Intro to Map Projection

8/14/2019 Intro to Map Projection

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INTRO TO MAP PROJECTION

In the process of map making ellipsoidal or spherical surfaces are used to represent the

surface of the Earth.These curved reference surfaces are transformed to the flat plane of

the map by means of a map projection. Since a map is a small-scale representation of the

Earth's surface it is necessary to apply some kind of scale reduction.

The geometric process of map making, an overview

1.1 Reference surfaces

In mapping different reference surfaces or earth figures are used. These include a geomet

or mathematical reference surface, the ellipsoid or the sphere, for measuring locations, an

an equipotential surface called the geoid or vertical datum for measuring heights.


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Ellipsoids are used for large scale mapping, spherescan be used for small-scale mapping.

Most commonly used ellipsoids are the International (also known as Hayford), Krasovsky,

Bessel, and the Clarke 1880.

A cross section of an ellipsoid, used to represent the Earth's surface, indicating the major

and minor axis radius

To measure locations accurately, the selected ellipsoid should fit the area of interest.

Therefor a horizontal (or geodetic) datum is established, which is an ellipsoid but positione

and oriented in such a way that it best fits to the area or country of interest. There are a fe

hundred of these local horizontal datums defined in the world.

Vertical datums are used to measure heights given on maps. The starting point for

measuring these heights are Mean Sea Level points established at coastal places. Starting

from these points the heights of points on the earth's surface can be measured usinglevelling techniques.

1.2 Map projections

To produce a map the curved reference surface of the Earth, approximated by an ellipsoid

or a sphere, is transformed to the flat plane of the map by means of a map projection. In

other words, each point on the reference surface of the Earth with geographic coordinates

( , ) may be transformed to set of cartesian coordinates ( x,y ) representing positions on thmap plane.


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Map projection principleSeveral hundreds of map projections have been described, but only a smaller part isactually used. Most commonly used map projections are:

Universal Transverse Mercator (UTM), Transverse Mercator (also known as Gauss-Kruger), Polyconic, Lambert Confomal Conic,

Stereographic projection.

Map projections are commonly classified according to the geometric surface from whichthey are derived: plane, cylinder or cone. The three classes of map projections are resp.azimuthal, cylindrical or conical:

Azimuthal Cylindrical Conical

The three classes of map projections


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Furthermore map projections are often described by means of their property: equivalent,

equidistant or conformal. On a conformal map projection angles and shapes are correct

represented. An equivalent map projection represents areas correctly, and an equidistant

projection correctly represents distances in certain directions.

1.3 Spatial reference systems

Most countries have defined their own local spatial reference system. We speak of a spatia

reference system if in addition to the selected reference surface (horizontal datum) and th

choosen map projection, the origin and axes of the map coordinate system have been

defined. The figures below show the Dutch system. It is called the "Rijks-Driehoeks" system

The system is based on the azimuthal stereographic projection, centred in the middle of th

country. The Bessel ellipsoid is used as reference surface. The origin of the coordinate

system has been shifted from the projection centre towards the south-west.

Obliques azimuthal stereographicprojection


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The Dutch National coordinate system.

The origin of the map coordinate system is

souteast of Paris, France

Fragment of the Dutch topo mapshowing the RD-coordinates

European Map Projections

European Reference Systems

Note that global spatial reference systems to measure the earth-as-a-whole - with the aid

of satellites- are becoming more in use. However, a re-adjustment of all existing local

spatial reference systems is not to be expected very soon. For the time being they retain

their practical importance for national mapping activities of many countries.

2.Coordinate Systems

Maps, whether analog or digital, and spatial data, whether in vector or raster format, arerelated to some location. We mostly refer to these locations using coordinate systems. Acoordinate system is a set of rules that specifies how coordinates are assigned tolocations.
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Three-dimensional spatial coordinate systems are used to locate data on the surface of

the Earth. For instance, any point on the earth can be located by means ofspatial

geographic coordinates( , , h ) or geocentric coordinates(x,y,z).


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Spatial geographic coordinates( , h )

Spatial cartesian or geocentriccoordinates ( x, y, z )


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The geographical coordinate system

The latitude of a point P (see figure below) is the angle between the ellipsoidal normal

through P' and the equatorial plane. Latitude is zero on the equator ( = 0 0) and increases

towards the two poles to maximum values of = +90 (N 90 0) at the North Pole and = -

90o (S 900) at the South Pole.

The longitude is the angle between the meridian ellipse which passes through Greenwic

and the meridian ellipse containing the point in question. It is measured in the equatorial

plane from the meridian of Greenwich = 0 0 either eastwards through = + 180o (E 180

or westwards through = -180 0 (W 1800).

Latitude and longitude representing the geographic coordinates , of a point P with

respect to the selected reference surface. They are always given in angular units (e.g. City

hall Enschede: f = 520 13' 13.5" N, = 6 0 53' 50.8" E).

Spatial geographic coordinates ( , , h) are obtained by introducing the ellipsoidal

height h to the system. The ellipsoidal height of a point is the vertical distance of the point

in question above the ellipsoid. It is measured in distance units along the ellipsoidal norma

from the point to the ellipsoid surface. The concept can also be applied to a sphere as the

reference surface.


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The spatial geographical coordinate system

2.1.2 Geocentric Coordinates (X,Y,Z)

An alternative and often more convenient method of defining a position is with spatial

cartesian coordinates. The system has its origin at the mass-center of the earth with the x

and y axes in the plane of the equator. The x-axis passes through the meridian of

Greenwich, and the z-axis coincides with the earth's axis of rotation. The three axes are

mutually orthogonal and form a right-handed system.

The spatial geocentric coordinate system

It should be noted that the rotational (spin) axis of the earth changes its position with the

time (catchword: polar motion). Due to this the mean position of the pole in the year 1903


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(based on observations between 1900 and 1905) was used to define the so-called

"Conventional International Origin" (CIO).

2.2 Plane Coordinate Systems

A flat map has only two dimension width (left to right) and length (bottom to top).

Transforming the three dimensional earth body into a two-dimensional map is subject of

map projections. Here, like in several other cartographic applications, two-dimensional

coordinates are needed to describe the location of any point in an unambiguous and uniqu

manner.

2.2.1 Cartesian Coordinates

One possibility of defining a point in a plane is to use plane rectangular coordinates. This is

a system of intersecting perpendicular lines, which contains two principal axes, called the

and Y--axis. The horizontal axis is usually referred to as the X-axis and the vertical the Y-ax

(Note that the X-axis is sometimes called Easting and the Y-axis Northing). The intersection

of the X- and Y-axis forms the origin. The plane is marked at intervals by equally spaced

coordinate lines.

The 2D cartesian coordinate system

Giving its two numerical coordinates Xp and Yp, one can precisely and objectively specify

any location P on the map. Normally, the coordinates Xp= 0 and Yp = 0 are given to the

origin. However, sometimes large positive values are added to the origin coordinates. This

to avoid negative values for the X - and Y -coordinates in case the origin of the coordinate

system is located inside the area of interest. The point which has then the coordinates Xp=


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0 and Yp= 0 is called the false origin. Rectangular coordinates are also called cartesian

coordinates after Descartes, a French mathematician of the seventeenth century.

2.2.2 Polar Coordinates

Another possibility of defining a point in a plane is by polar coordinates. This is the distanc

d from the origin to the point concerned and the angle a between a fixed (or zero) directio

and the direction to the point.

The 2D polar coordinate system

The angle a is called azimuth or bearing and is counted clockwise. It is given in angular un

while the distance d is expressed in length units. Bearings are always related to a fixed

direction (initial bearing) or a datum line. In principle, this reference line can be chosen

freely. However, in practice three different directions are widely in use: True North, Grid

North and Magnetic North. The corresponding bearings are called: true bearing or geodetic

bearing, grid bearing and magnetic or compass bearing.

Polar coordinates are often used in land surveying. For some types of surveying instrumen

it is advantageous to make use of this coordinate system. Especially the development of

precise remote distance measurement techniques has led to the virtually universal

preference for the polar coordinate method in detail survey.

2.3 Map grid and graticule

The grid represents lines having constant rectangular coordinates (x, y). The grid is almos

always a rectangular system and is used on large and medium scale maps to enable

detailed calculations and positioning. Plane coordinates and therefor the grid, are usually


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not used on small-scale maps, maps smaller than one to a million. The scale distortions tha

result from transforming the curved Earth surface to the map plane are so great on small-

scale maps that detailed calculations and positioning are difficult.

The grid and graticule of the Dutch National coordinate system (at small scale)

The graticule represents the projected position of the geographic coordinates at constant

intervals, or in other words the projected position of selected meridians and parallels. The

shape of the graticule depends largely on the characteristics and scale of the map

projection used.


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The world mapped in the Transverse Mercator projection with a 15 degrees graticule

The grid and graticule spacings on a map vary depending on the scale of the map. E.g. on

the 1: 50 000 topographic map of the Netherlands graticule lines or ticks are shown at eve

5 minutes and grid lines at every kilometer.

The map sheet limit or neat line (the line enclosing the mapped area) can either be forme

by the outline of the graticule or the grid. The grid as outline of the map has the advantag

of being rectangular, hence the map face of each map sheet will be exactly the same size.

The graticule as outline of the map might give a curved outline, but shows immediately the

extent of the map sheet in the geographical system

3. Reference surfaces for mapping

3.1 The figure of the Earth

The earth's surface is anything but uniform. Only a part of it, the oceans, can be treated as

reasonably uniform. But the surface or topography of the land masses show large verticalvariations between mountains and valleys which make it impossible to approximate theshape of the earth with any reasonably simple mathematical model.

We can simplify matters by the idealization of expanding the oceans below the landmasseand make the assumption that the water can flow freely also there. If we then neglect tidaand current effects on this "global ocean", the resultant water surface is remaining affecteonly by gravity. This has a certain consequence on the shape of this surface because thedirection of gravity - more commonly known as plumb line - is dependent on the mass


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distribution inside the earth. Due to irregularities or mass anomalies in this distribution the"global ocean" is forced to be an undulated surface. This surface is called the geoid or the"physical figure of the earth". The plumb line through any surface point is alwaysperpendicular to it.

Perspective view of the Geoid (Geoid

undulations 15000:1)

If the earth was of uniform density and the earth's topography didn't exist, the geoid would

have the shape of an oblate ellipsoid centered on the earth's center of mass. Unfortunately

the situation is not this simple. Where a mass deficiency exists, the geoid will dip below th

mean ellipsoid. Conversely, where a mass surplus exists, the geoid will rise above the mea

ellipsoid. These influences cause the geoid to deviate from a mean ellipsoidal shape by up

to +/- 100 meters (see figure). The deviation between the geoid and an ellipsoid is called

the Geoid undulation (N).

Relationships between the earth's surface, the geoid and a reference ellipsoid


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Fragment of the Dutch topo map showing the border ofBelgium and the Netherlands. The Mean Sea Level of

Belgium differ -2.34m from the MSL of The Netherlands.As a result , contour lines are abruptly ending at the

border.

The same care must be taken when using GPS measurements. GPS measurements are

taken relative to the WGS84 ellipsoid. GPS heights have to be adjusted before they can be

compared to heights given on topographic maps, which are related to a MSL point.

3.3 Approximations of the Earth's figureThe curvature of the geoid displays discontinuities at abrupt density variations inside theearth. Consequently, the geoid is not an analytic surface and it is thereby not suitable as a

reference surface for the determination of locations. If we are to carry out computations ofpositions, distances, directions, etc. on the earth's surface, we need to have somemathematical reference frame. The most convenient geometric reference is the oblateellipsoid as it provides a relatively simple figure which fits the geoid to a first orderapproximation. For small scale mapping purposes we can also use the sphere which fits thgeoid to a second order approximation.

3.3.1 The Ellipsoid

An ellipsoid is formed when an ellipse is rotated about its minor axis. This ellipse which

defines an ellipsoid or spheroid is called a meridian ellipse (Note that ellipsoid and spheroid

are being treated as equivalent and interchangeable words).


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A cross section of an ellipsoid, used to represent the Earth's surface,indicating what ismajor and minor axis radius

The shape of an ellipsoid may be defined in a number of ways, but in geodetic practice the

definition is usually by its semi-major axis and flattening. Flattening fis dependent on boththe semi-major axis a and the semi-minor axis b.

f = (a - b) / a

The ellipsoid may also be defined by its semi-major axis b and eccentricity e, which is give

by:

Given one axis and any one of the other three parameters, the other two can be derived.

Typical values of the parameters for an ellipsoid are:

a = 6378135.00m b = 6356750.52m

f= 1/298.26 e = 0.08181881066

3.3.2 The Sphere

As can be seen from the dimensions of the earth ellipsoid, the semi-major axis a and the

semi-minor axis b differ only by a bit more than 21 km. A better impression on the earth's

dimensions may be achieved if we refer to a more "human scale". Considering a sphere of

approximately 6 m in diameter then the ellipsoid is derived by compressing the sphere at

each pole by 1 cm only. This compression is rather small compared to the dimension of the

semi-major axis a.


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The ellipsoid and the sphere, a comparison

The consequence is that instead of using the ellipsoid, the sphere might be sufficient for

certain mapping tasks.

The sphere as reference surface for small-scale mapping

In practice 1:5,000,000 is recommended as the largest scale at which the spherical

approximation can be made.

3.4 The Ellipsoid as Reference surface for Locations

3.4.1 Local Reference Ellipsoids

It is important to realize that topographic maps are drawn and geodetic positions aredefined with respect to a horizontal datum (also referred to as geodetic datum or reference

datum). A horizontal (or geodetic) datum is defined by the size and shape of an ellipsoid as

well as several known positions on the physical surface at which latitude and longitude

measured on that ellipsoid are known to fix the position of the ellipsoid. In the United State

we use the North American Datum, in Japan the Tokyo Datum, in some European countries

the European Datum, in Germany the Potsdam Datum, etc.


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Horizontal datums have been established to fit the geoid well over the area of local interes

which in the past was never larger than a continent. As a consequence, the differences

between the geoid and the reference ellipsoid may be ignored. This allows accurate maps

be drawn in the vicinity of the datum. The figure below shows that a position on the geoid

will have a different set of latitude and longitude coordinates in each reference datum. In

this figure the North American datum is extrapolated to Europe. Even though the datum fit

the geoid in the North American continent well, it does not fit the European geoid.

Conversely, if the European datum is extrapolated to the North American continent, the

similar result is found.

The geoid and two best fitting local ellipsoids for a chosen region

Widely in use are the following ellipsoids generally named after their generator:


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Horizontal datums are defined by the size and shape of an ellipsoid, as well as its positionand orientation. There are a few hundred of these local horizontal datums defined in the

world. The table below shows some examples of local datums , which use the same ellipso

(Clarke 1866 or Hayford), but in different positions (referred as datum shifts).

Examples of reference datums, with its reference ellipsoid and datum shift values
http://kartoweb.itc.nl/geometrics/Reference%20surfaces/DATUM.txthttp://kartoweb.itc.nl/geometrics/Reference%20surfaces/DATUM.txt


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Two reference ellipsoids in different position.

3.4.2 Global Reference Ellipsoids

With increasing demands for global surveying activities are going on to establish also glob

reference ellipsoids. Especially the International Union for Geodesy and Geophysics (IUGG)

is involved in establishing those reference figures. The motivation is to make geodetic

results mutually comparable and to provide coherent results also to other disciplines like

astronomy and geophysics.

The geoid and a globally best fitting ellipsoid

In 1924 in Madrid, the general assembly of the IUGG introduced the ellipsoid determined b

Hayford in 1909 as the International Ellipsoid. In contrary to local reference ellipsoids, whic

apply only to a region or local area of the earth's surface, global reference systems

approximating the geoid as a mean earth ellipsoid. However, according to present

knowledge, the values for this earth model only give an insufficient approximation. At the


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general assembly 1967 of the IUGG in Luzern, the 1924 reference system was replaced by

the Geodetic Reference System 1967 (GRS 1967). It represents a good approximation (as o

1967) to the mean earth figure. The geometric ellipsoidal parameters a, b and f are given

the table below. The Geodetic Reference System 1967 has found application especially in

the planning of new geodetic surveys.

At its general assembly 1979 in Canberra the IUGG recognized that the Geodetic Referenc

System 1967 no longer represents the size and shape of the earth to an adequate accurac

Consequently, it was replaced by the Geodetic Reference System 1980 (GRS 1980 see

table). The World Geodetic System 1984 (WGS84) is based on the GRS 1980 and provides

the basic reference frame for GPS (Global Positioning System) measurements.

Note Nowadays, geodesists are able to measure the Earth-as-a-whole with the aid of

artificial satellites. However, a re-adjustment of all existing local geodetic survey reference

systems is not to be expected for the time being due to the great efforts in applying

coordinate transformation and changing existing maps. For the time being, local reference

systems retain their practical importance for national mapping activities.3.5 Relationships between reference surfaces

In summary, when speaking of the size and shape of the earth and positions on it, there ar

three surfaces to be considered:

The topography - the physical surface of the earth. The Geoid - the level surface (also a physical reality). The Ellipsoid - the mathematical surface for computations.

Surveying observations are made on the earth surface relative to the geoid. Before using

observations in geodetic computations, they must be corrected for locational differences

between the geoid and the reference ellipsoid. These corrections are small and may for

some purposes be ignored if a reference ellipsoid is chosen so as to closely fit the geoid in

the area of concern.


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Mean Sea Level (MSL) points, an approximation to the geoid, are used as reference surface

for height measurements ( orthometric heights).

The earth surface, and two reference surfaces, the geoid and a reference ellipsoid.Orthometric heights are measured from a Mean Sea Level point, an approximation to the

geoid.Ellipsoidal heights have to be adjusted before they can be compared to the orthometricheights given on topographic maps.The deviation between the geoid and an reference

ellipsoid is called Geoid undulation (N). Geoid undulations can be used to adjust theellipsoidal heights (H = h +/- N).

........

Ellipsoidal height h above the reference ellipsoid and the orthometric height H above the

Geoid for two points on the earth surface. The ellipsoidal height is measured orthogonal t

the ellipsoid. The orthometric height is measured orthogonal to the geoid.

4. Map projections

4.1 What is a Map Projection


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To produce a map of the world in a convenient way we make use of map projections. A maprojection is any transformation between the curved reference surface of the earth and theflat plane of the map.

We can as well define a map projection as a set of equations which allows us to transform set of Ellipsoidal Geographic Coordinates ( representing positions on the reference surfac of the earth to a set of Cartesian Coordinates (x, y) representing positions on the two-

dimensional surface of the map (see figure above) .

For each map projection the following equations are available:

X,Y = f ( , ) Forwardequation

, = f ( X,Y ) Inverseequation


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ActivityA point P is located on the Stereographic projection at 60o N and 130o E. The

sphere is taken as the reference surface of the earth. Use the equations given above to

obtain the Cartesian coordinates for point P. The origin of the coordinate system is located

on the North Pole (Radius (R) = 6371000 m, Central Meridian ( o) = 0o, equal to the

Greenwich meridian).

4.2 Scale distortions on a Map

The transformation from the curved reference surface of the earth to the flat plane of the

map is never completely successful. Look at the diagram below. By flattening the curved

surface of the sphere onto the map the curved surface is stretched in a non-uniform

manner.

It appears that it is impossible to project the Earth on a flat piece of paper without any

locational distortions, therefore without any scale distortions.

Projection plane tangent to

the reference surface

The distortions increase as the distance from the central point of the projection increases.

Placing the map plane so that it intersects the reference surface will reduce and mean out

the scale errors.


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Projection plane secant tothe reference surface

Since no map projection maintains correct scale throughout the map, it may be important

know the extent to which the scale varies on a map.

On a world map, the distortions are evident where landmasses are wrongly sized or out of

shape and the meridians and parallels do not intersect at right angles or are not spaced

uniformly. Some maps have a scale reduction diagram, which indicates the map scale at

different locations, helping the map-reader to become aware of the distortions.

On maps at larger scales, maps of countries or even city maps, the distortions are not

evident to the eye. However, the map user should be aware of the distortions if he or she

computes distances, areas or angles on the basis of measurements taken from these maps

Scale distortions can be measured and shown on a map by ellipses of distortion. The ellips

of distortion, which is also known as Tissot's Indicatrix, shows the shape of an infinitesimal

small circle with a fixed scale on the earth as it appears when plotted on the map. Every

circle is plotted as circle or an ellipse or, in extreme cases, as a straight line.

The size and shape of the ellipse shows how much the scale is changed and in what

direction. On map projections where all indicatrices remain circles, but the sizes change, th

scale change is the same in all directions at each location. These conformal projections

represent angles correctly and have no local shape distortion ( e.g. the Mercator

projection ).

The indicatrices on the diagram below are circles along the equator. There are no scale

distortions along the equator. The indicatrices elsewhere are ellipses with varying degreesof flattening. The projection represents areas correctly - all ellipses have the same area - b

angles and, consequently, shapes are not represented correctly.


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An equivalent map projection, also known as an equal-area map projection, correctly

represents areas sizes of the sphere on the map. When this type of projection is used for

small-scale maps showing large regions, the distortion of angles and shapes is considerabl

The Lambert cylindrical equal-area projection is an example of an equivalent map

projection.

The Lambert cylindrical equal-area projection as an example of an equivalent, cylindrical

projection

An equidistant map projection correctly represents distances. An equidistant map

projection is possible only in a limited sense. That is, distances can be shown at the nomin

map scale -the given map scale- only from one or two points to any other point on the map

or in certain directions. If the scale on a map is correct along all meridians, the map is

equidistant along the meridians (e.g. the Plate Carree projection). If the scale on a map is

correct along all parallels, the map is equidistant along the parallels.

The Plate Carree projection as an example of an equidistant, cylindrical projection


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A conformal map projection represents angles and shapes correctly at infinitely small

locations. Shapes and angles are only slightly distorted, as the region becomes larger. At

any point the scale is the same in every direction. On a conformal map projection meridian

and parallels intersect at right angles (e.g. Mercator projection).

The Mercator as an example of a conformal, cylindrical projection

NoteA map projection may possess one of the three properties, but can never have all

three properties. It can be proved that conformality and equivalence are mutually exclusiv

of each other and that a projection can only be equidistant (true to scale) in certain places

or directions.

There are map projections with rather special properties:

On a minimum-errormap projection the scale errors everywhere on the map as a whole ar

a minimum value (e.g. the Airy projection ).On the Mercator projection, all rumb lines, or lines of constant direction, are shown as

straight lines. A compass course or a compass bearing plotted on to a Mercator projection

a straight line, even though the shortest distance between two points on a Mercator

projection - the great circle path - is not a straight line.


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all rumb lines, or lines of constant direction, are shown as straight lines.

On the Gnomonic projection, all great circle paths - the shortest routes between points on

sphere - are shown as straight lines.

all great circles - the shortest routes between points on a sphere - are shown as straight

lines

4.4 The classification of Map Projections

Next to their property (equivalence, equidistance, conformality), map projections can be

discribed in terms of their class (azimuthal, cylindrical, conical) and aspect (normal,

transverse, oblique).

The three classes of map projections are cylindrical, conical and azimuthal.The earth'ssurface projected on a map wrapped around the globe as a cylinder produces the

cylindrical map projection. Projected on a map formed into a cone gives a conical map

projection. When projected on a planar map it produces an azimuthal orzenithal map

projections.


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The three classes of map projections

Projections can also be described in terms of their aspect: the direction of the projection

plane's orientation (whether cylinder, plane or cone) with respect to the globe. The three

possible apects of a map projection are normal, transverse and oblique. In a normal

projection, the main orientation of the projection surface is parallel to the earth's axis (as in

the second figure below). A transverse projection has its main orientation perpendicular to

the earth's axis. Oblique projections are all other, non-parallel and non-perpendicular, case

The figure below provides two examples.

A transverse cylindrical and an oblique conical map projection. Both are tangent to the

reference surface

The termspolar, oblique and equatorial are also used. In a polar azimuthal projection the

projection surface is tangent or secant at the pole. In a equatorial azimuthal or equatorial

cylindrical projection, the projection surface is tangent or secant at the equator. In anoblique projection the projection surface is tangent or secant anywhere else.

A map projection can be tangent to the globe, meaning that it is positioned so that the

projection surface just touches the globe. Alternatively, it can be secant to the globe,

meaning that the projection surface intersects the globe. The figure below provides

illustrations.


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Three normal secant projections: cylindrical, conical and azimuthal

A final descriptor may be the name of the inventor of the projection, such as Mercator,

Lambert, Robinson, Cassini etc., but these names are not very helpful because sometimes

one person invented several projections, or several people have invented the same

projection. For example J.H.Lambert described half a dozen projections. Any of these might

be called 'Lambert's projection', but each need additional description to be recognized.

It is now possible to describe a certain projection as, for example,

Polar stereographic azimuthal projection with secant projection plane Lambert conformal conic projection with two standard parallels Lambert cylindrical equal-area projection with equidistant equator Transverse Mercator projection with secant projection plane.

The question may arise here 'Why are there so many map projections?'.The main reason i

that there is no one projection best overall (see section 4.5 selecting a suitable mapprojection )

ActivityThe diagram below shows the developable surface of the Lambert conformal coni

projection with two standard parallels.


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Answer the following questions:

1. Which developable surface is used?

2. Is it a tangential or a secant projection?

3. What is the position of the developable surface?

4. Describe some of the scale distortion characteristics.

5. Are areas correctly represented?

4.5 Selecting a suitable Map ProjectionEvery map must begin, either consciously or unconsciously, with the choice of a map

projection and its parameters. The cartographer's task is to ensure that the right type ofprojection is used for any particular map. A well choosen map projection takes care thatscale distortions remain within certain limits and that map properties match to the purposeof the map.

The selection of a map projection has to be made on the basis of:

shape and size of the area position of the area purpose of the map

The choice of the classof a map projection should be made on the basis of the shape andsize of the geographical area to be mapped. Ideally, the general shape of a geographicarea should match with the distortion pattern of a specific projection. For example, if anarea is small and approximately circular it is possible to create a map that minimisesdistortion for that area on the basis of anAzimuthal projection. The Cylindrical projectionshould be the basis for a large rectangular area and a Conic projection for a triangular area


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Theposition of the geographical area determines the aspect of a projection. Optimal

when the projection centre coincides with centre of the area, or when the projection plane

located along the main axis of the area to be mapped (see example figure below).

Choice of position and orientation of the projection plane for a map of Alaska

Once the class and aspectof a map projection have been selected, the choice of the

property of a map projection has to be made on the basis of thepurposeof the map.

In the 15th, 16th and 17th centuries, during the time of great transoceanic voyaging, there

was a need for conformal navigation charts. Mercator's projection -conformal cylindrical-

met a real need, and is still in use today when a simple,straight course is needed for

navigation.

Because conformal projections show angles correctly, they are suitable for sea, air, and

meteorological charts. This is useful for displaying the flow of oceanic or atmospheric

currents, for instance.

For topographic and large-scale maps, conformality and equidistance are important

properties. The equidistant property, possible only in a limited sense, however, can be

improved by using secant projection planes.


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The Universal Transverse Mercator (UTM) projection is a conformal cylindrical projection

using a secant cylinder so it meets conformality and reasonable equidistance for

topographic mapping.

Other projections currently used for topographic and large-scale maps are the Transverse

Mercator ( the countries of . Argentina, Colombia, Australia, Ghana, S-Africa, Egypt use it )

and the Lambert Conformal Conic (in use for France , Spain, Morocco, Algeria ). Also in use

are the stereographic (the Netherlands ) and even non-conformal projections such as

Cassini or the Polyconic (India).

Suitable equal-area projections for distribution maps include those developed by Lambert,

whether azimuthal, cylindrical, or conical. These do, however, have rather noticeable shap

distortions. A better projection is the Albers equal-area conic projection, which is nearly

conformal. In the polar aspect, they are excellent for mid-latitude distribution maps and do

not contain the noticeable distortions of the Lambert projections.

An equidistant map, in which the scale is correct along a certain direction, is seldom

desired. However, an equidistant map is a useful compromise between the conformal and

equal-area maps. Shape and area distortions are moderate.

The projection which best fits a given country is always the minimum-error projection of th

selected class. The use of minimum-error projections is however exceptional. Their

mathematical theory is difficult and the equidistant projections of the same class will

provide a very similar map.

In conclusion, the ideal map projection for any country would either be an azimuthal,

cylindrical, or conic projection, depending on the shape of the area, with a secant projectio

plane located along the main axis of the country or the area of interest. The selected

property of the map projection depends on the map purpose.

Nevertheless for each country to use its own projection would make international co-

operation in data exchange difficult. There are strong arguments in favour of using an

international standard projection for mapping.

ActivityYou have been asked to produce a small-scale thematic map of your country

showing the distribution of the population. Which projection class, aspect and property

would you choose considering the location, size and shape of the country and the purpose

of the map? Justify your answer!


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4.6 Map Projections in common useSeveral hundreds of map projections have been described, but only a smaller part isactually used. Most commonly used map projections are:

Universal Transverse Mercator (UTM), Transverse Mercator (also known as Gauss-Kruger), Polyconic, Lambert Confomal Conic, Stereographic projection.

These projections and a few other well-known map projections are briefly described and

illustrated.

4.6.1 Cylindrical projections

Mercator projection The Mercator projection is a conformal cylindrical projection. Paralle

and meridians are straight lines intersecting at right angles, a requirement for conformality

Meridians are equally spaced. The parallel spacing increases with distance from the Equato

Mercator: conformal cylindrical projection

The ellipses of distortion appear as circles (indicating conformality) but increase in size

away from the equator (indicating area distortion). This exaggeration of area as latitude


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increases makes Greenland appear to be as large as South America when, in fact, it is only

quarter of the size.

The Mercator projection is used for long distance navigation because of the straight rhumb

lines. It is more convenient to steer a rumb-line course if the extra distance travelled is

small. Often and inappropriately used as a world map in atlases and for wall charts. It

presents a misleading view of the world because of excessive area distortion towards the

poles.

Transverse Mercator projection The Transverse Mercator projection is a transverse

cylindrical conformal projection.

The Transverse Mercator projection is based on a transverse cylinder

Versions of the Transverse Mercator Projection are used in many countries as national

projection on which the topographic mapping is based. The Transverse Mercator projection

is also known as the Gauss-Kruger or Gauss Conformal projection. The figure below shows

the World map in Transverse Mercator projection.


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The world mapped in the Transverse Mercator projection (at a small scale)

The Transverse Mercator is the basis for the Universal Transverse Mercator projection, as

well as for the State Plane Coordinate System in some of the states of the U.S.A.

Universal Transverse Mercator (UTM) The UTM projection is a projection accepted

worldwide-accepted for topographic mapping purposes. It is a version of the Transverse

Mercator projection, but one with a transverse secant cylinder.

The UTM is a secant, cylindrical projection in a transverse position

The UTM projection is designed to cover the world, excluding the Arctic and Antarctic

regions. To keep scale distortions within acceptable limits, 60 narrow, longitudinal zones of

six degrees longitude in width are defined and numbered from 1 to 60. The figure below


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shows the UTM zone numbering system. Shaded in the figure is UTM grid zone 3 N which

covers the area 168o - 162o W (zone number 3), and 0o - 8o N (letter N of the latitudinal

belt).

The UTM zone numbering system (click to enlarge)

Each zone has it's own central meridian. Along each central meridian, the scale is 0.9996.

The central meridian is always given an Easting value of 500,000 m; to avoid negative

coordinates sometimes large values are added to the origin coordinates, called false

coorinates. For positions north of the equator, the equator is given a Northing value of 0m

For positions south of the equator, the equator is given a (false) Northing value of

10,000,000 m.
http://kartoweb.itc.nl/geometrics/Bitmaps/utm%20zones1.gifhttp://kartoweb.itc.nl/geometrics/Bitmaps/utm%20zones1.gifhttp://kartoweb.itc.nl/geometrics/Bitmaps/utm%20zones1.gif


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2 adjacent UTM-zones of 6 degrees longitude

Other Cylindrical projections Pseudo-cylindrical projections are projections in which the

parallels are represented by parallel straight lines, and the meridians by curves. Examples

are the Sinusoidal, Eckert, Winkel, Mollweide, DeNoyer and the Robinson projection.

The Mollweide projection as an example of a pseudo-cylindrical projection

The Robinson projection is neither conformal nor equal-area and no point is free of

distortion, but the distortions are very low within about 45o of the center and along the

Equator and therefore recommended and frequently used for thematic world maps. The


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projection provides a more realistic view of the world than rectangular maps such as the

Mercator.

The Robinson projection as an example of a pseudo-cylindrical projection

4.6.2 Conic projections

Three well-known conical projections are the Lambert Conformal Conic projection, the

Albers equal-area projection and the Polyconic projection.

The Lambert Conformal Conic projection in normal position is an example of a conic

projection


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Polyconic projection The Polyconic projection is neither conformal nor equal-area. The

polyconic projection is projected onto cones tangent to each parallel, so the meridians are

curved, not straight.

The polyconic projection is an example of a conic projection, equidistant along the paralle

The scale is true along the central meridian and along each parallel. The distortion increas

away from the central meridian in East or West direction.

The polyconic projection is used for early large-scale mapping of the United States until the

1950's, early coastal charts by the U.S. Coast and Geodetic Survey, early maps in the

International Map of the World (1:1,000,000 scale) series and for topographic mapping in

some countries.

4.6.3 Azimuthal projections

The five common azimuthal (also known as Zenithal) projections are the Stereographic

projection, the Orthographic projection, the Lambert azimuthal equal-area projection, the

Gnomonic projection and the azimuthal equidistant(also called Postel ) projection.

For the Gnomonic projection, the perspective point (like a source of light rays), is the centr

of the Earth. For the Stereographic this point is the opposite pole to the point of tangency,

and for the Orthographic the perspective point is an infinite point in space on the opposite

side of the Earth.


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The projection principle for the Gnomonic, Stereographic and Orthographic projection

Stereographic projection The Sterographic projection is a conformal azimuthal projectio

All meridians and parallels are shown as circular arcs or straight lines. Since the projection

conformal, parallels and meridians intersect at right angles.

In the polar aspect the meridians are equally spaced straight lines, the parallels are

unequally spaced circles centered at the pole. Spacing gradually increases away from the

pole.


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The transverse (or equatorial) stereographic projection is an example of a conformal

azimuthal projection

The scale is contant along any circle having its centre at the projection centre, but scale

increases moderately with distance from the centre. The areas increase with distance from

the projection center. The ellipses of distortion remain circles (indicating conformality).The Stereographic projection is commonly used in the polar aspect for topographic maps o

polar regions. Recommended for conformal mapping of regions approximately circular in

shape (e.g. The Netherlands)

Gnomonic projection The Gnomonic (also known as central azimuthal) projection is

neither conformal nor equal-area. The scale increases rapidly with the distance from the

center. Area, shape, distance and direction distortions are extreme, but all great circles - t

shortest routes between points on a sphere - are shown as straight lines.

all great circles - the shortest routes between points on a sphere -

are shown as straight lines on the Gnomonic projection

In combination with the Mercator map where all lines of constant direction, are shown as

straight lines it assist navigators and aviators to determine appropriate courses. Since scal

distortions are extreme the projection should not be used for regular geographic maps or f

distance measurements.

4.6.4 Other map projections


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The table below gives an overview of other commonly used map projections.


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5. Coordinate transformations

5.0 Introduction

Coordinate transformations are used to bring spatial data into a common reference system

For instance spatial data that are related to the Lambert Conformal Conical projection

system may need to be transformed to UTM coordinates if the UTM projection is the

common reference system used.

5.1 Projection Change

Spatial data with co-ordinates of a known projection are normally transformed from one

projection co-ordinate system to another using the forward and inverse projection

equations.

The inverse equations of the source projection are used to transform the source projection

co-ordinates (projection A or system A ) to geographic latitude and longitude co-ordinates.


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The forward equations of the target projection are used to transform the geographic co-

ordinates to target projection co-ordinates (projection B or system B).

Refer to ' Map projections used by the U.S. Geological Survey, John P. Snyder' for acomplete overview of projection equations.

5.2 Datum transformations

Spatial data can have co-ordinates with different underlying ellipsoids or the underlying

ellipsoids have different datums. The latter means that, apart from different ellipsoids, the

centres or the rotation axes of the ellipsoids do not coincide. To relate these data one may

need a so-called datum transformation. For example, spatial data that are related to the

European 1950 (ED 50) datum may need to be transformed to the datum underlying theDutch RD system (this implies the Bessel 1841 ellipsoid).

In such a case the projection transformation must be combined with a datum transformatio

step in between as is illustrated in the figure below. The inverse equations take us from

some projection (System A) into geographic co-ordinates. Then we apply a datum

transformation (from Datum A to Datum B), and finally move into another map projection

(System B).


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5.2.1 Datum transformation via geocentric coordinates


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Mathematically a datum transformation is feasible via 3 dimensional geocentric co-

ordinates, implying a 3D similarity transformation defined by 7 parameters: 3 shifts, 3

rotations and a scale difference. This transformation is combined with transformations

between the geocentric co-ordinates and ellipsoidal latitude and longitude co-ordinates in

both datum systems.

The transformation from the latitude and longitude co-ordinates into the geocentric co-

ordinates is rather straightforward and turns ellipsoidal latitude ( ), longitude ( ) and

height (h) into X,Y and Z, using 3 direct equations that contain the ellipsoidal parameters a

and e.

The inverse equations are more complicated and require either an iterative calculation of

the latitude and ellipsoidal height, or it makes use of approximating equations like those o

Bowring. These last have millimetre precision for 'earth-bound' points, i.e. points that are a

most 10 km away from the ellipsoidal surface (this is the case for all topographic points).

5.2.2 Datum transformation via geographic coordinates

However a good approximation of this datum transformation make use of the Molodensky

and the regression equations, relating directly the ellipsoidal latitude and longitude, and

case of Molodensky also the height, of both datum systems.


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A. Molodensky equations The standard Molodensky equations relate ellipsoidal latitude

and longitude co-ordinates and ellipsoidal height of a local geodetic datum to those of the

WGS84 datum (NIMA report, 1997).

Molodensky formula:

( = geodetic latitude in local system; = geodetic longitude in local system; h = the distance of a point above or below the localellipsoid measured along the ellipsoid normal through the point; a= semi-major axis of local ellipsoid; f = flattening of the localellipsoid; X Y, Z = shifts between the centers of the local geodetic system and the WGS84 ellipsoid; a f = differences between the semi-major axis and the flattening of the local andWGS84 system; all quantities are obtained by subtracting local geodetic system ellipsoid values from WGS84 ellipsoid values )

Simplified:


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Molodensky.exe (program that finds X, Y and Z, given a, f and 3 points (lat, lon, height) in both the

WGS84 and the local system )

Molodensky(generalised).exe (program that ....)

Molodensky(inverse).exe (program that .....)

B. Regression equations Themultiple regression equations relate ellipsoidal latitude an

longitude co-ordinates of continental size datums to those of the WGS84 datum and involv

polynomial expressions in the two ellipsoidal co-ordinates which go up to degree 9 for the

time being. The coefficients (transformation parameters) are determined on the basis of

coordinate differences of a set of selected points whose coordinates are known in both

datum systems. The main advantage of this method over Molodensky formula (implementin most geo-software) is that better fits over continental size land areas can be achieved.

Regression formula:

Simplified:

Note: to apply a datum transformation, the datum transformation parameters have to be

estimated on the basis of a set of selected points whose co-ordinates are known in both

datum systems. If the coordinates of these points are not correct - this is often the case for

points measured on a local datum system - the estimated parameters may be inaccurate.

As a result the datum transformation will be inaccurate.
http://kartoweb.itc.nl/geometrics/Coordinate%20transformations/Molodensky.exehttp://kartoweb.itc.nl/geometrics/Coordinate%20transformations/Molodensky%20(generalised).exehttp://kartoweb.itc.nl/geometrics/Coordinate%20transformations/Molodensky%20(inverse).exehttp://kartoweb.itc.nl/geometrics/Coordinate%20transformations/Molodensky.exehttp://kartoweb.itc.nl/geometrics/Coordinate%20transformations/Molodensky%20(generalised).exehttp://kartoweb.itc.nl/geometrics/Coordinate%20transformations/Molodensky%20(inverse).exe


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5.3 Application of projection change (incl. datum transformation)

Forward and inverse projection equations - as discussed earlier - are normally used to

transform spatial data from one projection co-ordinate system to another. Some

transformation programs, however, only include the equations that relate to a sphere as

model of the earth. A spherical model can be used at small-scale but for larger scales an

ellipsoid should be chosen. A spherical model assumption may result in unacceptable

differences in co-ordinates.

Projection transformation programs do not always combine a projection change with a

datum transformation. If one neglects the difference in datums there will be no perfect

match between adjacent maps of neighbouring countries or between overlaid maps

originating from different projections. It may result in differences in co-ordinates in the

range of the datum shifts, which can go up to several hundred meters.

To apply the required datum transformation we need the ellipsoidal latitude, longitude and

height in one datum system and the shift and rotation of the ellipsoidal axes of one datum

system with respect to the other. However, datum transformation programs implemented

GIS and Cartographic software often simplify this transformation: i.e. ellipsoidal heights (h)

are taken equal to 0 or the rotation differences of the ellipsoidal axes are ignored.

Illustration of a datum shift and rotation

of one datum system with respect toanother.The centres of the ellipsoids donot coincide and the axes are rotated.

Datum transformations using Molodensky equations (implemented in most geo-software)

are becoming increasingly important, because of the growing use of GPS data. Very often

the data is captured using the WGS84 ellipsoid and datum, and have to be transformed to


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local projection with its own ellipsoid and datum. Moreover, heights measured with GPS

have to be transformed to heights related to the height reference point (vertical datum)

used in a particular country (this implies for the Netherlands the N.A.P.).

List of datums and datum shift values

In the list a datum is defined as: Datum_Name = Ellipsoid_Name, Shift_X, Shift_Y, Shift_Z,

Name_Geographic_Area or as: Datum_Name = Ellipsoid_Name (In this case a subsection

[Datum_Name] provides the shift). The given datum shift values are compared to WGS84 !

5.4 Direct transformations

If the underlying projection of a co-ordinate system is unknown we may relate the co-

ordinate system to a known co-ordinate system on the basis of a set of selected points

whose co-ordinates are known in both systems (given in the figure Overview of Coordinate

Transformations as direct transformations). These points may be ground control points or

common points such as corners of houses or road intersections, as long as they have know

co-ordinates in both systems.

co-ordinate transformation performed on the basis of selected points. Here six points werchosen.

Image and scanned data are usually transformed by this method. The transformations may

be conformal, affine, projective, polynomial or of another type, depending on the geometri

errors in the data set. Linear conformal or affine transformations can be used to rectify

distortions such as a shift (or translation), a rotation or a linear scale difference. Non-linearpolynomial transformations can be used to correct variable scale differences.

Direct transformations are also used to match vector data layers that don't fit exactly by

stretching or rubber sheeting them over the most accurate data layer. Moreover, affine

transformations are used in map digitising for the registration of a paper or scanned map.
http://kartoweb.itc.nl/geometrics/Reference%20surfaces/DATUM.txthttp://kartoweb.itc.nl/geometrics/Reference%20surfaces/DATUM.txt


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A. a linear conformal transformation can be used to apply a Shift(Tx, Ty), a Rotation (

and/or a Scale change (s).

B. a linear affine transformation can be used to apply a Shift(Tx, Ty), a Rotation ( )

and/or a Scale change in Xand Ydirection (Sx, Sy ).


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C. a non-linear polynomial transformations can be used to apply a Shift, a Rotation and

Variable scale change.

Polynomial transformation functions can have an infinite number of terms. At least 6 contr

points (12 coefficients or unknowns : Xo ,a1 - a5 , Yo ,b1- b5 ) are required to solve a simp

second-order polynomial transformation.

When there are more control points, than actually needed for the estimation of the

coefficients (transformation parameters), the Root Mean Squares Error (RMSE) can be

calculated using the Least Squares Adjustment. The RMSE is an indication of the

transformation accuracy.

Polynomial transformations are often applied to correct variable scale differences, as

appearing in uncorrected satellite imagery or aerial photographs.


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Variable scale differences in an image

Note: two-dimensional direct transformations have a different accuracy compared to the

transformations based on projection equations. The latter take into account the earth

curvature. This is especially important in the case of large areas and small scale. However,

if the control points are coplanar and the extent of the area is not too large, the 2D direct

transformation could yield a better model of co-ordinate relations than the presumed set o

projection equations would do.

5.6 Application of direct transformations

Different methods are in use to rectify raw images. Raw images are built up by a

rectangular array of pixels with variable values, but these pixels don't have a correct

geometric position yet. Co-ordinates can be assigned to the uncorrected image as is

illustrated in the figure below, or the other way round, the uncorrected image can be

resampled to match it

co-ordinate assignment: here the image is not resampled.


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with the known co-ordinate system as is illustrated in next figure. Resampling is a process

which for each pixel in the new co-ordinate system, a new pixel value has to be determine

by means of an interpolation from surrounding pixels in the old image.

co-ordinate assignment: here the image is resampled

In our institute, the ITC, a frequently used GIS package is ILWIS, developed in house. A

typical feature with respect to co-ordinate transformations is the possibility in ILWIS to

match vector and raster data by an on-the-fly transformation of the vector data. One can

combine in such a way a raster image with several vector layers in one map window by

dragging the vector data without the need of resampling the raster image. The raster imag

can be a raw satellite image, an oblique or vertical aerial photo or a scanned topographic

map. For a correct matching, one needs a set of reliable control points in the image which

are linked to map co-ordinates in accordance with the method discussed earlier This link is

defined by a geometric correction model, for instance linear equations, projective equation

orthophoto correction equations, etc. After that, one can drag any vector map, even with

another datum, over the non-corrected image.
http://www.itc.nl/personal/ilwisdev/index.htmlhttp://www.itc.nl/personal/ilwisdev/index.html


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Line map (kadaster en topography) overlayed on an oblique aerial photograph (Enschede

1997)

Same line map overlayed on a scanned topographic map from 1995


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Because there is no need to rectify (resample) the image, one can save time and disk spac

in case of combined analysis of raster and vector data. The degradation of the image quali

due to resample errors is also avoided. Once the vector editing or analysis is completed, o

can still decide to rectify the image to the well-defined co-ordinate system of the vector

data.

note: a common frustration for users of spatial data is the loss of co-ordinate and projectio

information in the process of data translation from one software program to another. To

convert spatial data, a data format should be used that embeds co-ordinate and projection

information. Vector data formats, however, often don't include projection information.

Moreover, many raster data formats such as bitmaps (BMP) and most Tagged Image File

Formats (TIFF) don't facilitate co-ordinate information. In recent years an extension of the

Tiff format, called Geo-Tiff, has been developed. Geo-Tiff files contain co-ordinates of at

least two opposite corner pixels and (if applicable) also the parameters of the projection

(central meridian, false origin, ellipsoid, etc.) to which the co-ordinates pertain.

FAQ

This section intends to answer your Questionsconcerning Geometric Aspects of Mapping. Answersare given on a number of commonly asked questions related to Coordinate systems, Referencesurfaces, Map projections, and Coordinate transformations.

FAQ:on Coordinate systems

What is a coordinate system? Coordinate systems as a basic method for georeferencing are

used to locate the position of objects in two or three dimensions into correct relationship with

respect to each other.

What kind of coordinate systems are used in mapping?

Coordinate systems are often classified in spatial coordinate systems: e.g. spatial geographicand geocentric coordinate systems and in plane coordinate systems: e.g. 2D cartesian and polarcoordinate systems.

Generally two types of coordinate systems are given on maps: cartesian coordinates (or X,Ymap projection coordinates) and projected geographic coordinates.

Satellite positioning systems (e.g. GPS) make use of 3-dimensional spatial coordinatesystems to define positions on the earth surface, with reference to a mean reference surface for theearth (e.g. GPS measurements use the WGS84 ellipsoid).

2D Polar coordinates are often used in land surveying. For some types of surveyinginstruments it is advantageous to make use of this coordinate system.


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5. Instructions on the use of the grid: instructions on the use of the grid reference system shouldshow clearly how to give a standard map reference on the sheet. A typical grid reference panel isshown below:

6. Unit of Elevation: the note ELEVATIONS IN METRES or ELEVATIONS IN FEET is to appear in aconspicuous position normally in the lower margin. Wherever possible a conspicuous color should beused. The normal and preferred unit of elevation is the meter.

7. Contour interval: this is to be shown in the lower margin near the graphic scales. It should be in theform: "Contour interval ... metres (or feet)". When necessary the note "Supplementary contours at ...metres (or feet)" is to be added.8. Information on True, Grid and Magnetic North: each map sheet is to contain the informationnecessary to determine the true, grid and magnetic bearings of any line within the sheet. Thisinformation is to be provided in the form of a diagram with explanatory notes.


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The line denoting Grid North should be drawn parallel to the N-S grid lines on the map. The anglesmade with Grid North by the other two lines are usually too small to be shown accurate on thediagram; therefore they may be arbitrarily exaggerated for clarity. This exaggeration should besufficiently large to prevent any confusion with true angles. In all cases the relative positions of thethree lines must represent their true positions.The arcs of the Grid Magnetic Angle and of the Grid Convergence are to be marked on the

diagram.The values of these two angles are to be shown alongside the diagram in the followingforms, for example:

1st of January 1999 GRID MAGNETIC ANGLE for CENTRE OF SHEET 230' (45 Mils)ANNUAL MAGNETIC CHANGE 3' or 1 Mil East

GRID CONVERGENCE at CENTRE OF SHEET 141' (30 Mils)The Grid Convergence is normally given to the nearest minute and half mil. The Grid Magnetic Angleis normally given to the nearest 15 minutes and 5 mils. The Annual Magnetic Change is given to anaccuracy sufficient to allow the accuracy of the Grid Magnetic Angle to be maintained. Where thevalues of these angles vary significantly over the map sheet value for the various portions of thesheet are to be given in tabulated form. When the map includes more than one grid zone separatediagrams are required for each and must be clearly labeled with the grid or grid zone designation towhich they refer (see example below). In regions where large magnetic irregularities occur specialtreatment may be required.

Where space allows the following additional note should be placed next to the diagram:TO CONVERT A MAGNETIC BEARING TO A GRID BEARING ADD (OR SUBTRACT) GRID

MAGNETIC ANGLE

Geometric information given on a German Topographic map at scale 1:25 000 (TK 25)


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Geometric information given on all maps distributed by the NIMA

What is True North, Magnetic North, and Grid North?True North (TN): the direction of themeridian to the North Pole at any point on the map. Magnetic North (MN): the direction of theMagnetic North Pole as shown on a compass free from error or disturbance. Grid North: the northerndirection of the north-south grid lines on a map.Magnetic Declination: the angle between magnetic north and true north at any point. Sometimes theterm Magnetic Variation is used and this is mainly on Nautical and Aeronautical Charts. Normally,

however, magnetic variation is taken to refer to regular or irregular changes with time of the magneticdeclination, dip or intensity.Grid Convergence: the angle between grid north and true north.Grid Magnetic Angle: this is the angle between grid north and magnetic north. This is the anglerequired for conversion of grid bearings to magnetic bearings or vice versa.

Annual Magnetic Change: the amount by which the magnetic declination changes annually becauseof the change in position of the magnetic north pole.The diagram used on the sheet to show this information is illustrated below:

8.4 FAQ on Coordinate transformationsWhat is a coordinate transformation? A coordinate transformation is aconversion of coordinates from one to another coordinate system.

Transformations can be between plane coordinate systems, between geographicand plane coordinate systems, between geographic coordinates and geocentriccoordinate systems, etc.What is a map projection change?The transformation of coordinates from aplane system based on one projection type into a plane system based on


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another type. For example X,Y coordinates can be transformed from the UTMprojection coordinate system into the Lambert Conformal Conical projectionsystem.How do you assign coordinates to a data set if the coordinate system ofa data set is unknown?You may use control points, such as the corners ofhouses, and road intersections, to determine the relationship between the

unknown and a known coordinate system. The transformation may beconformal, affine or polynomial depending on the systematic errors in the dataset.How do you digitize a map in map projection coordinates, if the maponly shows geographic coordinates? If you have a map in a certainprojection you select at least two graticule intersection points on the map, thenuse the projection forward equations to calculate their X,Y rectangularcoordinates. These X,Y coordinates you use to reference your map on thedigitizer.What is a datum transformation? Numerous maps are projected onto variousellipsoids or reference datums. Very often, it is needed to transform one datumto another.

Intro to Map Projection

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