Intro To Map Projection

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 geometric or mathematical reference surface, the ellipsoid or the sphere, for measuring locations, and an equipotential surface called the geoid or vertical datum for measuring heights.

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Ellipsoids are used for large scale mapping, spheres can 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 positioned and oriented in such a way that it best fits to the area or country of interest. There are a few 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 using levelling 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 the map plane.

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Map projection principle Several hundreds of map projections have been described, but only a smaller part is actually 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 which they 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 correctly 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 spatial reference system if in addition to the selected reference surface (horizontal datum) and the 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 the 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.

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Obliques azimuthal stereographic projection

The Dutch National coordinate system. The origin of the map coordinate system is souteast of Paris, France Fragment of the Dutch topo map showing 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

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Maps, whether analog or digital, and spatial data, whether in vector or raster format, are related to some location. We mostly refer to these locations using coordinate systems. A coordinate system is a set of rules that specifies how coordinates are assigned to locations.

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

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Spatial geographic coordinates ( , h )

Spatial cartesian or geocentric coordinates ( x, y, z )

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Plane coordinate systems are used to locate data on the map plane. E.g. any point on the map plane can be located by means of two-dimensional cartesian (or rectangular) coordinates (x,y) or two-dimensional polar coordinates ( ,d ).

Plane rectangular coordinates (x, y)

2D polar coordinates ( , d)

2.1 Spatial coordinate systems 2.1.1 Geographic Coordinates The most widely used global coordinate system consists of lines of geographic latitude and longitude. Lines of equal latitude are called parallels. They form circles on the surface of the ellipsoid. Lines of equal longitude are called meridians and they form ellipses (meridian ellipses) on the ellipsoid.

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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 ( = 00) and increases towards the two poles to maximum values of

= +90 (N 900) at the North Pole and

=-

90o (S 900) at the South Pole. The longitude

is the angle between the meridian ellipse which passes through Greenwich

and the meridian ellipse containing the point in question. It is measured in the equatorial plane from the meridian of Greenwich = 00 either eastwards through or westwards through

= + 180o (E 1800)

= -1800 (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,

= 60 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 normal 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 unique 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 Xand Y--axis. The horizontal axis is usually referred to as the X-axis and the vertical the Y-axis (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 is 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 distance d from the origin to the point concerned and the angle a between a fixed (or zero) direction 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 units 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 instruments 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 almost 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 that result from transforming the curved Earth surface to the map plane are so great on smallscale 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 every 5 minutes and grid lines at every kilometer. The map sheet limit or neat line (the line enclosing the mapped area) can either be formed by the outline of the graticule or the grid. The grid as outline of the map has the advantage 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 vertical variations between mountains and valleys which make it impossible to approximate the shape of the earth with any reasonably simple mathematical model. We can simplify matters by the idealization of expanding the oceans below the landmasses and make the assumption that the water can flow freely also there. If we then neglect tidal and current effects on this "global ocean", the resultant water surface is remaining affected only by gravity. This has a certain consequence on the shape of this surface because the direction 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 always perpendicular 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 the mean ellipsoid. Conversely, where a mass surplus exists, the geoid will rise above the mean 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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The biggest presently known undulations are the minimum in the Indian Ocean with N = -100 meters and the maximum in the northern part of the Atlantic Ocean with N = +70 meters. Surveying observations are usually made with instruments levelled by means of spirit bubbles. Since these bubbles follow the influence of the earth's gravity the observations are made relative to the geoid. 3.2 The Geoid as reference surface for Heights In order to establish the geoid as reference for height measurements, the ocean's water level is registered at coastal places over several years using tide gauges (mareographs). Averaging the registrations, as long as they are periodic, largely eliminates variations of the sea level with time. The resulting water level represents an approximation to the geoid and is called the Mean Sea Level (MSL). Every nation or groups of nations have established those observation points, which are normally located close to the area of concern. For the Netherlands the geodetic tide gauge station is in Amsterdam, for France in Marseille, for Greece in Saloniki, etc. Starting from these stations the heights of points on the earth's surface can be measured using geodetic levelling techniques.

Differential leveling for height measurements (Mean Sea Level is the starting point for the height measurements) Since all the height values within a country are related to a particular reference point, called levelling datum or vertical datum, care must be taken when using heights from another system. This might be the case in the border area of adjacent nations. Even within the territory of a state, heights may differ depending on to which tide gauge they are related. As an example, the MSL from the Atlantic to the Pacific coast of the USA increases by 0.6 to 0.7 m. The MSL from the Netherlands differs -2.34 meters from the MSL from Belgium. As consequence heights in the border area differ (see figure below; note that the contours end at the border).

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Fragment of the Dutch topo map showing the border of Belgium 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 figure The curvature of the geoid displays discontinuities at abrupt density variations inside the earth. 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 of positions, distances, directions, etc. on the earth's surface, we need to have some mathematical reference frame. The most convenient geometric reference is the oblate ellipsoid as it provides a relatively simple figure which fits the geoid to a first order approximation. For small scale mapping purposes we can also use the sphere which fits the geoid 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 is major 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 f is dependent on both the 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 given 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 f = 1/298.26

b = 6356750.52m

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 are defined 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 States 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 interest, 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 to 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 fits 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 position and 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 ellipsoid (Clarke 1866 or Hayford), but in different positions (referred as datum shifts).

Examples of reference datums, with its reference ellipsoid and datum shift values

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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 global 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 by Hayford in 1909 as the International Ellipsoid. In contrary to local reference ellipsoids, which 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 of 1967) to the mean earth figure. The geometric ellipsoidal parameters a, b and f are given in 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 Reference System 1967 no longer represents the size and shape of the earth to an adequate accuracy. 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 are 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 surfaces 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 orthometric heights 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 the ellipsoidal 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 to 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 map projection is any transformation between the curved reference surface of the earth and the flat plane of the map.

We can as well define a map projection as a set of equations which allows us to transform a set of Ellipsoidal Geographic Coordinates (  representing  positions on the reference surface 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 ( , ) equation ,

= f ( X,Y )

Forward Inverse e q u a ti o n

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The forward equations are used to transform geographic coordinates - latitude ( ) and longitude ( ) - into Cartesian coordinates (X,Y), while the inverse equations of a map projection are used to transform Cartesian coordinates into geographic coordinates. These equations have a significant role in projection change (see section on Coordinate transformations ). Some examples of map projection equations are given below:

Map projection equations can be considerably more complicated than those introduced here, for example, when an ellipsoid is introduced as reference surface. J. P. Snyder gives an overview of map projection equations in his book entitled 'Map Projections used by the U.S. Geological Survey'. A number of equations are given at World of Mathematics. Map projection equations have a number of parameters such as o o o o o o

radius of the sphere (R) or equatorial (a) and polar radius (b) of the reference ellipsoid; geodetic datum; origin of the coordinate system; false easting and northings; central meridian ( o ), standard parallels ( 1, 2 ) or centre of projection ( 1, o ); scale factor at the central meridian or standard parallels.

Information about the projection parameters is required to define a countries spatial reference system.

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Activity A 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 to the reference surface

Since no map projection maintains correct scale throughout the map, it may be important to 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 ellipse of distortion, which is also known as Tissot's Indicatrix, shows the shape of an infinitesimally 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, the 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 degrees of flattening. The projection represents areas correctly - all ellipses have the same area - but angles and, consequently, shapes are not represented correctly.

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Lambert Cylindrical equal-area projection with ellipses of distortion

Scale distortions can also be shown on a map by a scale factor. A scale factor smaller than 1 indicates that the scale is smaller than the nominal scale, the scale given on the map. A scale factor larger than one indicates that the scale is larger than the nominal scale. For example, on the UTM projection a scale factor of 0.99960 has been given to the central meridian of a UTM zone. This means that 1000m measured on the ground becomes 999.6m on the map surface along the central meridian. E.g. the actual map scale along the central meridian will be 1:10,004 (10000 / 0.9996) at a nominal map scale of 1:10,000, so smaller than the nominal scale. Note Scale distortions can remain within certain limits by choosing the right map projection (see section 4.5)

4.3 Properties of Map Projections The following properties would be present on a map projection without any scale distortions: •

Areas are everywhere correctly represented

•

All distances are correctly represented.

•

All directions on the map are the same as on Earth

•

All angles are correctly represented.

•

The shape of any area is correctly represented

It is, unfortunately, impossible to have all these properties together in one map projection.

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

The Mercator as an example of a conformal, cylindrical projection Note A 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 exclusive 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-error map projection the scale errors everywhere on the map as a whole are 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 is 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 a 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's surface 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 or zenithal 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, cases. The figure below provides two examples.

A transverse cylindrical and an oblique conical map projection. Both are tangent to the reference surface The terms polar, 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 an oblique 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 is that there is no one projection best overall (see section 4.5 selecting a suitable map projection ) Activity The diagram below shows the developable surface of the Lambert conformal conic projection with two standard parallels.

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Answer the following questions:

1. 2. 3. 4. 5.

Which developable surface is used? Is it a tangential or a secant projection? What is the position of the developable surface? Describe some of the scale distortion characteristics. Are areas correctly represented?

4.5 Selecting a suitable Map Projection Every 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 of projection is used for any particular map. A well choosen map projection takes care that scale distortions remain within certain limits and that map properties match to the purpose of 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 class of a map projection should be made on the basis of the shape and size of the geographical area to be mapped. Ideally, the general shape of a geographical area should match with the distortion pattern of a specific projection. For example, if an area is small and approximately circular it is possible to create a map that minimises distortion for that area on the basis of an Azimuthal projection. The Cylindrical projection should be the basis for a large rectangular area and a Conic projection for a triangular area.

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The position of the geographical area determines the aspect of a projection. Optimal is when the projection centre coincides with centre of the area, or when the projection plane is 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 aspect of a map projection have been selected, the choice of the property of a map projection has to be made on the basis of the purpose of 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 cylindricalmet 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 shape 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 the 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 projection 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 cooperation in data exchange difficult. There are strong arguments in favour of using an international standard projection for mapping. Activity You 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 use Several hundreds of map projections have been described, but only a smaller part is actually 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. Parallels 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 Equator.

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 increases

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makes Greenland appear to be as large as South America when, in fact, it is only a quarter of the size. The Mercator projection is used for long distance navigation because of the straight rhumblines. 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.

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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 parallels The scale is true along the central meridian and along each parallel. The distortion increases 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 centre 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 projection. All meridians and parallels are shown as circular arcs or straight lines. Since the projection is 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 of 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 - the 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 scale distortions are extreme the projection should not be used for regular geographic maps or for 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 coordinates 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 a complete 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 the Dutch RD system (this implies the Bessel 1841 ellipsoid). In such a case the projection transformation must be combined with a datum transformation 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 coordinates, 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 coordinates 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 of Bowring. These last have millimetre precision for 'earth-bound' points, i.e. points that are at 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 in 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 local ellipsoid measured along the ellipsoid normal through the point; a = semi-major axis of local ellipsoid; f = flattening of the local ellipsoid; 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 and WGS84 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 )

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

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

B. Regression equations The multiple regression equations relate ellipsoidal latitude and longitude co-ordinates of continental size datums to those of the WGS84 datum and involve 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 (implemented in 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.

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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 in 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 to another. The centres of the ellipsoids do not 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 a

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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 coordinate 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 were chosen. 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 geometric 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-linear polynomial 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.

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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 X and Y direction (Sx, Sy ).

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C. a non-linear polynomial transformations can be used to apply a Shift, a Rotation and a Variable scale change. Polynomial transformation functions can have an infinite number of terms. At least 6 control points (12 coefficients or unknowns : Xo ,a1 - a5 , Yo ,b1- b5 ) are required to solve a simple 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 of 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 in which for each pixel in the new co-ordinate system, a new pixel value has to be determined 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 image 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 equations, orthophoto correction equations, etc. After that, one can drag any vector map, even with another datum, over the non-corrected image.

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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 space in case of combined analysis of raster and vector data. The degradation of the image quality due to resample errors is also avoided. Once the vector editing or analysis is completed, one 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 projection 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 Questions concerning Geometric Aspects of Mapping. Answers are given on a number of commonly asked questions related to Coordinate systems, Reference surfaces, 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 geographic and geocentric coordinate systems and in plane coordinate systems: e.g. 2D cartesian and polar coordinate systems. Generally two types of coordinate systems are given on maps: cartesian coordinates (or X,Y map projection coordinates) and projected geographic coordinates. Satellite positioning systems (e.g. GPS) make use of 3-dimensional spatial coordinate systems to define positions on the earth surface, with reference to a mean reference surface for the earth (e.g. GPS measurements use the WGS84 ellipsoid). 2D Polar coordinates are often used in land surveying. For some types of surveying instruments it is advantageous to make use of this coordinate system.

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What is a graticule? The graticule represents the projected position of selected meridians (lines with constant longitude λ) and parallels (lines with constant latitude ϕ). What is a grid? The grid represents the lines having constant X or Y coordinates and situated at constant intervals depending on map scale.

8.3 FAQ on Reference surfaces Why do we need a reference surface? The physical surface of the Earth is a complex shape. In order to represent it on a plane it is necessary to move from the physical surface to a mathematical one, close to the former. What kinds of reference surfaces are used in mapping? In mapping different surfaces or earth figures are used. These include a geometric or mathematical reference surface, the ellipsoid or the sphere, for measuring locations, and an equipotential surface called the geoid or vertical datum for measuring heights. What is a vertical datum? The vertical datum, an approximation of the geoid, is defined as natural reference surface for land surveying. A vertical datum fits the Mean Sea Level surface throughout the area of interest and provides the surface to which height ground control measurements are referred.. What is a sphere and when do you use the sphere to approximate the Earth's surface? The surface of Earth may be taken mathematically as a sphere instead of ellipsoid for maps at smaller scales. In practice 1:1 000 000 - 1:5 000 000 is recommended as the largest scale at which the spherical approximation can be made. A sphere can be derived from the certain ellipsoid corresponding either to the semi-major or semi-minor axis, or average of both axes or can have equal volume or equal surface than the ellipsoid. The Sphere represents a worse approximation than ellipsoid but reduces a mathematical difficulty. What is a geodetic datum? The geodetic (or horizontal) 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. Why are there so many ellipsoids and datums defined? An ellipsoid and a datum serve as geometric models of the Earth surface. They have been established to fit the earth's surface well over the area of local interest. This is important to minimize distortions on maps. About 15 different reference ellipsoids and many more local datums may be encountered in world mapping. Most commonly used ellipsoids are the International, Krasovsky, Bessel, Clark 1880 and the WGS84 ellipsoid. Local ellipsoids serve as reference only for a local area of the earth's surface. Global ellipsoids (e.g. WGS84 ) serve as mean reference for the entire earth surface. What is WGS84? WGS84 is one of the World Geodetic Systems which provides the basic reference frame and geometric figure for the earth. WGS84 provides a positional relation of various local geodetic systems to an earth-centered, earth-fixed coordinate system, through reports of the DMA of U.S. Defense (D.O.D) GPS measurements use the WGS84 as reference surface for their measurements.

8.4 FAQ on Map projections What is a map projection? A map projection is any transformation between the curved reference surface of the earth and the flat plane of the map. You can as well define a map projection as a mathematical formula by which you can transform Geographic coordinates ( latitude φ and longitude λ angles ) into Cartesian projection coordinates ( X and Y ) X, Ymap projection = f ( φ, λ ) Forward equation The inverse equations of a map projection are used to transform Cartesian coordinates into geographic coordinates. An overview of map projections equations is given by J.P Snyder ' Map projections used by the U.S. Geological Survey'. What are the parameters of a map projection? Map projection equations contain map projection parameters. The most common parameters are: R = radius of the sphere; a = equatorial radius or semi-major axes of the ellipsoid of reference; b = polar radius or semi-minor axes of the ellipsoid of reference; e = eccentricity of the ellipsoid; f = flattening of the ellipsoid; ho = scale factor at central meridian; h = relative scale factor along a meridian of longitude; ko = scale factor at standard

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parallel(s); k = relative scale factor along a parallel of latitude; λo = central meridian or longitude of origin; ϕo = latitude of origin; Xo = false Easting; Yo = false Northing. Projection parameters have a significant role in defining a coordinate system. See list of parameters described by J.Snyder 'Map Projections Used by the United States', p.xi-xii. Why do we need a map projection? If you are mapping a significant portion of the Earth's surface it is impossible to project it on a flat piece of paper without scale distortions. Map projections take care that the scale distortions remain within certain limits and the distortion pattern of a map projection determines the property of the projection. Each projection has its own characteristics. For example a map projection may have the property that all angles are correctly represented (conformal projection property). A map projection is not of major importance for city or street maps, which cover a relatively small surface of the earth. How do we classify map projections? There are three map projection classes: cylindrical, conical and azimuthal. Map projections can be subdivided into three aspects: the polar or normal aspect, which centers the map at one pole of the globe; the equatorial aspect, which centers the map at the equator; and the oblique aspect that centers the map anywhere else. Map projections can have the properties: conformal, equal-area or equidistant. A further descriptor is whether the projection has a secant or tangent projection plane. Why are some map projections using mapping zones? Some map projections divide the mapping area into zones in order to keep scale distortions within acceptable limits. For example sixty longitudinal zones are used for the UTM grid system. All these zones are exactly 6° wide in longitude and 164° extent in latitude. Zoning system have the disadvantage of working with different coordinate systems, each zone has its own coordinate system.

How do we match adjacent maps? In order to fit two or more separate maps exactly along their edges, a number of parameters must be maintained: 1. the maps must be constructed with the same projection and projection parameters; 2. they must be at the same scale; and 3. they should be based on the same reference datum. How to select a suitable map projection? The choice of a map projection class depends on the size and shape of the geographical area to be mapped. - Cylindrical projections for large rectangular areas; - Conic projections for medium size triangular areas; - Azimuthal projections small-size circular areas. The choice of a map projection property has to be made on the basis of the purpose of map. Conformal projections for sea, air and meteorological charts, topographic and large scale maps; Equidistant projections for topographic and large scale maps; - Equal-area projections for historical, population, geological and soil maps. What kind of map projection information should be mentioned on a map? Map projection information should at least include the projection name, reference ellipsoid, and the reference datum.It is placed on the map sheet outside the map frame as marginal information. To define a map coordinate system in a GIS detailed information on the projection parameters are required. Other geometric What kind of geometric information should be given in the margins of a topographic map? 1. Numerical scale: this is to appear near the graphic scale, and in the upper margins next to the area of coverage. 2. Graphic scale: these will normally be in kilometers and statute miles, with the addition of meters and yards when the scale of the map requires it. If it is necessary a scale of nautical miles is added. These scales are to be placed in the center of the lower margin.

3. Projection, Spheroid, Geodetic datum, Levelling datum, notes relating to the basis geodetic data 4. Notes concerning the grid(s): information is to be given as to the grids to which lines, ticks and figures refer. The projection, spheroid(s), datum(s) origin and false co-ordinates of origin will be stated for each grid.

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

6. Unit of Elevation: the note ELEVATIONS IN METRES or ELEVATIONS IN FEET is to appear in a conspicuous position normally in the lower margin. Wherever possible a conspicuous color should be used. 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 the form: "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 information necessary to determine the true, grid and magnetic bearings of any line within the sheet. This information 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 angles made with Grid North by the other two lines are usually too small to be shown accurate on the diagram; therefore they may be arbitrarily exaggerated for clarity. This exaggeration should be sufficiently large to prevent any confusion with true angles. In all cases the relative positions of the three 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 following forms, for example: 1st of January 1999 GRID MAGNETIC ANGLE for CENTRE OF SHEET 2°30' (45 Mils) ANNUAL MAGNETIC CHANGE 3' or 1 Mil East GRID CONVERGENCE at CENTRE OF SHEET 1°41' (30 Mils) The Grid Convergence is normally given to the nearest minute and half mil. The Grid Magnetic Angle is normally given to the nearest 15 minutes and 5 mils. The Annual Magnetic Change is given to an accuracy sufficient to allow the accuracy of the Grid Magnetic Angle to be maintained. Where the values of these angles vary significantly over the map sheet value for the various portions of the sheet are to be given in tabulated form. When the map includes more than one grid zone separate diagrams are required for each and must be clearly labeled with the grid or grid zone designation to which they refer (see example below). In regions where large magnetic irregularities occur special treatment 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 the meridian to the North Pole at any point on the map. Magnetic North (MN): the direction of the Magnetic North Pole as shown on a compass free from error or disturbance. Grid North: the northern direction of the north-south grid lines on a map. Magnetic Declination: the angle between magnetic north and true north at any point. Sometimes the term 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 magnetic declination, 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 angle required for conversion of grid bearings to magnetic bearings or vice versa. Annual Magnetic Change: the amount by which the magnetic declination changes annually because of 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 transformations What is a coordinate transformation? A coordinate transformation is a conversion of coordinates from one to another coordinate system. Transformations can be between plane coordinate systems, between geographic and plane coordinate systems, between geographic coordinates and geocentric coordinate systems, etc. What is a map projection change? The transformation of coordinates from a plane 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 UTM projection coordinate system into the Lambert Conformal Conical projection system. How do you assign coordinates to a data set if the coordinate system of a data set is unknown? You may use control points, such as the corners of houses, and road intersections, to determine the relationship between the unknown and a known coordinate system. The transformation may be conformal, affine or polynomial depending on the systematic errors in the data set. How do you digitize a map in map projection coordinates, if the map only shows geographic coordinates? If you have a map in a certain projection you select at least two graticule intersection points on the map, then use the projection forward equations to calculate their X,Y rectangular coordinates. These X,Y coordinates you use to reference your map on the digitizer. What is a datum transformation? Numerous maps are projected onto various ellipsoids or reference datums. Very often, it is needed to transform one datum to another.

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