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Map projections and parameters

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Printable PDF map of Minnesota Coordinate System Zones (PDF 551 KB)

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:

Roy Graff
Minnesota Department of Transportation, Office of Land Management
Surveys Research and Support Unit
Transportation Building Mail Stop 643
395 John Ireland Boulevard
Saint Paul, MN 55155-1800
(651)366-3443
roy.graff@dot.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)

NAD83 UTM Parameters
Zone False Northing (meters) False Easting (meters) Central Meridian Latitude of Grid Origin Grid Scale Factor at Central Meridian West Edge of Zone East Edge of Zone
UTM 14 0.000 500000.000 -99 00 00 00 00 00 0.999600000000 -102 00 00 -96 00 00
UTM 15 0.000 500000.000 -93 00 00 00 00 00 0.999600000000 -96 00 00 -90 00 00
UTM 16 0.000 500000.000 -87 00 00 00 00 00 0.999600000000 -90 00 00 -84 00 00

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

NAD83 Minnesota State Plane Parameters
State Plane Zone Semi-major Axis (meters) Semi-minor Axis (meters) Southern Standard Parallel Northern Standard Parallel Longitude of Origin Latitude of Grid Origin
North 6378137.000 6356752.314 47 02 00 48 38 00 -93 06 00 46 30 00
Central 6378137.000 6356752.314 45 37 00 47 03 00 -94 15 00 45 00 00
South 6378137.000 6356752.314 43 47 00 45 13 00 -94 00 00 43 00 00


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

 

NAD83 Lambert Conformal Conic County Parameters
 County Semi-major Axis (meters) Semi-minor Axis (meters) Southern Standard Parallel Northern Standard Parallel Longitude of Origin Latitude of Grid Origin Ellipsoid Height (meters) Scale Factor (PPM) *
 Anoka 6378418.941 6357033.310 45 04 00 45 22 00 -93 16 00 45 02 07 281.941 44.20428724
 Becker 6378586.581 6357200.388 46 47 00 47 05 00 -95 41 00 46 43 04 449.581 70.48782427
 Beltrami/North ** 6378505.809 6357119.886 48 07 00 48 28 00 -95 01 00 48 01 12 368.809 57.82393824
 Beltrami/South 6378544.823 6357158.770 47 30 00 47 55 00 -94 51 00 47 24 45 407.823 63.94077142
 Benton 6378490.569 6357104.698 45 35 00 45 47 00 -94 03 00 45 33 33 353.569 55.43452579
 Big Stone 6378470.757 6357084.952 45 13 00 45 32 00 -96 03 00 45 09 08 333.757 52.32828959
 Blue Earth 6378403.701 6357018.121 43 56 00 44 22 00 -94 16 00 43 50 53 266.701 41.81487478
 Brown 6378434.181 6357048.499 44 10 00 44 28 00 -94 44 00 44 06 29 297.181 46.59369970
 Carlton 6378454.907 6357069.155 46 28 00 46 44 00 -92 41 00 46 25 02 317.907 49.84323792
 Carver 6378400.653 6357015.083 44 41 00 44 54 00 -93 46 00 44 38 23 263.653 41.33699229
 Cass/North 6378567.378 6357181.249 46 55 00 47 19 00 -94 13 00 46 48 13 430.378 67.47707050
 Cass/South 6378546.957 6357160.896 46 16 00 46 44 00 -94 28 00 46 09 23 409.957 64.27535188
 Chippewa 6378476.853 6357091.028 44 50 00 45 12 00 -95 51 00 44 45 10 339.853 53.28405458
 Chisago 6378411.321 6357025.715 45 20 00 45 40 00 -93 05 00 45 17 47 274.321 43.00958101
 Cook/North 6378647.541 6357261.143 47 56 00 48 10 00 -90 15 00 47 53 00 510.541 80.04547409
 Cook/South 6378647.541 6357261.143 47 33 00 47 49 00 -90 15 00 47 26 20 510.541 80.04547409
 Cottonwood 6378514.953 6357129.000 43 54 00 44 10 00 -94 55 00 43 50 53 377.953 59.25758572
 Crow Wing 6378546.957 6357160.896 46 16 00 46 44 00 -94 28 00 46 09 23 409.957 64.27535188
 Dakota 6378421.989 6357036.347 44 31 00 44 55 00 -93 19 00 44 28 19 284.989 44.68216973
 Dodge 6378481.425 6357095.584 43 53 00 44 08 00 -92 55 00 43 50 02 344.425 54.00087831
 Douglas 6378518.001 6357132.038 45 48 00 46 03 00 -96 03 00 45 45 32 381.001 59.73546821
 Faribault 6378521.049 6357135.075 43 34 00 43 48 00 -93 57 00 43 30 00 384.049 60.21335070
 Fillmore 6378464.661 6357078.876 43 33 00 43 48 00 -92 05 00 43 30 00 327.661 51.37252461
 Freeborn 6378521.049 6357135.075 43 34 00 43 48 00 -93 57 00 43 30 00 384.049 60.21335070
 Goodhue 6378434.181 6357048.499 44 18 00 44 40 00 -93 08 00 44 11 41 297.181 46.59369970
 Grant 6378518.001 6357132.038 45 48 00 46 03 00 -96 03 00 45 45 32 381.001 59.73546821
 Hennepin 6378418.941 6357033.310 44 53 00 45 08 00 -93 23 00 44 47 28 281.941 44.20428724
 Houston 6378436.619 6357050.928 43 34 00 43 48 00 -91 28 00 43 30 00 299.619 46.97594298
 Isanti 6378411.321 6357025.715 45 20 00 45 40 00 -93 05 00 45 17 47 274.321 43.00958101
 Itasca/North 6378574.389 6357188.237 47 34 00 47 49 00 -93 44 00 47 30 00 437.389 68.57629430
 Itasca/South 6378574.389 6357188.237 47 05 00 47 25 00 -93 44 00 47 01 35 437.389 68.57629430
 Jackson 6378521.049 6357135.075 43 34 00 43 48 00 -93 57 00 43 30 00 384.049 60.21335070
 Kanabec 6378472.281 6357086.471 45 49 00 46 20 00 -92 54 00 45 43 48 335.281 52.56723084
 Kandiyohi 6378498.189 6357112.292 44 58 00 45 20 00 -94 45 00 44 53 29 361.189 56.62923202
 Kittson 6378449.421 6357063.688 48 36 00 48 56 00 -96 09 00 48 32 38 312.421 48.98311215
 Koochiching 6378525.621 6357139.632 48 00 00 48 37 00 -93 45 00 47 50 45 388.621 60.93017444
 Lac Qui Parle 6378476.853 6357091.028 44 50 00 45 12 00 -95 51 00 44 45 10 339.853 53.28405458
 Lake of the Woods/North 6378466.185 6357080.395 49 11 00 49 20 00 -94 59 00 49 09 00 329.185 51.61146586
 Lake of the Woods/South 6378496.665 6357110.773 48 27 00 48 53 00 -94 53 00 48 21 58 359.665 56.39029077
 Le Sueur 6378434.181 6357048.499 44 18 00 44 40 00 -93 08 00 44 11 41 297.181 46.59369970
 Lincoln 6378643.579 6357257.195 44 17 00 44 37 00 -96 16 00 44 11 48 506.579 79.42428957
 Lyon 6378559.758 6357173.655 44 15 00 44 35 00 -95 51 00 44 11 44 422.758 66.28236427
 McLeod 6378414.369 6357028.753 44 32 00 44 55 00 -94 38 00 44 27 22 277.369 43.48746350
 Mahnomen 6378586.581 6357200.388 47 12 00 47 27 00 -95 49 00 47 09 06 449.581 70.48782427
 Marshall 6378441.801 6357056.093 48 14 00 48 29 00 -96 23 00 48 10 23 304.801 47.78840592
 Martin 6378521.049 6357135.075 43 34 00 43 48 00 -93 57 00 43 30 00 384.049 60.21335070
 Meeker 6378498.189 6357112.292 44 58 00 45 20 00 -94 45 00 44 53 29 361.189 56.62923202
 Morrison 6378502.761 6357116.849 45 51 00 46 16 00 -94 12 00 45 46 26 365.761 57.34605575
 Mower 6378521.049 6357135.075 43 34 00 43 48 00 -93 57 00 43 30 00 384.049 60.21335070
 Murray 6378617.061 6357230.765 43 55 00 44 10 00 -95 46 00 43 50 53 480.061 75.26664918
 Nicollet 6378403.701 6357018.121 43 56 00 44 22 00 -94 16 00 43 50 53 266.701 41.81487478
 Nobles 6378624.681 6357238.360 43 34 00 43 48 00 -95 57 00 43 30 00 487.681 76.46135541
 Norman 6378468.623 6357082.825 47 12 00 47 27 00 -96 27 00 47 09 02 331.623 51.99370913
 Olmsted 6378481.425 6357095.584 43 53 00 44 08 00 -92 55 00 43 50 02 344.425 54.00087831
 Ottertail 6378525.621 6357139.632 46 11 00 46 39 00 -95 43 00 46 06 23 388.621 60.93017444
 Pennington 6378445.763 6357060.042 47 36 00 48 05 00 -96 22 00 47 29 56 308.763 48.40959045
 Pine 6378472.281 6357086.471 45 49 00 46 20 00 -92 54 00 45 43 48 335.281 52.56723084
 Pipestone 6378670.401 6357283.927 43 53 00 44 09 00 -96 15 00 43 50 57 533.401 83.62959278
 Polk 6378445.763 6357060.042 47 36 00 48 05 00 -96 22 00 47 29 56 308.763 48.40959045
 Pope 6378502.761 6357116.849 45 21 00 45 42 00 -95 09 00 45 16 58 365.761 57.34605575
 Ramsey 6378418.941 6357033.310 44 53 00 45 08 00 -93 23 00 44 47 28 281.941 44.20428724
 Red Lake 6378445.763 6357060.042 47 36 00 48 05 00 -96 22 00 47 29 56 308.763 48.40959045
 Redwood 6378438.753 6357053.055 44 16 00 44 34 00 -95 14 00 44 11 41 301.753 47.31052343
 Renville 6378414.369 6357028.753 44 32 00 44 55 00 -94 38 00 44 27 22 277.369 43.48746350
 Rice 6378434.181 6357048.499 44 18 00 44 40 00 -93 08 00 44 11 41 297.181 46.59369970
 Rock 6378624.681 6357238.360 43 34 00 43 48 00 -95 57 00 43 30 00 487.681 76.46135541
 Roseau 6378449.421 6357063.688 48 36 00 48 56 00 -96 09 00 48 32 38 312.421 48.98311215
 St Louis/North 6378543.909 6357157.859 47 59 00 48 32 00 -92 27 00 47 50 00 406.909 63.79746939
 St Louis/ Central 6378605.783 6357219.525 47 20 00 47 45 00 -92 27 00 47 15 00 468.783 73.49842125
 St Louis/South 6378540.861 6357154.821 46 47 00 47 08 00 -92 27 00 46 39 00 403.861 63.31958690
 Scott 6378421.989 6357036.347 44 31 00 44 55 00 -93 19 00 44 28 19 284.989 44.68216973
 Sherburne 6378443.325 6357057.612 45 02 00 45 28 00 -93 53 00 44 58 39 306.325 48.02734717
 Sibley 6378414.369 6357028.753 44 32 00 44 55 00 -94 38 00 44 27 22 277.369 43.48746350
 Stearns 6378502.761 6357116.849 45 21 00 45 42 00 -95 09 00 45 16 58 365.761 57.34605575
 Steele 6378481.425 6357095.584 43 53 00 44 08 00 -92 55 00 43 50 02 344.425 54.00087831
 Stevens 6378502.761 6357116.849 45 21 00 45 42 00 -95 09 00 45 16 58 365.761 57.34605575
 Swift 6378470.757 6357084.952 45 13 00 45 32 00 -96 03 00 45 09 08 333.757 52.32828959
 Todd 6378548.481 6357162.415 45 52 00 46 17 00 -94 54 00 45 46 24 411.481 64.51429312
 Traverse 6378463.746 6357077.964 45 38 00 45 58 00 -96 33 00 45 35 08 326.746 51.22906579
 Wabasha 6378426.561 6357040.904 44 09 00 44 25 00 -92 16 00 44 06 25 289.561 45.39899347
 Wadena 6378546.957 6357160.896 46 16 00 46 44 00 -94 28 00 46 09 23 409.957 64.27535188
 Waseca 6378481.425 6357095.584 43 53 00 44 08 00 -92 55 00 43 50 02 344.425 54.00087831
 Watonwan 6378514.953 6357129.000 43 54 00 44 10 00 -94 55 00 43 50 53 377.953 59.25758572
 Winona 6378453.688 6357067.940 43 54 00 44 08 00 -91 37 00 43 50 50 316.688 49.65211628
 Wright 6378443.325 6357057.612 45 02 00 45 28 00 -93 53 00 44 58 39 306.325 48.02734717
 Yellow Medicine 6378530.193 6357144.189 44 40 00 44 57 00 -95 54 00 44 32 30 393.193 61.64699818

* 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

NAD83 Transverse Mercator County Parameters
 County False Northing (meters) False Easting (meters) Central Meridian Latitude of Grid Origin Grid Scale Factor at Central Meridian Ellipsoid Height (meters)
 Aitkin 30481.8640 152409.3196 -93 25 57 46 09 15 1.000059152669 377.953
 Clay 30481.4423 152407.2110 -96 42 00 46 37 48 1.000045317862 289.561
 Clearwater 30482.2708 152411.3547 -95 22 33 47 09 06 1.000072505661 463.297
 Hubbard 30482.2416 152411.2097 -94 55 14 46 48 13 1.000071553661 457.201
 Lake 30482.3728 152411.8636 -91 24 33 47 04 00 1.000075844621 484.633
 Mille Lacs 30481.7112 152408.5566 -93 37 14 45 33 32 1.000054146138 345.949
 Washington  30481.2751 152406.3759 -92 50 00 44 44 45 1.000039836799 254.509
 Wilkin 30481.5511 152407.7567 -96 31 28 46 01 18 1.000048901066 312.421


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):

YF = YS * (( R + Ellipsoid Height ) / R )
XF = XS * (( R + Ellipsoid Height ) / R )

where:     YF= Final northing
                YS= Stem's northing
                XF= Final easting
                XS= 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:

YS = YO /  (( R + Ellipsoid Height ) / R )
XS = XO /  (( R + Ellipsoid Height ) / R )

where:     YO= Original northing
                XO= 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.

NAD83 Oblique Mercator County Parameters
 County Latitude of Local Origin Longitude of Local Origin Azimuth of Positive Skew Elevation Shift (meters) Added Northing (meters) Added Easting (meters)
 Cook/ North Shore 46 30 00 89 19 40 62 00 00 -1693.167 -3505207.011 -6187452.376
 Lake/ North Shore 46 10 00 89 58 34 46 00 00 -1697.739 -4389128.779 -4206248.413
 St.Louis/NorthShore  45 45 00 90 41 30 45 00 00 -1703.835 -4358648.718 -4114808.230


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 = kc = 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):

YF = YS * ( ( R + Elevation Shift ) / R ) + N1
XF = XS * ( ( R + Elevation Shift ) / R ) + E1

where:    YF= Final northing
              YS= Stem's northing
              XF= Final easting
              XS= 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:

YS = ( R * ( YO - N1 ) ) / ( R + Elevation Shift )
XS = ( R * ( XO - E1 ) ) / ( R + Elevation Shift )

where:     YO= Original northing
               XO= Original easting

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


NAD27 Universal Transverse Mercator (UTM)

NAD27 UTM Parameters
Zone False Northing (meters) False Easting (meters) Central Meridian Latitude of Grid Origin Grid Scale Factor at Central Meridian West Edge of Zone East Edge of Zone
UTM 14
0.000
500000.000
-99 00 00
00 00 00
0.999600000000
-102 00 00
-96 00 00
UTM 15
0.000 500000.000 -93 00 00 00 00 00 0.999600000000 -96 00 00
-90 00 00
UTM 16
0.000
500000.000
-87 00 00
00 00 00
0.999600000000
-90 00 00
-84 00 00


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

NAD27 Minnesota State Plane Parameters
State Plane Zone Semi-major Axis (meters) Semi-minor Axis (meters) Southern Standard Parallel Northern Standard Parallel Longitude of Origin Latitude of Grid Origin
 North 6378206.4 6356583.8 47 02 00 48 38 00 -93 06 00 46 30 00
 Central 6378206.4 6356583.8 45 37 00 47 03 00 -94 15 00 45 00 00
 South 6378206.4 6356583.8 43 47 00 45 13 00 -94 00 00 43 00 00


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 Parameters for Alternate Computations
Zone North Central South
L 1 2000000.00  2000000.00  2000000.00 
L 2 335160.00  339300.00  338400.00 
L 3 18984319.62  20006679.72  21327006.06 
L 4 19471398.75  20493457.15  21874349.14 
L 5 0.9999028166  0.9999220223  0.9999220448 
L 6 0.7412196637  0.7233880702  0.7009277824 
L 7 2861  2771  2661 
L 8 24.63011  20.89747  20.12517 
L 9 3.80362  3.80497  3.80662 
L 10 5.01609  4.76197  4.46959 
L 11


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 North State Plane Project Zone Parameters
Project Zone Y Constant X Constant Combined Factor
96-4845 800000 600000 1.0000272
96-4830 709000 600000 0.9999524
96-4815 618000 600000 0.9999016
96-4800 527000 600000 0.9998700
96-4745 435000 600000 0.9998568
96-4730 344000 600000 0.9998639
96-4715 253000 600000 0.9998893
96-4700 162000 600000 0.9999335
95-4845 800000 800000 1.0000177
95-4830 709000 800000 0.9999477
95-4815 618000 800000 0.9998969
95-4800 527000 800000 0.9998605
95-4745 435000 800000 0.9998521
95-4730 344000 800000 0.9998449
95-4715 253000 800000 0.9998703
95-4700 162000 800000 0.9999097
94-4900 893000 1100000 1.0001119
94-4845 800000 1100000 1.0000177
94-4830 709000 1100000 0.9999477
94-4815 618000 1100000 0.9998969
94-4800 527000 1100000 0.9998605
94-4745 435000 1100000 0.9998425
94-4730 344000 1100000 0.9998449
94-4715 253000 1100000 0.9998703
93-4830 709000 1300000 0.9999477
93-4815 618000 1300000 0.9998873
93-4800 527000 1300000 0.9998605
93-4745 435000 1300000 0.9998425
93-4730 344000 1300000 0.9998449
93-4715 253000 1300000 0.9998655
93-4700 162000 1300000 0.9999097
93-4645 162000 1300000 0.9999019
92-4830 709000 1600000 0.9999381
92-4815 618000 1600000 0.9998873
92-4800 527000 1600000 0.9998510
92-4745 435000 1600000 0.9998378
92-4730 344000 1600000 0.9998401
92-4715 253000 1600000 0.9998655
92-4700 162000 1600000 0.9999145
92-4645 162000 1600000 0.9999019
92-4630 162000 1600000 0.9998845
91-4800 527000 1800000 0.9998510
91-4745 435000 1800000 0.9998283
91-4730 344000 1800000 0.9998305
91-4715 253000 1800000 0.9998655
91-4700 162000 1800000 0.9999097
91-4645 162000 1800000 0.9998971
90-4800 527000 2100000 0.9998462
90-4745 435000 2100000 0.9998330
90-4730 344000 2100000 0.9998401




NAD27 Project Zones Lying in the Central State Plane

NAD27 Central State Plane Project Zone Parameters
Project Zone Y Constant X Constant Combined Factor
96-4700 618000 600000 0.9999335
96-4645 618000 600000 0.9999209
96-4630 527000 600000 0.9998845
96-4615 435000 600000 0.9998666
96-4600 344000 600000 0.9998783
96-4545 253000 600000 0.9999037
96-4530 162000 600000 0.9999479
96-4515 162000 600000 0.9999881
95-4715 618000 800000 0.9998703
95-4700 618000 800000 0.9999097
95-4645 618000 800000 0.9998924
95-4630 527000 800000 0.9998702
95-4615 435000 800000 0.9998571
95-4600 344000 800000 0.9998640
95-4545 253000 800000 0.9998941
95-4530 162000 800000 0.9999431
95-4515 162000 800000 0.9999833
94-4715 618000 1100000 0.9998703
94-4700 618000 1100000 0.9999097
94-4645 618000 1100000 0.9999019
94-4630 527000 1100000 0.9998702
94-4615 435000 1100000 0.9998618
94-4600 344000 1100000 0.9998735
94-4545 253000 1100000 0.9998989
94-4530 162000 1100000 0.9999431
94-4515 162000 1100000 0.9999833
93-4715 618000 1300000 0.9998655
93-4700 618000 1300000 0.9999097
93-4645 618000 1300000 0.9999019
93-4630 527000 1300000 0.9998749
93-4615 435000 1300000 0.9993618
93-4600 344000 1300000 0.9998735
93-4545 253000 1300000 0.9998989
93-4530 162000 1300000 0.9999526
93-4515 162000 1300000 0.9999928
92-4645 618000 1600000 0.9999019
92-4630 527000 1600000 0.9998845
92-4615 435000 1600000 0.9998666
92-4600 344000 1600000 0.9998783
92-4545 253000 1600000 0.9999132
92-4530 162000 1600000 0.9999574
92-4515 162000 1600000 0.9999928




NAD27 Project Zones Lying in the South State Plane

NAD27 South State Plane Project Zone Parameters
Project Zone Y Constant X Constant Combined Factor
96-4530 800000 600000 0.9999479
96-4515 800000 600000 0.9999881
96-4500 709000 600000 0.9999357
96-4445 618000 600000 0.9998929
96-4430 527000 600000 0.9998598
96-4415 435000 600000 0.9998503
96-4400 344000 600000 0.9998595
96-4345 253000 600000 0.9999072
96-4330 162000 600000 0.9999686
95-4515 800000 800000 0.9999881
95-4500 709000 800000 0.9999310
95-4445 618000 800000 0.9998976
95-4430 527000 800000 0.9998741
95-4415 435000 800000 0.9998693
95-4400 344000 800000 0.9998768
95-4345 253000 800000 0.9999072
95-4330 162000 800000 0.9999686
94-4530 800000 1100000 0.9999431
94-4515 800000 1100000 0.9999833
94-4500 709000 1100000 0.9999310
94-4445 618000 1100000 0.9998976
94-4430 527000 1100000 0.9998788
94-4415 435000 1100000 0.9998788
94-4400 344000 1100000 0.9998978
94-4345 253000 1100000 0.9999262
94-4330 162000 1100000 0.9999829
93-4530 800000 1300000 0.9999526
93-4515 800000 1300000 0.9999928
93-4500 709000 1300000 0.9999357
93-4445 618000 1300000 0.9998976
93-4430 527000 1300000 0.9998836
93-4415 435000 1300000 0.9998788
93-4400 344000 1300000 0.9998978
93-4345 253000 1300000 0.9999262
93-4330 162000 1300000 0.9999829
92-4515 800000 1600000 0.9999928
92-4500 709000 1600000 0.9999357
92-4445 618000 1600000 0.9998976
92-4430 527000 1600000 0.9998836
92-4415 435000 1600000 0.9998788
92-4400 344000 1600000 0.9998978
92-4345 253000 1600000 0.9999262
92-4330 162000 1600000 0.9999829
91-4415 435000 1800000 0.9998788
91-4400 344000 1800000 0.9998978
91-4345 253000 1800000 0.9999357
91-4330 162000 1800000 0.9999924

 

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 North Shore Project Zone Parameters
Project Zone Y Constant X Constant Combined Factor
91-4715 253000 1800000 0.9998941
91-4700 162000 1800000 0.9999383
91-4645 162000 2100000 0.9999266
90-4745 435000 2100000 0.9998616
90-4730 344000 2100000 0.9998687



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:

NAD27 County Project Zone Parameters
 County Project Zone Y Constant X Constant Combined Factor
 Anoka, North 700000 1300000 1.0000073
 Anoka, Central 700000 1300000 0.9999644
 Anoka, South 700000 1300000 0.9999321
 Carver 300000 1000000 0.9998906
 Dakota 527000 1300000 0.9998836
 Hennepin 500000 1300000 0.9999129
 Ramsey 500000 1300000 0.9999129
 Scott 527000 1300000 0.9998836
 Washington, North 709000 1600000 0.9999490
 Washington, Central 618000 1600000 0.9999490
 Washington, South 618000 1600000 0.9999064



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