Michigan Georef Projection Problems

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Michigan Georef Projection Problems

Clever, Max
>From reading the previous posts on proj's support for RSO's it sounds like it is already in the works.  Just to make sure though, I just want to let everyone know that the Michigan Georef Projection is not supported by proj.  The only proj definition that I have seen for it is the one described under the ESRI listing that describes it as an <omerc> which is not really correct.  
 
Michigan Georef is a Hotine Oblique Mercator.  The official parameters are listed at this website http://www.michigan.gov/documents/DNR_Map_Proj_and_MI_Georef_Info_20889_7.pdf  The parameters as given at this site require the computation of gamma, hopefully that helps you figure out whether to have gamma as a mandatory or optional parameter requirement.
 
I appreciate any help that could be given.  Please see the history of emails below for reference to this problem.  Thanks.
 
-Max
 
 
Frank,

First, to answer your question, the parameters Melita offered are not accurate enough and are not the true parameters of the Michigan Georef Projection.  Also, the translation from EPSG to PROJ4 is not correct.  I don't think Proj 4 supports a point azimuth method transformation for Hotine oblique Mercator (Rectified Skew Orthomorphic Projection)

Here is a link that contains the formulas for oblique mercator and Hotine Oblique Mercator (Michigan Georef is Hotine Oblique).

 http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs34i.html <http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs34i.html>

please note though, that in the forward case, I believe the small "v" should be computed using Log not natural log.

Also, if you want it for reference, below is the code of a couple of VB forms that I wrote 3 years ago with help from my professor in college to perform the forward and reverse cases.  It is only accurate to about 7 cm in the Northing and 3 mm in the Easting when converting a point from Michigan Georef to Geographic and back.  I think there must be some lack of precision in the code that is the reason for the accuracy problem.  I hope this helps.


Const a = 6378137                         ' semi major axis of the GRS80 ellipsoid
Const e2 = 0.0066943800229          ' excentricity^2 of the GRS80 ellipsoid
Dim Pi As Double
Dim E As Double                            ' excentricity of the GRS80 ellipsoid

Public Function G2GREF(CoordIN() As Double) As Double()
    Dim M1 As Double ' Paramter
    Dim m2 As Double ' Paramter
    Dim t1 As Double ' Paramter
    Dim t2 As Double ' Paramter
    Dim t0 As Double ' Paramter
    Dim n As Double ' Paramter
    Dim F As Double ' Paramter
    Dim Eb As Double ' Paramter
    Dim Nb As Double ' Paramter
    Dim R As Double ' Paramter
   
    MS_GeoRef = "Projection Oblique Mercator; Datum:  NAD83; Ellipsoid: GRS80 " & vbCrLf & _
        "Standard Units: Meters" & vbCrLf & _
        "Origin = 86° 00' 00 W and 45° 18' 33 N" & vbCrLf & _
        "Scale factor at projection's center: k= 0.9996" & vbCrLf & _
        "Azimuth at center of projection: 337° 15' 20" & vbCrLf & _
        "False Easting: 2546731.496, False Northing: -4354009.816"
       
    Lon1 = -86#                 ' Longitude of projection's origin: 86° 00' 00" W
    Lon1 = Lon1 * Pi / 180#
    Lat1 = 45.30916666667       ' Latitude of projection's origin: 45° 18' 33" N
    Lat1 = Lat1 * Pi / 180#
    Az = 337.255555555556       ' Azimuth at center of projection: 337° 15' 20
    Az = Az * Pi / 180#
    SF = 0.9996                 ' Scale factor at projection's center
    Eb = 2546731.496            ' False Easting ( Eb = 528600.24)
    Nb = -4354009.816           ' False Northing (Nb = 499839.834)
   
    B = Sqr(1 + e2 * Cos(Lat1) ^ 4 / (1 - e2))
    A1 = a * B * SF * Sqr(1 - e2) / (1 - e2 * (Sin(Lat1) ^ 2))
    Temp = ((1 - E * Sin(Lat1)) / (1 + E * Sin(Lat1))) ^ (0.5 * E)
    t0 = Tan(Pi / 4 - (Lat1) / 2) / Temp
    D = B * Sqr(1 - e2) / (Cos(Lat1) * Sqr(1 - e2 * Sin(Lat1) ^ 2))
    F = D + Lat1 / Abs(Lat1) * Sqr(D ^ 2 - 1)       ' Eq. 4.110
    E1 = F * t0 ^ B                                 ' Eq. 4.111
    G = 0.5 * (F - 1 / F)                           ' Eq. 4.112
    Gamma0 = Isin(Sin(Az) / D)                      ' Eq. 4.113
    Lon0 = Lon1 - Isin(G * Tan(Gamma0)) / B  ' Lambda 0 at Eq. 4.114
    U0 = (Lat1 / Abs(Lat1)) * (A1 / B) * Atn(Sqr(D ^ 2 - 1) / Cos(Az))
    V0 = 0
   
    LatIN = DMS2R(CoordIN(1))
    LonIN = DMS2R(CoordIN(2))
   
    Temp = ((1 - E * Sin(LatIN)) / (1 + E * Sin(LatIN))) ^ (0.5 * E)
    t = Tan(Pi / 4 - (LatIN) / 2) / Temp            ' Eq. 4.117
    Q = E1 / (t ^ B)
    S = 0.5 * (Q - 1 / Q)
    Tc1 = 0.5 * (Q + 1 / Q)
    V1 = Sin(B * (LonIN - Lon0))
    U1 = (-V1 * Cos(Gamma0) + S * Sin(Gamma0)) / Tc1
    vl = A1 * Log((1 - U1) / (1 + U1)) / (2 * B)
    temp1 = (S * Cos(Gamma0) + V1 * Sin(Gamma0)) / Cos(B * (LonIN - Lon0))
    ul = (A1 / B) * Atn(temp1)
    x = vl * Cos(Az) + ul * Sin(Az) + Eb
    y = ul * Cos(Az) - vl * Sin(Az) + Nb
   
    ReDim Preserve CTemp(1 To 2) As Double
   
    CTemp(1) = x
    CTemp(2) = y
    G2GREF = CTemp
       
End Function


Public Function GREF2G(CoordIN() As Double) As Double()
    Dim M1 As Double ' Paramter
    Dim m2 As Double ' Paramter
    Dim t1 As Double ' Paramter
    Dim t2 As Double ' Paramter
    Dim t0 As Double ' Paramter
    Dim n As Double ' Paramter
    Dim F As Double ' Paramter
    Dim Eb As Double ' Paramter
    Dim Nb As Double ' Paramter
    Dim PhiOut As Double ' Paramter
    Dim LonOut As Double ' Paramter
   
    MS_GeoRef = "Projection Oblique Mercator; Datum:  NAD83; Ellipsoid: GRS80 " & vbCrLf & _
        "Standard Units: Meters" & vbCrLf & _
        "Origin = 86° 00' 00 W and 45° 18' 33 N" & vbCrLf & _
        "Scale factor at projection's center: k= 0.9996" & vbCrLf & _
        "Azimuth at center of projection: 337° 15' 20" & vbCrLf & _
        "False Easting: 2546731.496, False Northing: -4354009.816"
       
    Lon1 = -86#                 ' Longitude of projection's origin: 86° 00' 00" W
    Lon1 = Lon1 * Pi / 180#
    Lat1 = 45.309166666667       ' Latitude of projection's origin: 45° 18' 33" N
    Lat1 = Lat1 * Pi / 180#
    Az = 337.255555555556       ' Azimuth at center of projection: 337° 15' 20
    Az = Az * Pi / 180#
    SF = 0.9996                 ' Scale factor at projection's center
    Eb = 2546731.496            ' False Easting ( Eb = 528600.24)
    Nb = -4354009.816           ' False Northing (Nb = 499839.834)
   
   
    ttemp = e2 * (Cos(Lat1) ^ 4) / (1 - e2)
    B = Sqr(1 + ttemp)
    A1 = a * B * SF * Sqr(1 - e2) / (1 - e2 * (Sin(Lat1)) ^ 2)
    Temp = ((1 - E * Sin(Lat1)) / (1 + E * Sin(Lat1))) ^ ( 0.5 * E)
    t0 = Tan(Pi / 4 - (Lat1) / 2) / Temp
    ttemp1 = Cos(Lat1) * Sqr(1 - e2 * (Sin(Lat1)) ^ 2)
    D = B * Sqr(1 - e2) / ttemp1
   
    F = D + Lat1 / Abs(Lat1) * Sqr(D ^ 2 - 1)       ' Eq. 4.110
    E1 = F * t0 ^ B                                 ' Eq. 4.111
    G = 0.5 * (F - 1 / F)                           ' Eq. 4.112
    Gamma0 = Isin(Sin(Az) / D)                      ' Eq. 4.113
    Lon0 = Lon1 - Isin(G * Tan(Gamma0)) / B  ' Lambda 0 at Eq. 4.114
   
    xIN = CoordIN(1)
    yIN = CoordIN(2)

'Actual Computations for Reverse case Hotine Oblique Mercator

    xr = xIN - Eb
    yr = yIN - Nb
   
    vs = xr * Cos(Az) - yr * Sin(Az)
    us = yr * Cos(Az) + xr * Sin(Az)
   
    temp1 = -B * vs / A1
    Qp = (2.71828182845905) ^ temp1
    Sp = 0.5 * (Qp - 1 / Qp)
    Tp = 0.5 * (Qp + 1 / Qp)
   
    Vp = Sin(B * us / A1)
    Up = (Vp * Cos(Gamma0) + Sp * Sin(Gamma0)) / Tp

    temp2 = (1 + Up) / (1 - Up)
    t = (E1 / Sqr(temp2)) ^ (1 / B)
   
    PhiOut = Pi / 2 - 2 * Atn(t)
'Iterative Solution for Phi Out
'    For i = 1 To 30
'        Temp = ((1 - E * Sin(PhiOut)) / (1 + E * Sin(PhiOut))) ^ ( 0.5 * E)
'        PhiOut = Pi / 2 - 2 * Atn(t * Temp)
'    Next i

'Single Line Solution for Phi Out

    PhiOut = PhiOut + Sin(2 * PhiOut) * (e2 / 2 + (5 * e2 ^ 2) / 24 + (e2 ^ 4) / 12 + (13 * e2 ^ 6) / 360) + Sin(4 * PhiOut) * ((7 * e2 ^ 2) / 48 + (29 * e2 ^ 4) / 240 + (811 * e2 ^ 6) / 11520) + Sin(6 * PhiOut) * ((7 * e2 ^ 4) / 120 + (81 * e2 ^ 6) / 1120) + Sin(8 * PhiOut) * ((4279 * e2 ^ 6) / 161280)
           
    temp3 = (Sp * Cos(Gamma0) - Vp * Sin(Gamma0)) / Cos(B * us / A1)
   
    LonOut = Lon0 - Atn(temp3) / B
     
    ReDim Preserve CTemp(1 To 2) As Double
   
    CTemp(1) = r2dms(PhiOut)
    CTemp(2) = r2dms(LonOut)
    GREF2G = CTemp
End Function






-----Original Message-----
From: Frank Warmerdam [mailto:[hidden email] <mailto:[hidden email]> ] On Behalf Of Frank Warmerdam
Sent: Saturday, July 15, 2006 10:01 PM
To: Clever, Max
Subject: Re: [UMN_MAPSERVER-USERS] FW: Michigan Georef Projection Problems in Proj4

Clever, Max wrote:
> Hi,
>
>
>
> Did anyone see this the last time I sent it?  It relates to Mapserver as
> well since Mapserver uses Proj4 for its projections.

Max,

I have skimmed this material, but frankly I'm not sure what action item
there is, and I find myself with limited time for work on PROJ.4.

Is the problem that the parameters Melita offered a couple years ago
aren't accurate enough?  Or that the underlying translation from EPSG
to PROJ.4 wasn't fixed so the epsg file keeps getting regenerated
wrong?  If it is a tranlation problem, then that is basically something
I ought to fix.  But I basically need some formulation to recognise
a distinct oblique mercator case for the michigan projection from the
EPSG codes (or parameters), and what that should map to in WKT format,
and in PROJ.4 format.  I'm happy to use the ESRI WKT representation if
there isn't an obvious existing form for this special case.

If you can walk me through what should be changed, I'm willing to work
on it.

Best regards,
...

> Two years ago, I ran into a problem with the Michigan Georef Projection
> and the way that proj identified it.  I had sent emails back and forth
> for a while until someone sent a temporary solution of providing false
> parameters that worked.. *for the most part*.  This temporary solution,
> of course, did not actually solve the problem, but instead delayed the
> fixing of the methods that proj identifies projections and translates
> them.  For that I am sorry for not remaining vigilant in seeing a true
> solution being devised.  But now, since I have just now installed the
> latest version of GRASS 6.1, I have come full circle and face this
> problem again.  To provide a quick access to the background of what has
> already been said on this projection please note the previous emails
> below.  I believe, at this time still, that the *omerc* projection and
> its parameters as used by proj cannot correctly describe or transform a
> omerc projection with a "natural origin".  From what I understand,
> Hotine oblique mercator and Rectified Skew Orthomorphic are one and the
> same or depend on where the skew is corrected.  Has there been a
> solution determined for this projection?  If not, maybe a solution to
> this problem would be to have Proj have oblique mercators split between
> "natural origins" and cartesian center point origins.  I hope, maybe,
> someone has been looking at this lately but I doubt it.  Any comments or
> solutions would be very welcome.  Thanks.


--
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I set the clouds in motion - turn up   | Frank Warmerdam, [hidden email]
light and sound - activate the windows | http://pobox.com/~warmerdam <http://pobox.com/~warmerdam>
and watch the world go round - Rush    | President OSGF, http://osgeo.org <http://osgeo.org>


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