worked forward from concepts to mathematics to software is hard to imagine but easy to appreciate. That many of them worked in more than one of those fields must have helped bridge many gaps.
Fortunately, once the ellipsoidal coordinates are converted to plane coordinates, coordinate geometry works exactly as it should. The result is a reliable system that can even be extended far beyond its intended boundaries.
Some helpful links
- AirNav data sheet for Honolulu International Airport
- Conformal Map Projections in Geodesy, E. J. Krakiwsky. Lecture notes that cover all of the mathematics that supports the derivation of conformal map projections. Includes good intuitive explanations for many of the difficult concepts.
- Corpscon 6.0.1 free download. Converts between geographic and state plane coordinates.
- Geometric Reduction of Measured Lines, T. Vincenty. Formulas with justification for reducing measured lines to the ellipsoid.
- NOAA Manual NOS NGS 5 State Plane Coordinate Systems, James Stem.