EPSG:9815

Hotine Oblique Mercator (variant B)

Attributes

Data source: OGP

Information source: EPSG guidance note #7-2, http://www.epsg.org

Revision date: 2018-08-29

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Definition: OGP XML

<div class="syntax"><pre><span class="cp">&lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt;</span> <span class="nt">&lt;gml:OperationMethod</span> <span class="na">xmlns:epsg=</span><span class="s">&quot;urn:x-ogp:spec:schema-xsd:EPSG:1.0:dataset&quot;</span> <span class="na">xmlns:gml=</span><span class="s">&quot;http://www.opengis.net/gml/3.2&quot;</span> <span class="na">xmlns:xlink=</span><span class="s">&quot;http://www.w3.org/1999/xlink&quot;</span> <span class="na">gml:id=</span><span class="s">&quot;iogp-method-9815&quot;</span><span class="nt">&gt;</span> <span class="nt">&lt;gml:metaDataProperty&gt;</span> <span class="nt">&lt;epsg:CommonMetaData&gt;</span> <span class="nt">&lt;epsg:alias</span> <span class="na">alias=</span><span class="s">&quot;Rectified Skew Orthomorphic (RSO)&quot;</span> <span class="na">code=</span><span class="s">&quot;14853&quot;</span> <span class="na">codeSpace=</span><span class="s">&quot;urn:ogc:def:naming-system:EPSG::7301&quot;</span><span class="nt">&gt;</span> <span class="nt">&lt;epsg:remarks&gt;</span>This same name is also used for variant A.<span class="nt">&lt;/epsg:remarks&gt;</span> <span class="nt">&lt;/epsg:alias&gt;</span> <span class="nt">&lt;epsg:alias</span> <span class="na">alias=</span><span class="s">&quot;Oblique Mercator&quot;</span> <span class="na">code=</span><span class="s">&quot;14854&quot;</span> <span class="na">codeSpace=</span><span class="s">&quot;urn:ogc:def:naming-system:EPSG::7301&quot;</span><span class="nt">&gt;</span> <span class="nt">&lt;epsg:remarks&gt;</span>This was the method name used prior to October 2010.<span class="nt">&lt;/epsg:remarks&gt;</span> <span class="nt">&lt;/epsg:alias&gt;</span> <span class="nt">&lt;epsg:informationSource&gt;</span>EPSG guidance note #7-2, http://www.epsg.org<span class="nt">&lt;/epsg:informationSource&gt;</span> <span class="nt">&lt;epsg:revisionDate&gt;</span>2018-08-29<span class="nt">&lt;/epsg:revisionDate&gt;</span> <span class="nt">&lt;epsg:changes&gt;</span> <span class="nt">&lt;epsg:changeID</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:change-request:EPSG::1999.811&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:changeID</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:change-request:EPSG::2004.430&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:changeID</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:change-request:EPSG::2004.600&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:changeID</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:change-request:EPSG::2007.044&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:changeID</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:change-request:EPSG::2010.058&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:changeID</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:change-request:EPSG::2017.024&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;/epsg:changes&gt;</span> <span class="nt">&lt;epsg:show&gt;</span>true<span class="nt">&lt;/epsg:show&gt;</span> <span class="nt">&lt;epsg:isDeprecated&gt;</span>false<span class="nt">&lt;/epsg:isDeprecated&gt;</span> <span class="nt">&lt;/epsg:CommonMetaData&gt;</span> <span class="nt">&lt;/gml:metaDataProperty&gt;</span> <span class="nt">&lt;gml:metaDataProperty&gt;</span> <span class="nt">&lt;epsg:CoordinateOperationMethodMetaData&gt;</span> <span class="nt">&lt;epsg:isOperationReversible&gt;</span>true<span class="nt">&lt;/epsg:isOperationReversible&gt;</span> <span class="nt">&lt;epsg:signReversal</span> <span class="na">changeSign=</span><span class="s">&quot;false&quot;</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8811&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:signReversal</span> <span class="na">changeSign=</span><span class="s">&quot;false&quot;</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8812&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:signReversal</span> <span class="na">changeSign=</span><span class="s">&quot;false&quot;</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8813&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:signReversal</span> <span class="na">changeSign=</span><span class="s">&quot;false&quot;</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8814&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:signReversal</span> <span class="na">changeSign=</span><span class="s">&quot;false&quot;</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8815&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:signReversal</span> <span class="na">changeSign=</span><span class="s">&quot;false&quot;</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8816&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:signReversal</span> <span class="na">changeSign=</span><span class="s">&quot;false&quot;</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8817&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;epsg:example&gt;</span>For Projected Coordinate System Timbalai 1948 / R.S.O. Borneo (m) Parameters: Ellipsoid: Everest 1830 (1967 Definition) a = 6377298.556 metres 1/f = 300.8017 then e = 0.081472981and e^2 = 0.006637847 Latitude Projection Centre fc = 4°00&#39;00&quot;N = 0.069813170 rad Longitude Projection Centre lc = 115°00&#39;00&quot;E = 2.007128640 rad Azimuth of central line ac = 53°18&#39;56.9537&quot; = 0.930536611 rad Rectified to skew gc= 53°07&#39;48.3685&quot; = 0.927295218 rad Scale factor ko= 0.99984 Easting at projection centre Ec = 590476.87 m Northing at projection centre Nc = 442857.65 m Forward calculation for: Latitude lat = 5°23&#39;14.1129&quot;N = 0.094025313 rad Longitude lon = 115°48&#39;19.8196&quot;E = 2.021187362 rad B = 1.003303209 F = 1.072121256 A =6376278.686 H = 1.000002991 to = 0.932946976 g0 = 0.927295218 D = 1.002425787 lon0 = 1.914373469 D2 =1.004857458 uc =738096.09 vc =0.00 t =0.910700729 Q =1.098398182 S =0.093990763 T = 1.004407419 V =0.106961709 U = 0.010967247 v =-69702.787 u = 163238.163 Then Easting E = 679245.73 m Northing N = 596562.78 m Reverse calculations for same easting and northing first gives : v’ = -69702.787 u’ = 901334.257 Q’ = 1.011028053 S’ = 0.010967907 T’ = 1.000060146 V’ = 0.141349378 U’ = 0.093578324 t’ = 0.910700729 c = 0.093404829 Then Latitude = 5°23&#39;14.113&quot;N Longitude = 115°48&#39;19.820&quot;E<span class="nt">&lt;/epsg:example&gt;</span> <span class="nt">&lt;/epsg:CoordinateOperationMethodMetaData&gt;</span> <span class="nt">&lt;/gml:metaDataProperty&gt;</span> <span class="nt">&lt;gml:identifier</span> <span class="na">codeSpace=</span><span class="s">&quot;IOGP&quot;</span><span class="nt">&gt;</span>urn:ogc:def:method:EPSG::9815<span class="nt">&lt;/gml:identifier&gt;</span> <span class="nt">&lt;gml:name&gt;</span>Hotine Oblique Mercator (variant B)<span class="nt">&lt;/gml:name&gt;</span> <span class="nt">&lt;gml:formula&gt;</span>Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided. The following constants for the projection may be calculated : B = {1 + [e^2 * cos^4(latc) / (1 - e^2 )]}^0.5 A = a * B * kc *(1 - e^2 )^0.5 / ( 1 - e^2 * sin^2(latc)) to = tan(pi/4 - latc/2) / ((1 - e*sin(latc)) / (1 + e*sin(latc)))^(e/2) D = B (1 - e^2)^0.5 / (cos(latc) * ( 1 - e^2*sin^2(latc))^0.5) if D <span class="ni">&amp;lt;</span> 1 to avoid problems with computation of F make D^2 = 1 F = D + (D^2 - 1)^0.5 * SIGN(latc) H = F*(to)^B G = (F - 1/F) / 2 gammao = asin(sin(alphac) / D) lonO = lonc - (asin(G*tan(gamma0))) / B vc = 0 In general: uc = (A / B) atan2((Dsq - 1)^0.5 , cos (alphac) ) * SIGN(latc) (see Gn7-2 implementation notes in preface for atan2 convention) but for the special cases where alphac = 90 degrees (e.g. Hungary, Switzerland) then uc = A*(lonc - lonO) Forward case: To compute (E,N) from a given (lat,lon) : t = tan(pi/4 - lat/2) / ((1 - e sin (lat)) / (1 + e sin (lat)))^(e/2) Q = H / t^B S = (Q - 1 / Q) / 2 T = (Q + 1 / Q) / 2 V = sin(B (lon - lonO)) U = (- V cos(gammao) + S sin(gammao)) / T v = A ln((1 - U) / (1 + U)) / 2 B In general: u = (A atan2((S cos(gammao) + V sin(gammao)) , cos(B (lon - lonO))) / B) - (ABS(uc) . SIGN(latc)) but when alphac = pi/2 rad if lon = lonc, u = 0 else u = (A atan((S cos(gammao) + V sin(gammao)) / cos(B (lon - lonO))) / B) - (ABS(uc) . SIGN(latc) . SIGN(lonc – lon)) The rectified skew co-ordinates are then derived from: E = v cos(gammac) + u sin(gammac) + Ec N = u cos(gammac) - v sin(gammac) + Nc Reverse case: Compute (lat,lon) from a given (E,N) : v’ = (E - Ec) cos(gammac) - (N - Nc) sin(gammac) u’ = (N - Nc) cos(gammac) + (E - Ec) sin(gammac) + (ABS(uc) . SIGN(latc)) Q’ = e- (B v ‘/ A) where e is the base of natural logarithms. S&#39; = (Q’ - 1 / Q’) / 2 T’ = (Q’ + 1 / Q’) / 2 V’ = sin (B u’ / A) U’ = (V’ cos(gammac) + S’ sin(gammac)) / T’ t’ = (H / ((1 + U’) / (1 - U’))^0.5)^(1 / B) chi = pi / 2 - 2 atan(t’) lat = chi + sin(2chi).( e^2 / 2 + 5*e^4 / 24 + e^6 / 12 + 13*e^8 / 360) + sin(4*chi).( 7*e^4 /48 + 29*e^6 / 240 + 811*e8 / 11520) + sin(6chi).( 7*e^6 / 120 + 81*e8 / 1120) + sin(8chi).(4279 e^8 / 161280) lon= lonO - atan2 ((S’ cos(gammao) , V’ sin(gammao)) / cos(B*u’ / A)) / B<span class="nt">&lt;/gml:formula&gt;</span> <span class="nt">&lt;gml:generalOperationParameter</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8811&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;gml:generalOperationParameter</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8812&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;gml:generalOperationParameter</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8813&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;gml:generalOperationParameter</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8814&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;gml:generalOperationParameter</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8815&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;gml:generalOperationParameter</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8816&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;gml:generalOperationParameter</span> <span class="na">xlink:href=</span><span class="s">&quot;urn:ogc:def:parameter:EPSG::8817&quot;</span> <span class="nt">/&gt;</span> <span class="nt">&lt;/gml:OperationMethod&gt;</span> </pre></div>
<?xml version="1.0" encoding="UTF-8"?>
 <gml:OperationMethod xmlns:epsg="urn:x-ogp:spec:schema-xsd:EPSG:1.0:dataset" xmlns:gml="http://www.opengis.net/gml/3.2" xmlns:xlink="http://www.w3.org/1999/xlink" gml:id="iogp-method-9815">
  <gml:metaDataProperty>
    <epsg:CommonMetaData>
      <epsg:alias alias="Rectified Skew Orthomorphic (RSO)" code="14853" codeSpace="urn:ogc:def:naming-system:EPSG::7301">
        <epsg:remarks>This same name is also used for variant A.</epsg:remarks>
      </epsg:alias>
      <epsg:alias alias="Oblique Mercator" code="14854" codeSpace="urn:ogc:def:naming-system:EPSG::7301">
        <epsg:remarks>This was the method name used prior to October 2010.</epsg:remarks>
      </epsg:alias>
      <epsg:informationSource>EPSG guidance note #7-2, http://www.epsg.org</epsg:informationSource>
      <epsg:revisionDate>2018-08-29</epsg:revisionDate>
      <epsg:changes>
        <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::1999.811" />
        <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2004.430" />
        <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2004.600" />
        <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2007.044" />
        <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2010.058" />
        <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2017.024" />
      </epsg:changes>
      <epsg:show>true</epsg:show>
      <epsg:isDeprecated>false</epsg:isDeprecated>
    </epsg:CommonMetaData>
  </gml:metaDataProperty>
  <gml:metaDataProperty>
    <epsg:CoordinateOperationMethodMetaData>
      <epsg:isOperationReversible>true</epsg:isOperationReversible>
      <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8811" />
      <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8812" />
      <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8813" />
      <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8814" />
      <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8815" />
      <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8816" />
      <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8817" />
      <epsg:example>For Projected Coordinate System  Timbalai 1948 / R.S.O. Borneo (m)

Parameters:
Ellipsoid:  Everest 1830 (1967 Definition)
a = 6377298.556 metres 1/f = 300.8017
then e = 0.081472981and e^2 = 0.006637847

Latitude Projection Centre fc = 4°00'00"N = 0.069813170 rad
Longitude Projection Centre lc = 115°00'00"E = 2.007128640 rad
Azimuth of central line ac = 53°18'56.9537" = 0.930536611 rad
Rectified to skew gc= 53°07'48.3685" = 0.927295218 rad
Scale factor ko= 0.99984
Easting at projection centre Ec = 590476.87 m
Northing at projection centre Nc = 442857.65 m

Forward calculation for: 
Latitude lat = 5°23'14.1129"N = 0.094025313 rad
Longitude lon = 115°48'19.8196"E = 2.021187362 rad

B = 1.003303209            F = 1.072121256
A =6376278.686            H = 1.000002991
to = 0.932946976          g0 = 0.927295218
D = 1.002425787           lon0 = 1.914373469
D2 =1.004857458
uc =738096.09              vc =0.00

t =0.910700729             Q =1.098398182
S =0.093990763            T = 1.004407419
V =0.106961709            U = 0.010967247
v =-69702.787                u = 163238.163

Then Easting E =      679245.73 m
        Northing N =     596562.78 m

Reverse calculations for same easting and northing first gives :
v’ =   -69702.787              u’ = 901334.257
Q’ = 1.011028053
S’  = 0.010967907          T’ = 1.000060146
V’ = 0.141349378           U’ = 0.093578324
t’ = 0.910700729             c = 0.093404829

Then Latitude = 5°23'14.113"N
         Longitude = 115°48'19.820"E</epsg:example>
    </epsg:CoordinateOperationMethodMetaData>
  </gml:metaDataProperty>
  <gml:identifier codeSpace="IOGP">urn:ogc:def:method:EPSG::9815</gml:identifier>
  <gml:name>Hotine Oblique Mercator (variant B)</gml:name>
  <gml:formula>Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The following constants for the projection may be calculated :

B = {1 + [e^2 * cos^4(latc) / (1 - e^2 )]}^0.5
A = a * B * kc *(1 - e^2 )^0.5 / ( 1 - e^2 * sin^2(latc))
to = tan(pi/4 - latc/2) / ((1 - e*sin(latc)) / (1 + e*sin(latc)))^(e/2)
D = B (1 - e^2)^0.5  / (cos(latc) * ( 1 - e^2*sin^2(latc))^0.5)
if D &lt; 1 to avoid problems with computation of F make D^2  = 1 
F = D + (D^2 - 1)^0.5  * SIGN(latc)
H = F*(to)^B
G = (F - 1/F) / 2
gammao = asin(sin(alphac) / D)
lonO = lonc - (asin(G*tan(gamma0))) / B
vc = 0
In general: uc = (A / B) atan2((Dsq - 1)^0.5 , cos (alphac) ) * SIGN(latc)  (see Gn7-2 implementation notes in preface for atan2 convention)
but for the special cases where alphac = 90 degrees (e.g. Hungary, Switzerland) then 
uc = A*(lonc - lonO)  

 
Forward case: To compute (E,N) from a given (lat,lon) :

t = tan(pi/4 - lat/2) / ((1 - e sin (lat)) / (1 + e sin (lat)))^(e/2)
Q = H / t^B
S = (Q - 1 / Q) / 2
T = (Q + 1 / Q) / 2
V = sin(B (lon - lonO))
U = (- V cos(gammao) + S sin(gammao)) / T
v = A ln((1 - U) / (1 + U)) / 2 B

In general:
u = (A atan2((S cos(gammao) + V sin(gammao)) , cos(B (lon - lonO))) / B) - (ABS(uc) . SIGN(latc))

but when alphac = pi/2 rad
if lon = lonc, u = 0
else u = (A atan((S cos(gammao) + V sin(gammao)) / cos(B (lon - lonO))) / B) - (ABS(uc) . SIGN(latc) . SIGN(lonc – lon))


The rectified skew co-ordinates are then derived from:
E = v cos(gammac) + u sin(gammac) + Ec
N = u cos(gammac) - v sin(gammac) + Nc

Reverse case: Compute (lat,lon)  from a given (E,N)  :

v’ = (E - Ec) cos(gammac) - (N - Nc) sin(gammac)
u’ = (N - Nc) cos(gammac) + (E - Ec) sin(gammac) + (ABS(uc) . SIGN(latc))

Q’ = e- (B v ‘/ A)  where e is the base of natural logarithms.
S' = (Q’ - 1 / Q’) / 2
T’ = (Q’ + 1 / Q’) / 2
V’ = sin (B u’ / A)
U’ = (V’ cos(gammac) + S’ sin(gammac)) / T’
t’ = (H / ((1 + U’) / (1 - U’))^0.5)^(1 / B)

chi = pi / 2 - 2 atan(t’)

lat = chi + sin(2chi).( e^2 / 2 + 5*e^4 / 24 + e^6 / 12 + 13*e^8 / 360) +  sin(4*chi).( 7*e^4 /48 + 29*e^6 / 240 + 811*e8 / 11520) +  sin(6chi).( 7*e^6 / 120 + 81*e8 / 1120) +  sin(8chi).(4279 e^8 / 161280)

lon=  lonO - atan2 ((S’ cos(gammao) , V’ sin(gammao)) / cos(B*u’ / A)) / B</gml:formula>
  <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8811" />
  <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8812" />
  <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8813" />
  <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8814" />
  <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8815" />
  <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8816" />
  <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8817" />
</gml:OperationMethod>