Land Management

Map projections and parameters

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Notice of Error in Beltrami County North Zone Parameters

Conad83 was originally written in 1986 to run on the HP-310 computer system.  In 1988, Conad83 was rewritten to run on IBM-compatible computers.  At that time there was a transcription error, and the south parallel parameter for Beltrami County North Zone was mistakenly set one degree too far south, which results in county coordinate positional errors on the order of one to three meters.  See below in the Lambert Zones table.  The erroneous value persisted in the MnCon program that replaced the functionality of the Conad programs in 1997 and also appeared in some documentation of the Minnesota County Coordinate System, including this page.

This error was repaired in MnCon Version 1.9.3, released on June 7, 2001. Beginning with version 1.9.3, MnCon will include both the correct and erroneous parameters for Beltrami County North Zone.  For lack of a better term, they are designated “good” and “bad” on the map projection lists and in the printed and file outputs.  The uncorrected (or bad) value appears at the bottom of the lists and is included for the convenience of those who may need to convert from or to the old erroneous county coordinates.

To summarize, any Beltrami County North Zone coordinate computed by the PC version of Conad83 or by MnCon prior to June 7, 2001 is incorrect with respect to the Minnesota County Coordinate System definition.  Any coordinate projections computed from Beltrami County North Zone coordinates are also suspect and should be examined.  No other counties, zones, or projections are affected.

Introduction

The following tables and examples provide the data needed to perform conversions of coordinates between map projections used in the State of Minnesota.

Required parameters, examples of usage and methods of parameter calculation are listed for all projections used in Minnesota.  Since different conversion packages give the parameters different names, all known aliases have been noted.

For more information, please contact:

Geoff Bitner
MnDOT, Office of Land Management
Survey & Mapping Section
395 John Ireland Boulevard, MS632
Saint Paul, MN 55155-1800
geoffrey.bitner@state.mn.us

Click on the following links to jump to the associated topic:

NAD83 Universal Transverse Mercator
NAD83 Minnesota State Plane
NAD83 Minnesota County Coordinate System

NAD83 Minnesota County Zones with Lambert Conformal Conic Projection
NAD83 Minnesota County Zones with Transverse Mercator Projection
NAD83 Minnesota County Zones with Oblique Mercator Projection

NAD27 Universal Transverse Mercator
NAD27 Minnesota State Plane
NAD27 Minnesota Project Coordinate System

NAD27 Minnesota Project Zones in North State Plane
NAD27 Minnesota Project Zones in Central State Plane
NAD27 Minnesota Project Zones in South State Plane
NAD27 Minnesota Project Zones Along North Shore of Lake Superior
NAD27 Minnesota Project Coordinate System -- Modified for Counties

Conversions Between Datums

NAD83 Universal Transverse Mercator (UTM)

When UTM coordinates are used in Minnesota, Zone 15 is usually extended east and west to encompass the entire state within a single zone, which may be denoted as "Zone 15E" or "Zone 15 Extended." For example, the MnDOT Basemap specifies "Universal Transverse Mercator (UTM) Minnesota extended Zone 15, NAD83 coordinates."

The following parameters apply to all NAD83 UTM zones:

Projection = Transverse Mercator
Spheroid = GRS 1980

UTM ARC/INFO Example

The parameters needed to convert to or from UTM Zone 15 are shown below. This can become part of a file used in ARC/INFO's PROJECT command.  More detailed examples showing complete conversions are shown in the NAD83 Minnesota County Coordinate System section of this document.

Projection UTM
Zone 15
Datum NAD83
Zunits NO
Units METERS
Xshift 0.0000000000
Yshift 0.0000000000
Parameters

NAD83 Minnesota State Plane

The Semi-major Axis can also be called the Equatorial Radius. It is represented by the letter a in equations. The Semi-minor Axis can also be called the Polar Radius. It is represented by the letter b in equations.

The Longitude of Origin may also be called the Central Meridian.  Within various software packages, the Southern Standard Parallel may be called either Standard Parallel 1 or Standard Parallel 2, making the Northern Standard Parallel either Standard Parallel 2 or Standard Parallel 1.  A process of trial and error may be necessary to determine the correct terminology.

The following constants apply for all Minnesota state plane zones:

Projection = Lambert Conformal Conic
False Northing = 100000.000 m
False Easting = 800000.000 m
Eccentricity = e = 0.08181919104283185
Flattening = f = 0.003352810681183637
Inverse Flattening = 1 / f = 298.2572221008827

State Plane ARC/INFO Example

The parameters needed to convert to or from the Minnesota North State Plane zone are shown below. This can become part of a file used in ARC/INFO's PROJECT command. More detailed examples showing complete conversions are shown in the NAD83 Minnesota County Coordinate System section of this document.

Projection STATEPLANE
Datum NAD83
Units METERS
Zone 4276
Parameters

NAD83 Minnesota County Coordinate System

Counties in the Minnesota County Coordinate System are defined using Lambert Conformal Conic, Transverse Mercator, or Oblique Mercator mapping projections. For the purpose of listing their parameters, they are separated into tables according to their mapping projections.

NAD83 Lambert Conformal Conic County Zones

NAD83 Lambert Conformal Conic Formulas

The Semi-major Axis, or Equatorial Radius, is represented by the letter a in equations. The Semi-minor Axis, or Polar Radius, is represented by the letter b in equations.

The Longitude of Origin may also be called the Central Meridian.  Within various software packages, the Southern Standard Parallel may be called either Standard Parallel 1 or Standard Parallel 2, making the Northern Standard Parallel either Standard Parallel 2 or Standard Parallel 1.  A process of trial and error may be necessary to determine the correct terminology.

The following constants apply for all Minnesota counties defined with Lambert projections:

False Northing = N₀ = 30480.0610 m
False Easting = E₀ = 152400.3048 m
Eccentricity = e = 0.08181919104283185
Flattening = f = 0.003352810681183637
Inverse Flattening = 1 / f = 298.2572221008827

Equations for computing both the forward and inverse calculations are available in NOAA Manual NOS NGS 5 titled State Plane Coordinate System of 1983 by James E. Stem. One change must be made to make those formulas work for the Minnesota County Coordinate System Lambert Conformal Conic counties. The ellipsoid height must be added to the standard GRS80 semi-major axis.

semi-major_axis = a = 6378137.000 + ellipsoid_height

If you are using a map projection conversion from a commercial software package you will need to create a user defined ellipsoid.  This generally requires the lengths of the semi-major axis and the semi-minor axis.  Both are available in the table above.  The length of the semi-minor axis of the ellipsoid was computed is as follows:

semi-minor_axis = b + ( ellipsoid_height * ( 1 - f ) )
semi-minor_axis = 6356752.314 + ( ellipsoid_height * 0.996647189318816363 )

where:
ellipsoid_height = County's ellipsoid height (rarely used in commercial packages)
                       a = GRS80 semi-major axis (equatorial radius) = 6378137.000 m
                       b = GRS80 semi-minor axis (polar radius) = 6356752.314 m
                       f  =  flattening of the geodetic ellipsoid = 0.003352810681183637

* Entry of a scale factor in commercial software packages may be required to achieve the same results as the MnCon program. The scale factor is entered typically as PPM, or parts per million, but a trial-and-error process may be necessary, since some programs expect positive values, while others expect negative values. The scale factor is computed by dividing the number in the "Ellipsoid Height" column by one millionth of the GRS80 semi-major axis length, for example in Anoka County, the scale factor = 281.941/6.378137 = 44.20428724 ppm.

** The latitude of the south parallel for Beltrami County North Zone was incorrectly shown as 47 07 00.  The correct value of 48 07 00 is now shown above.

NAD83 Lambert Projection ARC/INFO Example

ARC/INFO cannot convert files from a Lambert county coordinate system directly into UTM or State Plane coordinates.  Instead, a two step process must be used with a conversion to latitude/longitude being the extra step.  For example, to convert a coverage from Dakota County to UTM, one can run ARC/INFO's PROJECT command twice using these two projection files:

DAKOTA.PRJ

INPUT /* Dakota County to Lat/Long, metric units
Projection LAMBERT
Zunits NO
Units METERS
Xshift 0.0000000000
Yshift 0.0000000000
Parameters 6378421.989 6357036.347
44 31 00 /* 1st standard parallel
44 55 00 /* 2nd standard parallel
-93 19 00 /* central meridian
44 28 19 /* latitude of projection's origin
152400.3048 /* false easting (meters)
30480.06096 /* false northing (meters)
OUTPUT
Projection GEOGRAPHIC
Spheroid GRS1980
Zunits NO
Units DD
Xshift 0.0000000000
Yshift 0.0000000000
Parameters
END

UTM.PRJ

INPUT /* Lat/Long to UTM, metric units
Projection GEOGRAPHIC
Datum NAD83
Zunits NO
Units DD
Xshift 0.0000000000
Yshift 0.0000000000
Parameters
OUTPUT
Projection UTM
Zone 15
Datum NAD83
Zunits NO
Units METERS
Xshift 0.0000000000
Yshift 0.0000000000
Parameters
END

The commands needed are:

PROJECT COVER SOURCECOV INTERCOV DAKOTA.PRJ
PROJECT COVER INTERCOV TARGETCOV UTM.PRJ

where SOURCECOV = The source (input) coverage
               INTERCOV = The intermediate latitude/longitude coverage
            TARGETCOV = The target (output) coverage

ARC/INFO can also convert ASCII files containing one X Y coordinate set per record separated by a space. It uses the same two projection files:

PROJECT FILE SOURCEFIL INTERFIL DAKOTA.PRJ
PROJECT FILE INTERFIL TARGETFIL UTM.PRJ

where SOURCEFIL = The source (input) ASCII file
               INTERFIL = The intermediate latitude/longitude ASCII file
           TARGETFIL = The target (output) ASCII file

NAD83 Transverse Mercator County Zones

Note: St. Louis County has developed the St. Louis County Transverse Mercator Coordinate System 96 that provides a single map projection for the entire county.  Map scaling error is generally limited to less than one part in 40,000.  This projection is included in MnCon, and its parameters are provided below:

Central Meridian = 92 27 00 West
Latitude of Grid Origin = 46 37 00 North
False Northing = 1,000,000 meters
False Easting = 1,450,000 meters
Semi-Major Axis = 6,378,523 meters
Semi-Minor Axis = 6,357,138.3141403 meters
Scale Factor = 0.99998529
Inverse Flattening = 298.2752724012354

NAD83 Transverse Mercator Formulas

Equations for computing both the forward and inverse calculations are based on NOAA Manual NOS NGS 5 titled State Plane Coordinate System of 1983 by James E. Stem.  When the Minnesota County Coordinate System was designed by University of Minnesota surveying professor Gerald Johnson and some graduate students, the only parameters identified for transverse mercator counties were the central meridian, the latitude of grid origin, and an ellipsoid height.  They used the same false northing (30480.0610 m) and false easting (152400.3048 m) as for the lambert conformal conic counties.  They set the grid scale factor at the central meridian to 1.000.  Ellipsoid height is not used in James Stem's manual nor in map projection conversions in commercial software packages.

In the Minnesota County Coordinate System the ellipsoid height is used in some extra steps beyond those shown in Stem's equations.  These are the steps that must be added after finishing Stem's equations in the forward direction (latitude/longitude to county coordinate):

Y_F = Y_S * (( R + Ellipsoid Height ) / R )
X_F = X_S * (( R + Ellipsoid Height ) / R )

where:     Y_F= Final northing
                Y_S= Stem's northing
                X_F= Final easting
                X_S= Stem's easting
                R = Radius of curvature in the prime vertical
  and Ellipsoid Height = Average ellipsoid height for county

Similarly, when doing the inverse calculations (county to latitude/longitude), these steps must be applied before starting Stem's equations:

Y_S = Y_O /  (( R + Ellipsoid Height ) / R )
X_S = X_O /  (( R + Ellipsoid Height ) / R )

where:     Y_O= Original northing
                X_O= Original easting

NAD83 Transverse Mercator County Zones are based on the GRS80 ellipsoid.

Obviously, commercial software packages cannot be changed to handle this.  Instead you can account for the use of the ellipsoid height by making some minor adjustments to the grid scale factor at the central meridian and the false northing and easting.  The method of making these adjustments is shown below.  The results from these adjusted parameters give answers that match the original method to the nearest millimeter.  The adjusted parameters were listed in the above table of parameters for transverse mercator zones.

Grid Scale Factor at Central Meridian = ( R + ellipsoid_height ) / R

where:   ellipsoid_height = County's ellipsoid height
                                    R = Radius of curvature of the prime vertical
                                        = ( k * a ) / ( 1 - e² * sin² phi ) ^1/2

where:   k = scale factor = 1.000
             a = semi-major axis of the ellipsoid = 6378137 m
             e = first eccentricity = 0.08181919104283185
          phi = latitude at midpoint of county

False Northing and False Easting were computed on a trial and error basis to give the best results possible for points at the extreme edges of the county using the following procedure.  First, latitudes and longitudes for the corners of each county were converted  to true county coordinates to use for comparisons.

Then, using the map projection conversion from a commercial software package, a transverse mercator projection was defined.  The central meridian and latitude of origin were from the above table.  The scale factor was computed as shown in the equation above.  The false northing was 30480.0610 m and the false easting 152400.3048 m.

Next, the same latitudes and longitudes representing the corners of the counties were entered into the map projection conversion from the commercial package.  The difference between the true county coordinates and the commercial software package's results were averaged and added to the false northing and easting.  The transverse mercator projection was redefined using the new false northing and easting.  Finally, the conversions were re-run on the commercial software package.  In all cases, the results were within 1 mm of the true county coordinates.

NAD83 Transverse Mercator ARC/INFO Example

ARC/INFO can convert files between a transverse mercator county coordinate system and latitude/longitude, UTM or State Plane coordinates in a one step process.  It doesn't require two steps like in the case for the Lambert conformal conic counties because the ellipsoid is not changed.  For example, to convert a coverage from Washington County to UTM, one can run ARC/INFO's PROJECT command using this projection file:

WASH2UTM.PRJ

INPUT /* Washington County to UTM, metric units
Projection TRANSVERSE
Datum NAD83
Zunits NO
Units METERS
Xshift 0.0000000000
Yshift 0.0000000000
Parameters
1.000039836799 /* scale factor at central meridian
-92 50 00.000 /* longitude at central meridian
44 44 45.000 /* latitude of origin
152406.37590 /* false easting (meters)
30481.27510 /* false northing (meters)
OUTPUT
Projection UTM
Zone 15
Datum NAD83
Zunits NO
Units METERS
Xshift 0.0000000000
Yshift 0.0000000000
Parameters
END

The commands needed are:

PROJECT COVER SOURCECOV TARGETCOV WASH2UTM.PRJ

where:   SOURCECOV = The source (input) coverage
             TARGETCOV = The target (output) coverage

ARC/INFO can also convert ASCII files containing one X Y coordinate set per record separated by a space. It uses the same projection file:

PROJECT FILE SOURCEFIL TARGETFIL WASH2UTM.PRJ

where:   SOURCEFIL = The source (input) ASCII file
             TARGETFIL = The target (output) ASCII file

NAD83 Oblique Mercator County Zones

The Oblique Mercator parameters for map projection conversions in commercial software packages have not been determined.  Below are the parameters established by Professor Johnson and used by CONAD83 and MnCon.

The following constants apply for all Minnesota county zones defined with the Oblique Mercator projection:

False Northing = N₀ = 0.0000 m
False Easting = E₀ = 0.0000 m
Grid scale factor at the local origin = k_c = 1.0000000000000
First eccentricity = e = 0.08181919104283185
Computed flattening = f = 0.003352810681183637
Computed inverse flattening = 1 / f = 298.2572221008827

Oblique Mercator Formulas

Equations for computing both the forward and inverse calculations are based on NOAA Manual NOS NGS 5 titled State Plane Coordinate System of 1983 by James E. Stem.  When the Minnesota County Coordinate System was designed by University of Minnesota surveying professor Gerald Johnson and some graduate students, they added some extra steps for oblique mercator formulas to minimize the difference between ground distances and map distances.  As stated above, no one has yet determined how to make them work in map projection conversions in commercial software packages.

These are the steps that must be added after finishing Stem's equations in the forward direction (latitude/longitude to county coordinate):

Y_F = Y_S * ( ( R + Elevation Shift ) / R ) + N₁
X_F = X_S * ( ( R + Elevation Shift ) / R ) + E₁

where:    Y_F= Final northing
              Y_S= Stem's northing
              X_F= Final easting
              X_S= Stem's easting
              N₁= Added northing
              E₁ = Added easting
              R = Radius of curvature
     and Elevation Shift = County zone elevation shift

Similarly, when doing the inverse calculations (county to latitude/longitude), these steps must be applied before starting Stem's equations:

Y_S = ( R * ( Y_O - N₁ ) ) / ( R + Elevation Shift )
X_S = ( R * ( X_O - E₁ ) ) / ( R + Elevation Shift )

where:     Y_O= Original northing
               X_O= Original easting

NAD83 Oblique Mercator County Zones are based on the GRS80 Ellipsoid.

NAD27 Universal Transverse Mercator (UTM)

When UTM coordinates are used in Minnesota, Zone 15 is usually extended east and west to encompass the entire state within a single zone, which may be denoted as "Zone 15E" or "Zone 15 Extended."  For example, the MnDOT Basemap specifies "Universal Transverse Mercator (UTM) Minnesota extended Zone 15, NAD83 coordinates."

The following parameters apply to all NAD27 UTM zones:

Projection = Transverse Mercator
Spheroid = Clarke 1866

UTM ARC/INFO Example

The parameters needed to convert to or from UTM Zone 15 are shown below. This can become part of a file used in ARC/INFO's PROJECT command.  More detailed examples showing complete conversions are shown in the NAD83 Minnesota County Coordinate System section of this document.

Projection UTM
Zone 15
Datum NAD27
Zunits NO
Units METERS
Xshift 0.0000000000
Yshift 0.0000000000
Parameters

NAD27 Minnesota State Plane

The projection definitions under NAD27 are quite similar to those under NAD83.  The state plane coordinates from each datum, however, are different by about 100,000 m for Y and 190,380 m for X due to the use of different false northings and false eastings.

Another method for computing to or from NAD27 state plane coordinates used the following set of parameters.  The equations that make use of them are listed in Department of Commerce publication 62-4 titled State Plane Coordinates by Automatic Data Processing by Charles N. Claire.

NAD27 State Plane ARC/INFO Example

The parameters needed to convert to or from the Minnesota North State Plane zone are shown below. This can become part of a file used in ARC/INFO's PROJECT command. More detailed examples showing complete conversions are shown in the NAD83 Minnesota County Coordinate System section of this document.

Projection STATEPLANE
Datum NAD27
Units METERS
Zone 4276
Parameters

NAD27 Minnesota Project Coordinate System

In order to achieve a coordinate system in which map distances closely matched ground distances, the Minnesota Department of Highways developed the Minnesota Project Coordinate System.  In it, the State of Minnesota was divided into a grid of rectangles measuring 15 minutes of latitude by one degree of longitude.  The name for each zone was XX-YYYY where XX is the line of longitude marking the east edge of the rectangle and YYYY is the line of latitude forming the south edge of the rectangle.  For example zone 93-4445 is bounded by 94 degrees longitude to the west, 45 degrees latitude to the north, 93 degrees longitude to the east and 44 degrees, 45 minutes latitude to the south.  Parameters consisting of a combined factor, X constant and Y constant were established for each project coordinate system zone that was defined by one of these rectangles.  More information about this was written by Robert B. Roscoe, P.E., in a paper titled Project Coordinate System--A Ground Coordinate System for Surveyors and Engineers.

NAD27 Minnesota Project Coordinate System Formulas

State plane coordinates must always be computed as an intermediate step between the project coordinate system and latitude/longitude and between the project coordinate system and Universal Transverse Mercator.  The equations for converting project coordinate system coordinates to state plane are as follows:
           Ys = CF * ( Yp + Yc )
           Xs = CF * ( Xp + Xc )
           where   Ys = State Plane Y Coordinate
                  Yp = Project Y Coordinate
                  Yc = Y Constant
                  Xs = State Plane X Coordinate
                  Xp = Project X Coordinate
                  Xc = X Constant
                  CF = Combined Factor

To convert state plane coordinates to project coordinate system coordinates, these equations are used:
           Yp = ( Ys / CF ) - Yc
           Xp = ( Xs / CF ) - Xc

NAD27 Project Zones Lying in the North State Plane

NAD27 Project Zones Lying in the Central State Plane

NAD27 Project Zones Lying in the South State Plane

Other Minnesota Project Coordinate Zones

Project Zones Lying Along North Shore of Lake Superior

There are five Project Coordinate System zones that lie along the north shore of Lake Superior.  The portions of these zones that have a National Geodetic Vertical Datum of 1929 elevation below 1200 feet have been given different parameters than the rest of those zones to provide better results.
 

NAD27 Project Coordinate System -- Modified for Counties

Some counties have defined project zones that cover different, often larger, areas than the original rectangles.  This enabled all county mapping to be done in one or two or three zones rather than dealing with many of the rectangular one degree by 15 minute zones.  The formulas are identical to those used in the Minnesota Project Coordinate System.  The parameters for known project zones created for counties are as follows:

Conversions Between Datums

Converting data between the North American Datum of 1927 and the North American Datum of 1983 is not a simple task, since no direct mathematic relationship exists between them.  There are, however, several ways to convert data.  The appropriate method for any situation depends mostly on accuracy requirements.  If a high degree of accuracy is required, additional information is necessary.  For example, the coordinates of some points in a data set from both datums will be needed.  In addition, best results can be achieved if surveying measurements between data set points are available.

The most accurate way to convert data from one datum to the other is not always feasible.  It is the case where coordinates are known from both datums for one or more control points that are part of the data set and measurements from those control points(s) are available to all the other points in the data set.  In this situation, the way to convert the data set to the other datum is by starting with the other datum's coordinates on the control point(s) and recomputing the coordinates of all the other points based on the measurements from the control points.  This method will definitely provide data that is sufficiently accurate for engineering or surveying uses.

The next most accurate method is for the case where coordinates are known from both datums for three or more points in a data set, but measurements are not available.  In this situation, a transformation algorithm can be used to compute the relationship between the two datums and apply it to the other points in the data set.  This method may provide data that is sufficiently accurate for engineering or surveying uses, depending on the geographic extent of the data set being converted and the statistical results of the transformation.

If accuracy is not as important, there are several tools that can be used to convert between datums.  The most famous is the National Geodetic Survey's NADCON, which can convert between NAD27 latitude/longitude and NAD83 latitude/longitude.  MnDOT's MnCon program contains the NADCON functionality. The U.S. Army Corps of Engineers has a similar program called CORPSCON that extends the ability to state plane and Universal Transverse Mercator coordinates.  Both of these programs are available from the National Geodetic Survey.  ARC/INFO includes the NADCON algorithm in its PROJECT command, which can compute conversions within or between datums.

Datum Conversion ARC/INFO Example

ARC/INFO can convert between datums.  In this example, a coverage in NAD27 latitude/longitude will be converted to a coverage in NAD83 latitude/longitude.

DATUM.PRJ

INPUT /* NAD27 Lat/Long to NAD83 Lat/Long
Projection GEOGRAPHIC
Datum NAD27
Zunits NO
Units DD
Xshift 0.0000000000
Yshift 0.0000000000
Parameters
OUTPUT
Projection GEOGRAPHIC
Datum NAD83
Zunits NO
Units DD
Xshift 0.0000000000
Yshift 0.0000000000
Parameters
END

The command needed is:

PROJECT COVER SOURCECOV TARGETCOV DATUM.PRJ

where:   SOURCECOV = The source (input) coverage
               TARGETCOV = The target (output) coverage