projection and transformation calculations

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INTRODUCTION
	
ProjectionandTransformationCalculations.xls v3.36 23-December-2013
This spreadsheet which will carry out common calculations with coordinates. All of the concepts and formulae given in "A guide to coordinate systems in Great Britain", which is the user manual for these calculations, are detailed in this spreadsheet. "A guide to coordinate systems in Great Britain" is available from the Ordnance Survey GPS web site (www.ordnancesurvey.co.uk/gps). The guide gives some important warnings about the use of its formulae, which you should be aware of. There are also other projection functions such as scale factors, convergence and t-T in this spreadsheet that are not in the guide.
The Constants sheet is where ellipsoid, projection and Helmert transformation parameters are entered.
The Enter coordinates here sheet is for user input of coordinates to the calculations.
Use the buttons at the bottom left of the screen to scroll through the other sheets.
Each sheet carries out a different calculation. The calculations are shown in "step by step" format where all the intermediate calculation results are given and they are also shown as results in single cells computed by user defined functions written in Visual Basic. The code for the user defined functions can be viewed and if required copied to other applications.
The calculation sheets are as follows:-
Lat & Long Format Conversions converts between DMS format, decimal degrees and degrees & decimal minutes.
Helmert Transformation transforms the user entered cartesian coordinates to cartesian coordinates on a new datum using the transformation parameters from Constants.
XYZ to lat,long,H converts the user entered cartesian coordinates to geodetic coordinates in DMS format using the ellipsoidal parameters from Constants.
lat,long to E,N projects the user entered geodetic coordinates to easting and northings using a Transverse Mercator projection and the ellipsoid and projection parameters from Constants.
E,N to lat,long is the reverse of lat,long to E,N.
lat,long,H to XYZ is the reverse of XYZ to lat,long,H.
UD Funcs Transformation Example shows how a complete Helmert transformation from WGS84 geodetic coordinates to OSGB36 Eastings and Northings and vice versa can be carried out using the user defined functions.
convergence from lat & long computes the convergence at the geodetic coordinate entered by the user.
convergence from E,N computes the convergence at the easting, northing entered by the user.
local scale factor from lat,lon computes the local scale factor at the geodetic coordinate entered by the user.
local & line sf from E,N computes the local scale factor at the easting, northing entered by the user and also the line scale factor between easting, northing and easting2, northing2.
t-T correction from E,N computes the t-T (difference between true and projected directions) between easting, northing and easting2, northing2 entered by the user.
true azimuth from E,N computes the true direction (after allowing for t-T and convergence) between easting, northing and easting2, northing2 entered by the user.
Cells for user data entry are coloured Cells giving calculation results are coloured Cells showing intermediate calculation steps are coloured
Cells which repeat the user inputted data are coloured
Constants
	ELLIPSOID AND PROJECTION CONSTANTS
	
					°	'	"	Dec Degs	Rads
	Semi-major axis, a	6377563.396	True origin latitude, j0	N	49	0	0	49.00000000	0.855211333
	Semi-minor axis, b	6356256.909	True origin longitude, l0	W	2	0	0	-2.00000000	-0.034906585
	Central Meridan Scale, F0	0.999601271700					a for OSGB36 =	6377563.3960
	True origin Easting, E0	400000.000					b for OSGB36 =	6356256.9090
	True origin Northing, N0	-100000.000					a for GRS80 & WGS84 =	6378137.0000
	aF0	6375020.48098897					b for GRS80 =	6356752.3141	Slight difference in b is due to different flattening values
	bF0	6353722.48948831					b for WGS84 =	6356752.3142
						All other parameters same
	Projection Notes
	1. The constants above are correct for working with the Ordnance Survey National Grid.
	If you want to work with OS National Grid, do not change anything in the above box.
	
	2. It is also possible to use this spreadsheet to work with Universal Transverse Mercator (UTM) grids.
	The constants that must be entered in the yellow boxes if you want to work with UTM are given in Annex A of
	'A Guide to coordinate systems in Great Britain', available from the OS GPS website www.gps.gov.uk.
	
	HELMERT TRANSFORMATION CONSTANTS
							ETRS89 (WGS84) to OSGB36
	translation parallel to X	-446.448	metres					-446.448
	translation parallel to Y	125.157	metres					125.157
	translation parallel to Z	-542.060	metres					-542.060
	scale change	20.4894	parts per million					20.4894
	rotation about X	-0.1502	seconds of arc					-0.1502
	rotation about Y	-0.2470	seconds of arc					-0.2470
	rotation about Z	-0.8421	seconds of arc					-0.8421
					reverse signs to go from OSGB36 to ETRS89
	
	Transformation Notes
	1. It is very important to understand the limitations of simple datum transformations.
	This method does not take account of the local distortions present in coordinate reference frames (TRFs)
	This is especially important when one or both of the coordinate systems involved is based on a traditional triangulation network
	For instance, transforming between WGS84 GPS coordinates and OSGB36 National Grid will incur transformation errors
	of up to five metres, depending on location.
	Also, this method cannot convert ellipsoid heights to heights above mean sea level.
	For full details on the limitations of the method coded here, see the OS publication 'A guide to coordinate systems in
	Great Britain', available free from our GPS Website (www.gps.gov.uk) or OS customer helpline.
	
	2. The constants in the table above are suitable for transforming from WGS84 GPS coordinates to OSGB36 National Grid.
	
	
	By changing the constants above, this spreadsheet may be used to convert between any two geodetic datums
	Some ellipsoid constants are given in 'A guide to coordinate systems in Great Britain'
	Ordnance Survey cannot supply suitable transformation parameters for the geodetic systems of other countries -
	please make enquiries for these parameters to the national mapping agencies of the country concerned.
Enter coordinates here
	Enter your coordinates in the yellow boxes
	
	
			°	'	"	Decimal Deg	Radians
	latitude	N	52	39	27.253085	52.65757030139	0.9190479779		Enter either latitude and longitude
	longitude	E	1	43	4.517692	1.7179215810	0.0299833879
	Ellipsoidal height				24.700	metres			Ellipsoidal height is optional (see notes)
	Easting				651409.903	metres			or easting and northing
	Northing				313177.270	metres
	Easting 2				626238.249	metres			2nd pair of coordinates is optional (see notes)
	Northing 2				302646.415	metres
	Cartesian X				3909833.018	metres			or Earth centred Cartesian XYZ coordinates
	Cartesian Y				-147097.138	metres
	Cartesian Z				5020322.478	metres
	
	Notes
	1. Type your input data in the yellow boxes only on this page.
	Enter either Latitude, Longitude and optional Ellipsoidal height, Easting and Northing or Earth centred Cartesian coordinates.
	To convert from decimal latitude and longitude or degrees lat / long and decimal minutes to the degrees, minutes and seconds format
	see the "Lat & Long Format Conversions " sheet.
	Ellipsoidal height is only required for the Cartesian XYZ coordiante conversions.
	The Earth centred Cartesian coordinates are used as input to the Helmert datum transformation calculations.
	Look at the following sheets to see the results of the calculations.
	
	2. Check that the appropriate parameters are set on the "Constants" sheet.
	
	3. For two of the calculations (t-T correction and true azimuth), the eastings and northings of two points are required.
For these calculations only, you must enter Eastings 2 and Northings 2 in addition to Easting and Northing
	
	4. For background information on the calculations contained in this spreadsheet, download 'A Guide to Coordinate Systems
	in Great Britain' from our GPS website www.gps.gov.uk, or see any textbook on map projections or surveying computations.
Lat & Long Format Conversions
	Latitude & Longitude Format Conversions (Enter your coordinates in the yellow boxes)
	
	Degrees, Minutes & Seconds to Decimal Degrees and Degrees & Decimal Minutes
	Input Data					Output Data
			Degrees (°)	Minutes (')	Seconds (")	Decimal Degrees	Degrees & Decimal Minutes
	latitude	N	52	39	27.253100	52.657570305556	N	52	39.454218333
	longitude	E	1	43	4.517700	1.717921583333	E	1	43.075295000
	
	Decimal Degrees to Degrees, Minutes & Seconds and Degrees & Decimal Minutes
	Output Data					Input Data	Output Data
			Degrees (°)	Minutes (')	Seconds (")	Decimal Degrees	Degrees & Decimal Minutes
	latitude	N	52	39	27.253100	52.657570305556	N	52	39.454218333
	longitude	E	1	43	4.517700	1.717921583333	E	1	43.075295000
	
	Degrees & Decimal Minutes to Degrees, Minutes & Seconds and Decimal Degrees
	Output Data							Input Data
			Degrees (°)	Minutes (')	Seconds (")	Decimal Degrees	Degrees & Decimal Minutes
	latitude	N	52	39	27.253100	52.657570305556	N	52	39.454218333
	longitude	E	1	43	4.517700	1.717921583333	E	1	43.075295000
Helmert Transformation
	Helmert Datum Transformation
	
	X	3909833.018				This is the input data
	Y	-147097.1376
	Z	5020322.4777
	X	3909460.068				This is the result of the calculation
	Y	-146987.301
	Z	5019888.070
	Rotation and Scale Matrix (H)					These values are intermediate calculation steps for the
	1.0000204894	4.08261601E-06	-1.19748979E-06			formulae given in 'A Guide to Coordinate Systems in Great Britain'
	-4.08261601E-06	1.0000204894	7.28190149E-07
	1.19748979E-06	-7.28190149E-07	1.0000204894			See the Transformation Notes on the
						"Constants" page for information on the
	Translation Matrix (T)		Parameters (see "Constants" sheet)			limitations and accuracy of this transformation.
	-446.448		translation parallel to X	-446.448	m
	125.157		translation parallel to Y	125.157	m
	-542.06		translation parallel to Z	-542.060	m
			scale change	20.4894	ppm
	H * Input Cartesian coords		rotation about X	-0.1502	secs
	3909906.515806589		rotation about Y	-0.2470	secs
	-147112.4581295882		rotation about Z	-0.8421	secs
	5020430.130195189
	
	X	3909460.068				This is the result of the calculation using user defined functions.
	Y	-146987.301				To see the Visual Basic code of the functions -
	Z	5019888.070				Menu = Tools, Macro, Visual Basic Editor,
						then view the code in "Module 1".
XYZ to lat,long,H
	Cartesian XYZ to Latitude, longitude and Ellipsoidal height
	
	X	3909833.018						This is the input data
	Y	-147097.1376
	Z	5020322.4777
			°	'	"	Decimal Deg	Radians
	latitude	N	52	15	16.606464	52.25461290673	0.9120150446	This is the result of the calculation
	longitude	W	2	9	16.510138	-2.154586149489	-0.0376046223
	Ellipsoidal height	626.295	m
	p	3912599.110						These values are intermediate calculation steps for the
	e2	6.6705400741E-03						formulae given in 'A Guide to Coordinate Systems in Great Britain'
	Initial j	N	52	15	16.6722	52.2546311564	0.9120153631	The radians value of latitude is computed using a user defined function
	n	6.3909051552E+06						To see the Visual Basic code of the function -
								Menu = Tools, Macro, Visual Basic Editor,
								then view the code in "Module 1".
	
	latitude	N	52	15	16.606464	52.25461290673		This is the result of the calculation using user defined functions.
	longitude	W	2	9	16.510138	-2.154586149489		To see the Visual Basic code of the functions -
	Ellipsoidal height	626.295	m					Menu = Tools, Macro, Visual Basic Editor,
								then view the code in "Module 1".
lat,long to E,N
	Latitude and longitude to easting and northing
	
	latitude	N	52	39	27.253085	This is the input data
	longitude	E	1	43	4.517692
	Easting	651409.903	m			This is the result of the calculation
	Northing	313177.270	m
	e2	6.67054007E-03				These values are intermediate calculation steps for the
	n	1.67322033E-03				formulae given in 'A Guide to Coordinate Systems in Great Britain'
	n	6.38850233E+06
	r	6.37275644E+06				The value of M is computed using a user defined function
	h2	2.47081373E-03				To see the Visual Basic code of the function -
	P	6.48899729E-02				Menu = Tools, Macro, Visual Basic Editor,
	M	4.06688295E+05				then view the code in "Module 1".
	I	3.06688295E+05
	II	1.54040791E+06
	III	1.56068754E+05
	IIIA	-2.06711230E+04
	IV	3.87512058E+06
	V	-1.70000782E+05
	VI	-1.01344704E+05
	
	Easting	651409.903	m			This is the result of the calculation using user defined functions.
	Northing	313177.270	m			To see the Visual Basic code of the functions -
						Menu = Tools, Macro, Visual Basic Editor,
						then view the code in "Module 1".
E,N to lat,long
	Easting and Northing to Latitude and Longitude
	
	Easting	651409.903						This is the input data
	Northing	313177.270
			°	'	"	Decimal Deg	Radians
	latitude	N	52	39	27.253087	52.65757030193	0.91904797787	This is the result of the calculation
	longitude	E	1	43	4.517690	1.71792158065	0.02998338787
	j'	N	52	42	57.278445	52.7159106793	0.92006620954	These values are intermediate calculation steps for the
	e2	6.67054007E-03						formulae given in 'A Guide to Coordinate Systems in Great Britain'
	n	6.38852334E+06
	r	6.37281931E+06						The radians value of j' and the value of M are computed using user defined functions
	h2	2.46422064E-03						To see the Visual Basic code of the functions -
	M	4.13177270E+05						Menu = Tools, Macro, Visual Basic Editor,
	Et	2.51409903E+05						then view the code in "Module 1".
	VII	1.61305625E-14
	VIII	3.33955474E-28
	IX	9.41985617E-42
	X	2.58400625E-07
	XI	4.69859700E-21
	XII	1.61243166E-34
	XIIA	6.65773163E-48
	
	latitude	N	52	39	27.253087	52.65757030193		This is the result of the calculation using user defined functions.
	longitude	E	1	43	4.517690	1.71792158065		To see the Visual Basic code of the functions -
								Menu = Tools, Macro, Visual Basic Editor,
								then view the code in "Module 1".
lat,long,H to XYZ
	Latitude, longitude and Ellipsoidal height to Cartesian XYZ
	
	latitude	N	52	39	27.253085	This is the input data
	longitude	E	1	43	4.517692
	Ellipsoidal height	24.700	m
	X	3874938.850	m			This is the result of the calculation
	Y	116218.624	m
	Z	5047168.207	m
	e2	6.6705400741E-03				These values are intermediate calculation steps for the
	n	6.3910506267E+06				formulae given in 'A Guide to Coordinate Systems in Great Britain'
	
	X	3874938.850	m			This is the result of the calculation using user defined functions.
	Y	116218.624	m			To see the Visual Basic code of the functions -
	Z	5047168.207	m			Menu = Tools, Macro, Visual Basic Editor,
						then view the code in "Module 1".
UD Funcs Tranformation Example
	This sheet gives an example of how the user defined functions stored in this workbook can be used.
	To see the Visual Basic code of the functions, menu = Tools, Macro, Visual Basic Editor, then view the code in "Module 1".
	A WGS84 lat, long and height (e.g. from a handheld GPS receiver) is converted and transformed in stages to an OSGB36 easting, northing and approx ODN height.
	The reverse transformation from OSGB36 to WGS84 is then performed.
	The results of each stage are all produced by user defined functions in single cells (i.e. there are no intermediate steps done in other cells).
	The outputs of the previous stage are passed as inputs to the next stage. Other parameters used are on the right of each stage.
	Note : be aware when changing the layout of this sheet that all cell references are ABSOLUTE. This is so that function inputs can be easily identified.
	WGS 84 to OSGB36
	WGS84 Latitude, longitude and Ellipsoidal
height to WGS84 Cartesian XYZ
			°	'	"	Dec Degs		User inputs DMS and Height. Decimal degrees and
	latitude	N	52	39	28.723003	52.65797861194	Input	height are the input to the user defined functions.
	longitude	E	1	42	57.787253	1.71605201472		The DMS values are converted to decimal using
	Ellipsoidal height	69.391	m					standard Excel functions
	X	3875311.465	m				Output	Ellipsoid parameters (WGS84)
	Y	116103.234	m					a =	6378137.000
	Z	5047602.291	m					b =	6356752.314
	
	
	Helmert Datum Transformation (WGS84 to OSGB36)
								Transformation (WGS84 to OSGB36)
	X	3875311.465	m	Input				translation parallel to X =	-446.448
	Y	116103.234	m		See the Transformation Notes on the			translation parallel to Y =	125.157
	Z	5047602.291	m		"Constants" page for information on the			translation parallel to Z =	-542.060
	X	3874938.849	m	Output	limitations and accuracy of this			scale change =	20.4894
	Y	116218.624	m		transformation.			rotation about X =	-0.1502
	Z	5047168.210	m					rotation about Y =	-0.2470
								rotation about Z =	-0.8421
	
	
	OSGB36 Cartesian XYZ to OSGB36 Latitude, longitude and approx ODN height
								The decimal degrees and height are the output of
	X	3874938.849	m				Input	the user defined functions.
	Y	116218.624	m					The decimal values are converted to DMS using
	Z	5047168.210	m					standard Excel functions
			°	'	"	Dec Degs		Ellipsoid parameters (OSGB36)
	latitude	N	52	39	27.253135	52.657570315182	Output	a =	6377563.396
	longitude	E	1	43	4.517710	1.717921586082		b =	6356256.910
	~ODN height	24.701	m
	
	
	OSGB36 Latitude and longitude to OSGB36 easting and northing
			°	'	"	Dec Degs		The decimal degrees are the input to the
	latitude	N	52	39	27.253135	52.657570315182	Input	user defined functions.
	longitude	E	1	43	4.517710	1.717921586082		The decimal values are converted to DMS using
	Easting	651409.903	m				Output	standard Excel functions
	Northing	313177.271	m					Ellipsoid and projection parameters (OSGB36)
								a =	6377563.396
								b =	6356256.910
								f0 =	0.9996012717
								e0 =	400000
								n0 =	-100000
								j0 =	49.00000
								l0 =	-2.00000
	
	OSGB36 to WGS84
	OSGB36 easting and northing to OSGB36 latitude and longitude
								The eastings, northings and approx ODN height are
	easting	651409.903	m				Input	input by the user. User defined functions convert
	northing	313177.270	m					them to dec degrees. The decimal degrees are
			°	'	"	Dec Degs		converted to DMS using standard Excel functions
	latitude	N	52	39	27.253095	52.657570304133	Output	Ellipsoid and projection parameters (OSGB36)
	longitude	E	1	43	4.517699	1.717921583102		a =	6377563.396
	~ODN height	24.700	m					b =	6356256.910
								f0 =	0.9996012717
								e0 =	400000
								n0 =	-100000
								j0 =	49.00000
								l0 =	-2.00000
	
	
	OSGB36 latitude, longitude and approx ODN height to OSGB36 cartesian XYZ
			°	'	"	Dec Degs		Decimal degrees and height are the input to the
	latitude	N	52	39	27.253095	52.65757030413	Input	user defined functions.
	longitude	E	1	43	4.517699	1.71792158310		The DMS values are converted to decimal using
	~ODN height	24.700	m					standard Excel functions
	X	3874938.849	m				Output	Ellipsoid parameters (OSGB36)
	Y	116218.624	m					a =	6377563.396
	Z	5047168.208	m					b =	6356256.910
	
	
	Helmert Datum Transformation (OSGB36 to WGS84)
								Transformation (OSGB36 to WGS84)
	X	3874938.849	m	Input				translation parallel to X =	446.448
	Y	116218.624	m		See the Transformation Notes on the			translation parallel to Y =	-125.157
	Z	5047168.208	m		"Constants" page for information on the			translation parallel to Z =	542.060
	X	3875311.472	m	Output	limitations and accuracy of this			scale change =	-20.4894
	Y	116103.230	m		transformation.			rotation about X =	0.1502
	Z	5047602.299	m					rotation about Y =	0.2470
								rotation about Z =	0.8421
	
	
	WGS84 Cartesian XYZ to WGS84 Latitude, longitude and Ellipsoidal height
								The decimal degrees and height are the output of
	X	3875311.472	m				Input	the user defined functions.
	Y	116103.230	m					The decimal values are converted to DMS using
	Z	5047602.299	m					standard Excel functions
			°	'	"	Dec Degs		Ellipsoid parameters (WGS84)
	latitude	N	52	39	28.722980	52.657978605604	Output	a =	6378137.000
	longitude	E	1	42	57.787042	1.716051956076		b =	6356752.314
	Ellipsoidal height	69.402	m
convergence from lat & long
	Grid convergence from Latitude and Longitude
	
	latitude	N	52	39	27.253085		This is the input data
	longitude	E	1	43	4.517692
		°	'	"	Decimal Deg	Radians
	convergence	2	57	26.556114	2.95737669840	0.0516159606	This is the result of the calculation
	n	6.38850233E+06					These values are intermediate calculation steps
	r	6.37275644E+06
	h2	2.47081373E-03
	XIII	7.95024505E-01
	XIV	9.82300034E-02
	XV	2.02438381E-03
	
	convergence	2	57	26.556114	2.95737669840		This is the result of the calculation using a user defined function.
							To see the Visual Basic code of the function -
							Menu = Tools, Macro, Visual Basic Editor,
							then view the code in "Module 1".
	
	Notes
	Grid convergence is the angle FROM the meridian line through a point TO the North-South grid line at the same point. A positive angle is clockwise.
convergence from E,N
	Grid convergence from Easting and Northing
	
	Easting	651409.903					This is the input data
	Northing	313177.270
		°	'	"	Decimal Deg	Radians
	convergence	2	57	26.556147	2.9573767074	0.05161596077	This is the result of the calculation
	j'	52	42	57.278445	52.7159106793	0.92006620954	These values are intermediate calculation steps
	n	6.38852334E+06
	r	6.37281931E+06					The radians value of j' is computed using a user defined function
	h2	2.46422064E-03					To see the Visual Basic code of the function -
	Et	2.51409903E+05					Menu = Tools, Macro, Visual Basic Editor,
	XVI	2.05594320E-07					then view the code in "Module 1".
	XVII	4.57174659E-21
	XVIII	1.60898687E-34
	
	convergence	2	57	26.556147	2.957376707		This is the result of the calculation using a user defined function.
							To see the Visual Basic code of the function -
							Menu = Tools, Macro, Visual Basic Editor,
							then view the code in "Module 1".
	
	Notes
	Grid convergence is the angle FROM the meridian line through a point TO the North-South grid line at the same point. A positive angle is clockwise.
local scale factor from lat,lon
	Local scale factor from Latitude and Longitude
	
	latitude	N	52	39	27.253085	This is the input data
	longitude	E	1	43	4.517692
	factor	1.00037733				This is the result of the calculation
	n	6.38850233E+06				These values are intermediate calculation steps
	r	6.37275644E+06
	h2	2.47081373E-03
	XIX	1.84422569E-01
	XX	-1.03620198E-02
	
	factor	1.00037733				This is the result of the calculation using a user defined function.
						To see the Visual Basic code of the function -
						Menu = Tools, Macro, Visual Basic Editor,
						then view the code in "Module 1".
	
	Notes
	Local scale factor is the scale distortion inherent in the map projection at a point
local & line sf from E,N
	Local scale factor from Easting and Northing
	
	Easting	651409.903					This is the input data
	Northing	313177.270
	factor	1.00037732					This is the result of the calculation
		°	'	"	Decimal Deg	Radians	These values are intermediate calculation steps
	j'	52	42	57.278445	52.715910679	0.9200662095
	n	6.38852334E+06					The radians value of j' is computed using a user defined function
	r	6.37281931E+06					To see the Visual Basic code of the function -
	h2	2.46422064E-03					Menu = Tools, Macro, Visual Basic Editor,
	XXI	1.22811183E-14					then view the code in "Module 1".
	XXII	2.53854231E-29
	
	factor	1.00037732					This is the result of the calculation using a user defined function.
							To see the Visual Basic code of the function -
							Menu = Tools, Macro, Visual Basic Editor,
							then view the code in
"Module 1".
	Notes
	Local scale factor is the scale distortion inherent in the map projection at a point
	For a long line the scale factor at the mid point should be calculated. For lines up to 30km in length this will give a result with an error not exceeding 1 to 2 ppm.
	If still greater accuracy is required compute a line scale factor using Simpson's Rule as shown below.
	The line scale factor between Point 1 (East1, North1) and Point 2 (East2, North2)
	is the projected distance between the points divided by the true (geodesic) distance.
	Line scale factor = (1/6)(F1 + 4Fm + F2)
	Where F1 = scale factor at Point 1; F2 = scale factor at Point 2 and Fm = scale factor at mid point between Point 1 and Point 2.
	e.g.: -	1.00037732	= Scale factor at Point 1 (East1,North1)
		1.00022969	= Scale factor at Point 2 (East2,North2)
		1.0003015598	= Scale factor at mid point of Point 1 (East1,North1) and Point 2 (East2,North2)
		1.00030221	= Line scale factor for line between Point 1 (East1,North1) and Point 2 (East2,North2)
t-T correction from E,N
	t-T correction from Eastings and Northings
	
	Easting	651409.903					This is the input data
	Northing	313177.270
	Easting 2	626238.249
	Northing 2	302646.415
		°	'	"	Decimal Deg	Radians
	(t1 - T1)	0	0	6.482943	0.00180081737	0.00003143019240	This is the result of the calculation
	(t2 - T2)	-0	0	6.259111	-0.00173864187	-0.00003034502514
	Nm	307911.8425					These values are intermediate calculation steps
	M	4.07911842E+05
	j'	52	40	6.855172	52.668570881	0.9192399742	The radians value of j' and the value of M are computed using user defined functions
	n	6.38850630E+06					To see the Visual Basic code of the functions -
	r	6.37276830E+06					Menu = Tools, Macro, Visual Basic Editor,
	XXIII	4.09374978E-15					then view the code in "Module 1".
	
	
	(t1 - T1)	0	0	6.482943	0.00180081737		This is the result of the calculation using a user defined function.
	(t2 - T2)	-0	0	6.259111	-0.00173864187		To see the Visual Basic code of the function -
							Menu = Tools, Macro, Visual Basic Editor,
							then view the code in "Module 1".
	
	Notes
	t-T correction is the difference between true direction and projected direction due to the curving of straight lines on the projection
true azimuth from E,N
	True Azimuth from eastings and northings
	
	Easting	651409.903					This is the input data
	Northing	313177.270
	Easting 2	626238.249
	Northing 2	302646.415
		°	'	"	Decimal Deg	Radians
	True Azimuth(1 to 2)	250	15	10.839515	250.253010976	4.367739005	This is the result of the calculation
	Grid Bearing(1 to 2)	247	17	50.766311	247.2974350863	4.316154474	These values are intermediate calculation steps
	C1	2	57	26.556147	2.9573767074	0.0516159608
	(t1 - T1)	0	0	6.482943	0.001800817	0.000031430
	
	True Azimuth(1 to 2)	250	15	10.839515	250.253010976		This is the result of the calculation using a user defined function.
							To see the Visual Basic code of the function -
							Menu = Tools, Macro, Visual Basic Editor,
							then view the code in "Module 1".
	
	Notes
	The true azimuth from one point (easting, northing) to a second point (easting 2, northing 2)
	is computed by applying the convergence and t-T corrections to the grid bearing.