30 Coordinate System

Prof. Masood Ahsan Siddiqui

Cartography maps the Earth’s surface in selected areas on paper. A planimetric map depicts locations and features on the surface that are at different heights above or below a reference (datum) plane. In their relative positions, they appear as though projected onto an X-Y (horizontal) coordinate system. The co-ordinate system is defined as a frame of reference in which geometrical positions of point objects can uniquely be defined. It requires a reference point, a number of independent directions and a scale definition in each direction. Each point on the map corresponds to a point on the Earth’s surface. These surface points, however, are distributed on the curved surface of the Earth. Thus, to locate the points it is a must that these be referenced on to a geographic coordinate or grid system.

GIS gathers the information about the world as a collection of thematic layers that can be linked together with geography. It contains either an explicit geographic reference such as a latitude and longitude or national grid coordinate, or an implicit reference such as an address, postal code, forest identifier, or road name. implied references can be derived from explicit references using an automated process called “geocoding.” These geographic references allow locating features and events on the surface of the earth for interpretation and analysis.

The locations on the earth’s surface are shown with the help of points, lines, and polygons and are defined by a series of x, y co-ordinates. The co-ordinate systems must be consistent between map layers. Co-ordinate systems can be self-descriptive or in units that relate to the real world shown through decimal degrees; degrees, minutes, seconds; meters; and feet are all examples of units of measure in a co-ordinate system

A co-ordinate system consists of a Spheroid and a Map Projection. For any database to be useful for spatial analysis, the database must be connected to a recognized global co-ordinate system. What prevents the establishment of a standard coordinate system for use in GIS is that the most useful coordinate systems are orthogonal (composed of axes that cross at right angles, and units that are equal in all parts of the grid) yet the earth is round and must be flattened to fit into an orthogonal coordinate system. In order to perform actions such as spatial associations, entities must be referenced in a common coordinate system the selection of which is based on the specific requirement and the level of accuracy required. Consequently, there are hundreds of coordinate systems in common use based on a variety of geodetic datum, units, projections and reference system, and each of these has valid justification. There are many methods (projections) for making flat, orthogonal representations of the earth, but no single method is suitable for the entire earth.

The co-ordinate systems may be put into three broad categories:

1) Geographical coordinates

2) Rectangular coordinates

3) Non coordinates

Geographical Co-ordinates/Spherical Co-ordinates

Locations on the Earth’s surface are represented by latitude and longitude. The network of meridians and parallels is referred to as the graticule. Geographic co-ordinates are based on the Earth’s axis of rotation and the plane of the Equator. Parallels of latitude measure angular distance north or south of the equator, ranging from 0 degrees at the Equator to 90 degrees at the north and south poles while the meridians of longitude measure angular distance east or west of the prime meridian, which runs through the Royal Observatory in Greenwich, England. Each line of longitude represents a circle and divides the globe into two equal parts. All the longitudes are known as Great circles The world mapping community in 1884 agreed to use the Greenwich meridian in London as the prime meridian. Lines of latitudes lie at right angles to the lines of longitudes and run parallel to each other. The origin of this geographic coordinate system is the intersection of the equator and the prime meridian. A great circle is defined by the intersection of a sphere with a plane passing through the center of the sphere. These are the shortest possible paths on the spherical surface of the earth. The lines of latitudes also divide the globe but into unequal parts (except equator which divides the globe into two equal parts and consequently a great circle), and each has different circumference, are known as small circles. An infinite number of small circles are possible, depending on the position and orientation of the plane relative to the sphere.

The world data bank has listed the coordinates of the world’s coastline, rivers, islands and political boundaries. These coordinates are in decimal degrees and rounded to the nearest 0.001 degree. A degree is a unit of measurement equal to 1/360 of a circle. A degree of latitude on the earth’s surface is about 69 miles. A degree of longitude is about 69 miles at the equator and undefined at the poles. This way at equator one degree equals to 111.11 km. Thus 0.001-degree equals to 111 meters.

Figure 1: Geographical Coordinate

    Rectangular coordinates/Cartesian Co-ordinates

Latitude and longitude provide a convenient reference system for describing locations on the approximately spherical surface of the Earth. However, it is often difficult to work with latitude and longitude on a flat map, particularly if we are interested in measuring geometric properties of features on the map such as distances, directions or areas.

To overcome this problem, the French for the first time developed the rectangular coordinate system during the First World War. It is much easier to work with a Cartesian co-ordinate system on the map. Positions on the map are defined by their x, y co-ordinates, or eastings and northings. Distance and direction calculations are simplified since they are now based on right-angled triangles rather than on spherical trigonometry. The Cartesian co-ordinate system can be an arbitrary local co-ordinate system. A local co-ordinate system is often used for field survey work in which locations are measured relative to some fixed reference point or line. However, it is much more common to base the Cartesian co-ordinate system on a particular map projection. This provides a common frame of reference that allows data from a variety of sources to be integrated in a systematic way.

First of all a spherical surface is transformed to a plane and then an equi-spaced rectangular grid is placed over the map. To eliminate the possibility of repetition of the values and the occurrence of the negative values, the origin of the numbering is considered to be outside the map area in the southwestern corner. In the rectangular coordinates, the X value is always written first and called as Easting, followed by Y values called as Northing.

Figure: 2 Rectangular Coordinate System

Grid Convergence is the value to be subtracted from a Grid Bearing (as measured from the map) to obtain an azimuth (calculated from True North).

if Grid North is west of True North grid convergence is positive.

    if Grid North is east of True North grid convergence is negative.

The Grid Magnetic angle is calculated from the magnetic declination and the grid convergence, and is the angle between Grid North and Magnetic North – this value must be subtracted from a grid bearing  to obtain a magnetic bearing.

Figure: 3

The two values are always written in pairs. The rule of the thumb for reading the grid is “Read Right Up”. It is important that the projection for the grid should be orthomorphic. The grid is named after the type of the projection system used. The grid system is easy to understand and independent of scale of the map. These are designed for detailed calculations and positioning. Therefore, accuracy in a features absolute position and its relative position to other feature is more important than the preserved property of a map projection. To maintain the level of desired accuracy the coordinate system may be divided into different zones based on separate map projection. The grid origin coincides with the projection origin. All the countries of the world use rectangular grid co-ordinates on large-scale topographic maps. Many countries use the Universal

    Transverse Mercator projection as the frame of reference for their grid co-ordinate systems.

Some of the popular rectangular coordinate systems are discussed here.

Military Grid Reference System (MGRS)

Military grid reference system is an extension of UTM. The UTM zone number and zone character are used to identify an area of 6 degrees in east and west extent and 8 degrees in north and south extent. It uses a lettering system to reduce the number of digits for isolating a location. The 8-degree rows of areas are designated by letters C, 800-720 south and extending to X at 720-840 N. A single rectangle of 6×8 generally covers 1000 square km on ground. The squares are referenced by alphabets and numerals. Letters A, B, Y and Z are reserved for the UPS grid system. To locate an object with 6×8 quadrilaterals, a secondary lettered columns and rows are used, applying 100,000-m squares. Starting at the 180 meridians, and proceeding eastward, the characters A to Z (O and I are omitted) are used for identification. The alphabets are repeated after every 18 degrees and include about six full width columns per UTM zone. Towards north, from the equator, the characters A to V (omitting O and I) are used.

Figure 4: Military Grid reference system

    Northing designators normally begin with A at equator for odd numbered UTM easting zones. For even numbered easting zones, the Northing designations are offset by five characters; starting at equator with F. The characters continue the pattern to the north of the equator. UTM zone number, UTM zone and the two 100 square kilometer characters are followed by even number of numeric characters representing easting and northing values. If 10 numeric characters are used, a precision of 1 meter is assumed, 2 characters imply a precision of 10 kilometers. From 2to 10, the precision changes from 10km, 1km, 100m, 10m,and 1m. The polar areas are covered by the UPS grid system.

State Plane Coordinate System

The state plane coordinate system (SPS or SPCS) was developed in 1930 by the U.S. Coast and Geodetic Survey to provide a common reference system for surveyors and mappers. The goal was to design a conformal mapping system for the country with a maximum scale distortion of one part in 10,000, which at the time was considered the limit of surveying accuracy. The State Plane Coordinate System (SPCS) is used for local surveying and engineering applications, but isn’t used if crossing state lines.

This is an x, y coordinate system whose zones divide the U.S. into over 130 sections, each with its own projection surface and grid network. This technique ignored the curvature of the earth and assumed it simply a flat plane. It is accurate in terms of linear measurements. The State Plane grid system is very similar to that used with the UTM system, with the exception of where the origin for the grids is located. The easting origin for each zone is always placed an arbitrary number of feet west of the western boundary of the zone, eliminating the need for negative easting values. The northing origin, however, is not at the equator as in UTM, but rather it is placed at an arbitrary number of feet south of the state border. It has been designed to have a maximum linear error of 1 in 10,000. This is four times as accurate as the UTM system, whose maximum linear error is 1 in 2,500. With the exception of very narrow States such as New Jersey, and New Hampshire, most of the States are divided into two to ten zones. The Lambert Conformal projection is used for zones extending mostly in the east-west direction whereas the

Transverse Mercator projection is used for zones extending mostly in a north-south direction. Alaska, Florida, and New York use both Transverse Mercator and Lambert Conformal for different areas. The number of zones in a state is usually determined by the area of the state and ranges from one to as many as ten in Alaska. Each zone has a unique central meridian.

Figure 5: State Plane Coordinate System

Once a zone is mapped, a Cartesian coordinate system is created for the zone by establishing an origin some distance (usually, but not always, 2,000,000 feet) to the west of the zone’s central meridian and some distance (there is no standard; each zone uses its own unique distance) to the south of the zone’s southernmost point. This ensures that all coordinates within the zone will be positive. The X-axis running through this origin runs east west, and the Y-axis runs north south. Distances from the origin are generally measured in feet. In the beginning the SPCS was based on NAD 27 but later on switched to NAD 83. As a result now it is called as SPC 83. With the change in the measuring unit from feet to meter, the coordinate values of the SPC83 are published in meters.

National Grid System

Many nations have defined their own grid systems based on coordinates. The most popular and significant among them is the British National Grid (BNG). It is based on the national grid system of England and administrated by the British Ordnance Survey. It is rectangular grid system based on Transverse Mercators Projection since 1920and uses the Great Britain Datum 1936(Airy ellipsoid). The grid is 700x1300km covering all of Great Britain. This is divided into 500km squares. Which in turn are divided into 25 parts of 100 km squares. These are identified by two letters. Each 100km square is further divided into ten parts (10×10) and the 10km squares are divided into one hundred (1kmx1km) square. The grid references are commonly given in six figures prefixed by letters denoting 100km squares. The true origin of the system is at 49 degrees north latitude and 2 degrees west longitude (Fig.6). The false origin is 400 km west and 100 km north. The scale factor at the central meridian is 0.99960.

Figure 6: British National Grid 100 Km square

Indian Grid System

    The Indian system follows more or less British system. The Indian system has eight grid zones designated as 0, I, IIA, IIB, IIIA, IIIB, IVA, and IVB based on Lamberts conical orthomorphic projection with two standard parallels covering India, Pakistan, Burma, Afghanistan, and parts of Iran, Asiatic Russia, China, Tibet and Thailand. Each zone has a belt of 80 latitude.

The false origin for all the zones, except zone 0 is 3,000,000 yards easting and 1.000,000 yards northing. For the grid 0, the easting and northing is 2,355,000 and 2,590,000 yards respectively. The grid lines are drawn at 1000 grid yard apart on 1 inch to 1 mile and larger, whereas on 1 inch to 4 miles and smaller, the grid lines are 10,000 grid yards. With the change in the measuring unit from yards to meters, the same values are read as meters. As the topographical maps of India are prepared on Polyconic projection, and not on the Lambert projection, a small error occurs. As a result the grid squares are not perfect squares. On the Indian maps the grids are printed in purple colour. The grid letters are printed near the four corners of the topographical sheet. The primary letter is smaller than the secondary letter.

Figure 7 a: India Zone I

Figure 7 bIndia Zone II A

Figure 7 cIndia Zone II B

Figure 7 dIndia Zone III A

Figure 7 eIndia Zone III B

Figure 7 fIndia Zone IV A

Figure 7 gIndia Zone IV B

Figure 7 : Zones of Indian Grid System

Source: http://earth-info.nga.mil/GandG/publications/tm8358.1/tr83581c.html

    Non-coordinate systems

There are many other coordinate systems called as non-coordinates. These provide spatial references using a descriptive code rather than a coordinate. Some of them are standardized but many are not. The popular non coordinate systems are Postal codes, AT&TV and H coordinate system, navigation coordinate system, Maidenhead Grid squares for geographical position of amateur radio community, Metes and Bounds (identifies the boundaries of land parcels by describing length and direction of lines), Public Land Rectangular Survey (for identification of public land in USA) etc. the Postal Index Number (PIN) in India are the typical example.

Co-ordinate transformation

Geographical Information system acquires and handles data from different sources, which have been prepared on a variety of coordinate system. A digitized map is measured in the same measurement unit as the source map but before using it in a GIS project, the digitized map must be converted to the real world coordinates such as UTM. It helps in making such data comparable and useful in overlay operations and mosaicing. The coordinate transformation techniques are also helpful in removing geometric distortions with respect to accurately positioned feature.Coordinate transformation isoften needed to express one geometric data set in the coordinate system of another but there arises a problem if the coordinate system of one of the data sets is unknown. In such a situation, we can begin by finding features on the foreign map that we know represent the same features on the ‘standard’ coordinate system. These common points are called as ‘Control Points’. By choosing one control point, the unknown geometry could be transformed in a way such that the entire coordinate system shifts, making the coordinates of that one point the same — but probably failing to match anything else. In case of two defined control points, a coordinate transformation is specified whereby the foreign coordinate system is shifted, scaled and rotated such that a line on the foreign system matches with a line in the GIS. Where as with three control points, a triangle from the foreign geometry can be aligned to a triangle in the GIS. This may involve skewing or shearing the foreign geometry, effectively scaling x by a factor different than y. It either converts the coordinate system of the spatial data or changes the relative location of the feature in the same coordinate system by adjusting through shifting, rotations etc. In other words there are two ways to look at coordinate transformation i.e. either move objects on a fixed coordinate system so that the coordinates change diagram or hold the objects fixed and move the coordinate system. The transformation of coordinates involves seven transformation parameters- three translations due to shift of origin, three rotations due to change in orientation, and a scale factor due to different dimensions of the two reference ellipsoids. These transformation parameters must be estimated using coordinates of several well-distributed stations in both the systems in order to obtain the geodetic coordinates in local reference system. Based on the nature of the relationship between the coordinate systems, two methods of transformations are considered. These are Polynomial Transformation and Exact Transformation.

If the relationship between the coordinate system is derived through the control points in both the coordinate systems it is termed as polynomial transformation. The typical polynomial transformation is the Affine Transformation. It involves identification of control points on both the systems and after transformation the straight lines remain straight, parallel lines remain parallel, the angles go slight changes, and the process is capable of performing rotation, translation, skew and differential scaling operation on the rectangular object. The typical example of this kind of transformation is that the circle may be transformed into an ellipse. A common application of affine transformation is to translate the coordinates of the digitized map in digitizer units to those in the real world coordinate system of the map. Affine transformation is also used frequently in rectification of satellite imagery to ground truth.

Rotation rotates its x and y-axes from the origin that is; the origin is fixed, while axes move (rotate about origin). Translation shifts the origin to a new location but axes do not rotate diagram. Skew changes the shape to a parallelogram with slanted direction and differential scaling changes the scale by expansion or reduction in the x and/or y direction that is both origin and axes are fixed whereas the scale changes.

                                         Differential Scaling              Rotation           Skew              Translation

Figure 8: Scaling

The number of control points, depend on the degree of polynomial but the number should not be less than three. It is advised that the control points should be well distributed over the region. The affine transformation and its operation are first applied to the control points. All the GIS packages offer both affine and projective transformation methods but they are being used for different purposes. For example, the affine transformation is generally used on digital maps and satellite images, whereas projective transformation is used on aerial photographs with displacement

If the relationship could be expressed in terms of mathematical form, it is called as exact transformation. It has been divided into two categories, viz. Projective and Planer transformation. The projective transformation transforms the three-dimensional space to a two dimensional space. It applies the concept of map projection for transforming the three dimensional surface. It allows angular and length distortion, thus allowing the rectangle to be transformed into an irregular quadrilateral. Hence it can be used to relate two coordinate systems defined in two different map projections over a small area. The typical examples of projective transformation are geodetic to polyconic projection; geodetic to UTM etc. A planer transformation transforms the coordinate from one planer system to the other planer system. It is applicable if the two coordinate systems have known orientation and transitional shift.

Figure 9: Geometric Transformation

(RMS) Errors of Fit: If an Affine transformation is performed with more than three control points, a separate affine transformation can be attempted using all the possible triplets, then an average can be made that maximizes the ‘Fit’ of all the points. It is in fact an error report that shows which points are not so cooperative to being realigned satisfactorily with an affine transformation. A good transformation interface will return root-mean-squared errors for each of the points. This RMS error is an estimation of the error for that particular point assuming that the calculated transformation is the correct one. It must be remembered that the RMS errors should be within a tolerance limit i.e. 0.3.

Higher Order Transformations (Warps): An affine transformation can distort a planar coordinate system in all ways that preserve parallel lines; whereas the higher-order transformations maintain the earth curvature to a great extent. In other words higher polynomial order will lead to more curvy transformation. It has been observed that generally a large number of control points are selected and the highest order of transformation is used to get the best fit for all of the selected points.