# MERCATOR PROJECTION -cylindrical

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MERCATOR PROJECTION -cylindrical
-presented by Flemish geographer and cartographer Gerardus Mercator (original name Gerard Kremer) see:http://en.wikipedia.org/wiki/Gerardus_Mercator -this projection became very popular because of its ability to represent lines of constant course, which are called RHUMB LINES or LOXODROMES 1

Mercator -loxodrome or rhumb line is not the shortest route between two points -meridians are parallel to each other and the angles are preserved makes this an ideal for navigation -a straight line drawn between two points on the map cuts each meridian at the same angle 2

Mercator Derivation of the Mercator projection,
see 3

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http://mathworld.wolfram.com/Mercato rProjection.html
MercatorProjection 5

Mercator -conformal (look at Tissot's indicatrix) -> indicatrixes are circles -the scale factor is a function of latitude only -on the equator it is 1 and rises to infinity at the poles 6

Mercator -can be used latitudes approximately between +70º-70º
-not perspective DRAWBACK -scale factor starts to change significantly with increasing latitude -Arctic countries look much bigger than they really are 7

Transverse Mercator A Transverse Mercator projection is an adaptation of the Mercator projection. Both projections are cylindrical and conformal . However, in a Transverse Mercator projection, the cylinder is rotated 90° (transverse) relative to the equator so that projected surface is aligned with a meridian (or line of longitude) rather than the equator, as is the case with the regular Mercator projection.

Transverse Mercator -for those parts of the Earth that do not lie close to the equator an alternative is to turn the cylinder onto its side and make the line of contact a particular meridian -longitude can be chosen freely -usually narrow zones 3º or even 1º 9

Transverse Mercator projections result from projecting the sphere onto a cylinder tangent to a central meridian. Transverse Mercator maps are often used to portray areas with larger north-south than east-west extent. Distortion of scale, distance, direction and area increase away from the central meridian. 10

Regular/Transverse Mercator distortions
In both the regular and transverse form of the Mercator projection, there is very little distortion of scale in the narrow region near where the projected surface is tangent, or secant, to the sphere or ellipsoid representing the Earth. The scale 5° away from the equator is less than 0.4% larger than the scale at the equator and is approximately 1.53% at an angular distance of 10°. This low level of distortion, combined with the conformal property which it inherits from the Mercator projection, make the Transverse Mercator projection ideal for mapping areas with a narrow longitudinal range, e.g., a nation such as Chile

Spherical/ellipsoidal model of Mercator
The spherical form of the Transverse Mercator projection, which uses a sphere to represent the Earth, was first presented by Johann Heinrich Lambert in 1772. An elliptical form, which uses an ellipsoidal model of the Earth, was later presented by mathematician Carl Friedrich Gauss in 1822 and was further analyzed by L. Krüger in the early 20th century. In Europe, the Transverse Mercator projection is sometimes referred to as the Gauss-Krüger or Gauss Conformal projection.

Spherical model <> Ellipsoidal model
The spherical form of the Transverse Mercator projection is conformal. The distortion of scale increases entirely as a function of distance from the central meridian. The ellipsoidal form is also conformal, but scale distortion is affected to some degree by parameters of the ellipsoid and this distortion is not entirely a consistent function of distance away from the central meridian. The projected surface can be tangent to the model of the Earth in either case, which produces a map that is true to scale along this line. The scale factor can also be reduced in order to balance out the distortion over the mapped region. In this secant case, there are two lines of true scale on either side of the central meridian. These lines are parallel to the central meridian in the spherical model, but only approximately parallel in the ellipsoidal model.

Transverse Mercator, England
Many national grid systems are based on the Mercator projection The British National Grid (BNG) is based on the National Grid System of England, administered by the British Ordnance Survey. The true origin of the system is at 49 degrees north latitude and 2 degrees west longitude. The false origin is 400 km west and 100 km north. Scale at the central meridian is

Transverse Mercator -meridians and parallels are perpedicular to each other -scale factor at each point is the same in any direction: k =1 /cosθ = secθ 15

Transverse Mercator -central meridian is a straight line
-line along central meridian has right length (x-axis) -y-axis is perpendicular to central meridian -used in large scale maps 16

Transverse Mercator USGS uses many quadrangle maps at scales from 1:24000 to 1:250000 distances, directions, shapes and areas are reasonably accurate within 15 degrees of the central meridian, then distortions increases rapidly outside the 15 degrees band conformal graticule spacing increases away from central meridian 17

Case New Zealand In Land Information New Zealand (LINZ) approved the adoption and implementation of new geocentric datum, New Zealand Geodetic datum 2000 to replace the previous New Zealand Geodetic datum 1949 Influence on coordinates approximately 200 m Projection name:New Zealand Transverse Mercator Projection Datum: NZGD 2000

Origin latitude: 0º South
Origin longitude 173º East False Northing: m N False Easting: m E Scale factor:

Case New Zealand For mapping purposes since 1972, the New Zealand Map Grid (NZMG) was used. This was defined in terms of NZGD49 by an intrinsic set of formula. Because NZGD2000 uses a different reference ellipsoid a new projection was required to be defined in terms of NZGD2000. Following wide consultation, Land Information New Zealand announced on 1 July 2001 a new national mapping projection in terms of NZGD2000, New Zealand Transverse Mercator (NZTM).

Case Australia Datum: The Geodetic Datum of Australia realised by the coordinate set of 1994,hence GDA94. Projection: Transverse Mercator, utilising 6 degree Zones, Scale Factor at the Central Meridian in conformity with the Universal Transverse Mercator system. Extent: zones 47 to 58, and north-south from the equator - latitude 0° - encompassing designator M down to a latitude of 56°, or designator F. Units: International metre. False Coordinate Origin (All Zones): Northing 10,000,000 metres. Easting 500,000 metres.

Case Australia The true origin for each Zone is the Equator for Northings and the Central Meridian for Eastings. To avoid the problem of negative coordinates south of the Equator or west of the Central Meridian, the quantities above are added to the true coordinates. projected onto a vertical cylinder tangential at the Equator to the ellipsoid representing the earth. The Equator then becomes the standard parallel. This simple cylindrical projection is unsuitable for many mapping purposes as land mass shapes become greatly distorted quite quickly when moving away from theEquator

Case Australia The Geocentric Datum of Australia (GDA) is a coordinate reference system that best fits the shape of the earth as a whole. It has an origin that coincides with the centre of mass of the earth, hence the term 'geocentric'. Following a resolution of the Intergovernmental Committee on Surveying and Mapping (ICSM) in 1988, it is being progressively implemented throughout Australia as the preferred datum for all spatial information.

Case Australia It is considered to be the most effective datum as it provides: * compatibility with satellite navigation systems, such as the Global Positioning System (GPS) * compatibility with national mapping programmes already carried out on a geocentric datum, * single standard for the collection, storage and dissemination of spatial information at global, national and local levels. GDA replaces the Australian Geodetic Datum (AGD) which has been in place since The AGD provided a reference system that best fitted the shape of the earth in the Australian Region but its origin did not coincide with the centre of mass of the earth. National datums were commonly non-geocentric before satellite based navigation systems were established in the early 1970's.

Particular cases of the Transverse Mercator projection are used in several maps usually for a certain country or region. Besides UTM, GPSMan predefines the following ones: GPSMan User Manual © Miguel Filgueiras, mig_at_ncc.up.pt, LIACC & DCC Faculdade de Ciências, Universidade do Porto Some examples of parameters used: # the Austrian "Bundesmeldenetz" (BMN) projection. Parameters: central latitude 0, central longitude in three zones of 3 degrees named M28, M31 and M34 and centred at , and E, scale factor 1. The datum to be used is called "Austrian (MGI)". Coordinates in the BMN grid have a false northing of 0 and a false easting that depends on the zone: 150km in zone M28, 460km in M31, and 750km in M34. # the British National Grid (BNG) projection. Parameters: central latitude 49, central longitude -2, scale factor The datum to be used is called "Ordnance Survey Great Britain". Coordinates in this grid correspond to a false easting of 400km and a false northing of 100km.

Some examples... # the German Grid projection (GKK: Gauss-Krueger-Koordinatensystem). Parameters: central latitude 0, central longitude in zones of 6 degrees centred at 0, 3, 6, 9, 12, and 15E, scale factor 1. Coordinates in the GKK grid have a false easting of z× km, where z is the zone number. # the Irish Transverse Mercator Grid (ITM) projection. Parameters: central latitude 53.5, central longitude -8, scale factor The datum to be used is called "Ireland 1965". Coordinates in this grid correspond to a false easting of 200km and a false northing of 250km.

Some examples... # the Portuguese Military Maps projection, used in 1:25000 maps published by the Portuguese Army Geographic Institute. Parameters: central latitude , central longitude , scale factor 1. The datum to be used is called "Lisboa". Military coordinates in these maps correspond to a false easting of 200km and a false northing of 300km. # the Swedish Grid (SEG) projection. Parameters: central latitude 0, central longitude , scale factor 1. Coordinates in this grid correspond to a false easting of 1500km.

Some examples # the Portuguese Military Maps projection, used in 1:25000 maps published by the Portuguese Army Geographic Institute. Parameters: central latitude , central longitude , scale factor 1. The datum to be used is called "Lisboa". Military coordinates in these maps correspond to a false easting of 200km and a false northing of 300km. # the Swedish Grid (SEG) projection. Parameters: central latitude 0, central longitude , scale factor 1. Coordinates in this grid correspond to a false easting of 1500km.

Some examples.. # the Taiwan Grid projection (TWG). Parameters: central latitude 0, central longitude in 6 zones of 2 degrees centred at 115, 117, ..., 125, and scale factor Coordinates in the TWG grid have a false easting of 250km. This grid is usually employed with either the "Hu-Tzu-Shan" datum (also known as "TWD67"), or the "TWD97" datum (whose definition could not be found for inclusion in GPSMan). # the Uniform Finnish Grid (YKJ) projection. Parameters: central latitude 0, central longitude 27, scale factor 1. Coordinates in this grid correspond to a false northing of 500km. There is a single zone named 27E. # the Basic Finnish Grid (KKJ) projection. Parameters: central latitude 0, central longitude in zones of 6 degrees centred at 21, 24, 27, and 30E, scale factor 1. Coordinates in the KKJP grid have a false easting of z× km, where z is the zone number.

Transverse Mercator, Finland GAUSS-KRÜGER
-Finnish Geodetic Institute started carrying out triangulation in 1922 -Hayford ellipsoid was chosen as a reference ellipsoid a = m 1/f = 297.0 30

Case Finland for mapping purposes was taken transverse Mercator projection around 1920 it is also called Gauss-Krüger projection System of Helsinki until 1970 -datum point in the church tower of Kallio -atzimuth and astronomical coordinates were taken from I order triangulation -four zones, central meridians 21, 24, 27 and 30 degrees from Greenwich 31

Case Finland I order triangulation adjustment (European datum 1950) How to use better accuracy coordinates when basic mapping of Finland was done in this Helsinki system about 20 years 1970 desicion: four parameter transformation using 202 common points

Case Finland This new system was called Kartastokoordinaattijärjestelmä (KKJ) which was used until 2006 as an official map grid system Coordinate changes were about 3 metres in southern Finland and about 4 meters in northern Finland

Case Finland (YKJ) 34

Case Finland -the axis of the cylinder lies in the plane of the equator and forms the y-axis of the horizontal coordinate reference system(easting) -zones which central meridian are (18º), 21º, 24º,27º,30º and (33º) longitude -”YKJ” and ”KKJ” -The cylinder touches the reference ellipsoid along a great circle 35

Case Finland - practically zones 1,2,3 and 4 ( also 0 and 5),
-Number is put first and then y-coordinate value -false easting is , so that negative coordinates are avoided -This system is now old but there are lot of data in this system -> coordinate transformations are needed -National Land Survey will support this system until 36

Case Finland Realization of ETRS89 is called EUREF-FIN
About 100 points and all FinnRef stations defines the realization National Survey of Finland started to use this new system 2006

Case Finland NOW The coordinate system is ETRS89-TM35FIN
The Citizen’s MapSite introduces the coordinate system ETRS89-TM35FIN. The user interface, maps and aerial images will be shown in the new coordinate system. The coordinates according to the ETRS89-TM35FIN are shown in the upper right-hand corner below the logo. The Settings tab can be used to change the coordinate system in use. 38

Case Finland on the plane the central meridian is a straight line
central meridian's lenght scale is 0,9996 (true) the central meridian is the x-axis of the horizontal coordinate reference system (northing) y-axis (easting) 39

Case Finland -Also for local purposes:
-Finnish National Land Survey supports mapping local systems such a way, that central meridian can be chosen from 19, 20,..., 31 degrees from Greenwich -Gauss-Krüger -scale correction is 'acceptable' , enough small for large scale mapping 40

Properties -The axis of the cylinder lies in the plane of the equator and forms the y-axis of the horizontal coordinate reference system(easting) (zones) -The cylinder touches the reference ellipsoid along a great circle 41

But from 2005 Universal Transverse Mercator (UTM) in basic mapping
-properties nearly the same as in Gauss-Krüger, but -scale factor (cuts the cylinder) -also reference ellipsoid changed to GRS80 42

UTM -UTM is used all over the world
-originally developed for USA:s army purposes Standard properties: -valid from 80º southern latititude to 84º northern latitude -zone 6º wide -60 zones from 1-60 -scale factor 43

UTM in Finland -the scale of printed basic topographic maps changed from 1:20000 to 1:25000 -name ETRS-TM35FIN (whole country in one zone) -in standard UTM 6º degrees zones 44

Military Grid Reference System (from Wikipedia)
the Military Grid Reference System (MGRS) is the geocoordinate standard used by NATO militaries for locating points on the earth. The MGRS is derived from the UTM (Universal Transverse Mercator) grid system and the UPS (Universal Polar Stereographic) grid system, but uses a different labeling convention. The MGRS is used for the entire earth. An example of an MGRS coordinate, or grid reference, would be 4QFJ , which consists of three parts: * 4Q (grid zone designator, GZD), * FJ (the 100,000-meter square identifier), and * (numerical location; easting is 1234 and northing is 5678, in this case specifying a location with 10m resolution).

MGRS An MGRS grid reference is a point reference system. When the term 'grid square' is used, it can reference an area of 10 km × 10 km, 1 km × 1 km, 100 m × 100 m, 10 m × 10 m or 1 m × 1 m, depending on the precision of the coordinates provided. (In some cases, squares adjacent to a Grid Zone Junction (GZJ) are clipped, so polygon is a better descriptor of these areas.) The number of digits in the numerical location must be even: 0, 2, 4, 6, 8 or 10, depending on the desired precision. When changing precision levels, it is important to truncate rather than round the easting and northing values to ensure the more precise polygon will remain within the boundaries of the less precise polygon. Related to this is the primacy of the southwest corner of the polygon being the labeling point for an entire polygon. In instances where the polygon is not a square and has been clipped by a grid zone junction, the polygon keeps the label of the southwest corner as if it had not been clipped.

MGRS * 4Q GZD only, precision level 6° × 8° (in most cases) * 4QFJ GZD and 100 km SQ_ID, precision level 100 km * 4QFJ precision level 10 km * 4QFJ precision level 1 km * 4QFJ precision level 100 m * 4QFJ precision level 10 m * 4QFJ precision level 1 m Such an MGRS coordinate, standing alone, may be converted to latitude and longitude. But you still do not know the position on the Earth, unless you also know the geodetic datum that is used.

MGRSGridSouthPole See