Coordinate Systems & Map Projection

Dr. M.R.Sivaraman

Retd. Scientist

Space Applications Centre

Ahmedabad

Why Do You Need Coordinate Systems

• Geodesists basically assign addresses to points all over the Earth. If you were to stick pins in a model of the Earth and then give each of those pins an address, then you would be doing what a geodesist does.

• Using Spatial reference systems created by geodesists, any location on the earth can be located quickly and accurately.

• By looking at the height, angles, and distances between these locations, geodesists create a spatial reference system that everyone can use.

• Building roads and bridges, conducting land surveys, and making maps are some of the important activities that depend on Geodesy & spatial reference system.

What a Geodesist do ?

What is a Coordinate System ?

• Coordinate Systems are schemes for locating points in a given space by means of numerical quantities specified with respect to some frame of reference.

• These quantities are the coordinates of a point. • To each set of coordinates there corresponds just one point in any

coordinate system, but there are useful coordinate systems in which to a given point there may correspond more than one set of coordinates.

• A coordinate system is a mathematical language that is used to describe geometrical objects analytically; that is, if the coordinates of a set of points are known, their relationships and the properties of figures determined by them can be obtained by numerical calculations instead of by other descriptions.

• It is the province of analytic geometry, aided chiefly by calculus, to investigate the means for these calculations.

Cartesian Coordinate System

Polar Coordinate System

Size & Shape of The Earth

Size & Shape of The Earth

• The Topographical Surface of the earth (Physical Surface) is the actual surface of the earth, on which Geodesists can make measurements and define a useful Spatial Reference System.

• It is very irregular and mathematical calculations based on this surface are very difficult.

• It is necessary to define a smooth mathematical surface that fits this and relates it to measurements on the Physical Surface.

• The best mathematical surface that fits the physical surface of the earth, both in land and sea areas is a Geoid.

Size & Shape of The Earth

• A Geoid is defined as a surface where acceleration due to gravity is a constant (equipotential surface) and the direction of gravity is Perpendicular.

• Geoid is an equipotential surface of the Earth's gravity field which

best fits, in a least squares sense, global mean sea level • It is a surface to which the oceans would conform over the entire

Earth if free to adjust to the combined effect of the Earth’s mass attraction and the centrifugal force of the Earth’s rotation

• It is the mean sea level, the undisturbed sea level in ocean, in equilibrium between effect of the Earth’s mass attraction and the

centrifugal force of the Earth’s rotation • It is the undisturbed sea surface level in Land, in equilibrium

between effect of the Earth’s mass attraction and the centrifugal

force of the Earth’s rotation if canals were dug into the land to allow Ocean to enter the land.

GEOID

• Uneven distribution of the Earth’s mass makes the geoidal surface irregular. The surface of the geoid, with some exceptions, tends to rise under mountains and to dip above ocean basins. The geoid refers to the actual size and shape of the Earth, but such an irregular surface has serious limitations as a mathematical Earth model because:

• 1. It has no complete mathematical expression.• 2. Small variations in surface shape over time introduce small errors

in measurement.• 3. The irregularity of the surface would necessitate a prohibitive

amount of computations.

• For geodetic mapping, and charting purposes, it is necessary to use a regular or geometric shape which closely approximates the shape of the geoid either on a local or global scale and which has a specific mathematical expression. This shape is called the ellipsoid.

Definition of an Ellipsoid

• An ellipsoid is a 3-dimensional shape formed by rotating an ellipse either around its major axis or around its minor axis. The Earth's shape can be modeled by an ellipsoid formed by rotating an ellipse around its minor axis. This is because the centrifugal force resulting from its rotation has distorted it slightly. The figure below is enormously exaggerated; the Earth would appear to be spherical at the scale of the image.

• An ellipsoid is described by two numbers. In the figure below, a is the semi-major axis or equatorial radius, and b is the semi-minor axis or polar radius. Flattening and eccentricity, which are derivable from a and b, are convenient for use in the calculations involved in datums and surveying.

• f is the flattening, and is defined to be (a - b) / a.

• e is the eccentricity, and is defined as e2 = 2f – f2.

Ellipsoid

Different Ellipsoids

Reference ellipsoid name

Equatorial radius (m)

Polar radius (m)

Inverse flattening

Where used

Everest Spheroid 6,377,301.243 6,356,100.228 300.801694993 India

International (1924) 6,378,388 6,356,911.946 297 Europe

Krassovsky (1940) 6,378,245 6,356,863.019 298.3 Russia

Australian National (1966)

6,378,160 6,356,774.719 298.25 Australia

South American (1969) 6,378,160 6,356,774.719 298.25South America

GRS-80 (1979) 6,378,137 6,356,752.3141 298.257222101

NAD 83 6,378,137 6,356,752.3 298.257024899North America

WGS-84 (1984) 6,378,137 6,356,752.3142 298.257223563 US DOD

IERS (1989) 6,378,136 6,356,751.302 298.257

Conversion Formulas

• The conversion formula from Crtesian Coordinates to the Polar coordinates are given below.

Φ = atan((Z+e’2 b sin3θ)/(p-e2 a cos3θ))λ = atan2(Y,X)

h = (p/cos(Φ))-N(Φ)• Where• Φ,λ,h are Geodetic Latitude, Longitude and Height above Ellipsoid• X,Y,Z are the Earth Centered Earth Fixed Cartesian Coordinates

and• p = √X2+Y2 , θ = atan(Za/pb) , e’2 = (a2-b2)/b2

• N(Φ) = a/√1-a2 sin2Φ = Radius of curvature in prime vertical• a = Semi-major axis of the Ellipsoid, • b = Semi-minor axis of the Ellipsoid• f = (a-b)/a = Flattening• e2 = 2f – f2 = Eccentricity Squared

Conversion Formulas

• Conversely, the conversion formula from Polar Coordinates to Cartesian Coordinates are given below.

X = (N+h) cos Φ cos λY = (N+h) cos Φ sin λZ = [N(1-e2)+h] sin Φ

• Where• Φ,λ,h are Geodetic Latitude, Longitude and Height above Ellipsoid• X,Y,Z are the Earth Centered Earth Fixed Cartesian Coordinates• and ________• N(Φ) = a/√1-e2 sin2Φ = Radius of Curvature in Prime Vertical• a = Semi-major axis of the Ellipsoid, • b = Semi-minor axis of the Ellipsoid• f = (a-b)/a = Flattening• e2 = 2f – f2 = Eccentricity Squared.

Celestial Coordinate SystemsHorizontal Coordinate System

• The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane.

• This conveniently divides the sky into the upper hemisphere that you can see, and the lower hemisphere that you cannot (because the Earth is in the way).

• The pole of the upper hemisphere is called the zenith. The pole of the lower hemisphere is called the nadir.

Horizontal Coordinate System• The horizontal coordinates are:

• Elevation angle, also referred to as altitude, refers to the vertical angle measured from the geometric horizon (0°) towards the zenith (+90°). It can also take negative values for objects below the horizon, down to the nadir (-90°).

• Although some will use the term height instead of elevation, this is not recommended as height is usually understood to be a linear distance unit, to be expressed in meters (or any other length unit), and not an angular distance.

• The term zenith distance is more often used in astronomy and is the complement of the elevation. That is: 0° in the zenith, 90° on the horizon, up to 180° at the nadir.

• Azimuth (Az), that is the angle of the object around the horizon, usually measured from the north point towards the east.

• In former times, it was common to refer to azimuth from the south, as it was then zero at the same time the hour angle of a star was zero.

• This assumes, however, that the star (upper) culminates in the south, which is only true for most stars in the Northern Hemisphere.

Horizontal Coordinate System

• The horizontal coordinate system is fixed to the Earth, not the stars. Therefore, the altitude and azimuth of an object changes with time, as the object appears to drift across the sky.

• In addition, because the horizontal system is defined by your local horizon, the same object viewed from different locations on Earth at the same time will have different values of altitude and azimuth.

• Horizontal coordinates are very useful for determining the rise and set times of an object in the sky. When an object's altitude is 0°, it is on the horizon, if at that moment its altitude is increasing, it is rising, if its altitude is decreasing it is setting.

• However all objects on the celestial sphere are subject to the diurnal motion, which is always from east to west, so the inherent cumbersome determination whether altitude is increasing or decreasing can be easily found by considering the azimuth of the celestial object instead (referenced to North as 0°):

• if the azimuth is between 0° and 180° (north—east—south), it is rising.

• if the azimuth is between 180° and 360° (south—west—north), it is setting.

Equatorial Coordinate System

• The equatorial coordinate system is probably the most widely used celestial coordinate system, whose equatorial coordinates are:

• declination (δ) • right ascension (α) -also RA-, or hour angle (H) -also HA-

• It is the most closely related to the geographic coordinate system, because they use the same fundamental plane, and the same poles.

• The projection of the Earth's equator onto the celestial sphere is called the celestial equator.

• Similarly, projecting the geographic poles onto the celestial sphere defines the north and south celestial poles.

Equatorial Coordinate System

Equatorial Coordinate System

• There are two systems to specify the longitudinal (longitude-like) coordinate:

• the hour angle system is fixed to the Earth like the geographic coordinate system

• The right ascension system is fixed to the stars, thus, during a night or a few nights, it appears to move across the sky as the Earth spins and orbits under the fixed stars.

• Over long periods of time, precession and nutation effects alter the earth's orbit and thus the apparent location of the stars.

• When considering observations separated by long intervals, it is necessary to specify an epoch (frequently J2000.0, for older data B1950.0) when specifying coordinates of planets, stars, galaxies, etc.

Equatorial Coordinate System

• The latitudinal (latitude-like) angle of the equatorial system is called declination (Dec for short). It measures the angle of an object above or below the celestial equator.

• The longitudinal angle is called the right ascension (RA for short). It measures the angle of an object east of the vernal equinox point.

• Unlike longitude, right ascension is usually measured in hours instead of degrees, because the apparent rotation of the equatorial coordinate system is closely related to sidereal time and hour angle. Since a full rotation of the sky takes 24 hours of sidereal time to complete, there are (360 degrees / 24 hours) = 15 degrees in one hour of right ascension.

• The equatorial coordinate system is commonly used by telescopes equipped with equatorial mounts by employing Setting circles. Setting circles in conjunction with a star chart or ephemeris allow a telescope to be easily pointed at known objects on the celestial sphere

Conversion from Equatorial to Horizontal Coordinates

• Let δ be the declination and H the hour angle, φ be the observer's latitude, El be the elevation angle and Az the azimuth angle. Let θ be the zenith (or zenith distance, i.e. the 90° complement of Alt).

• Then the equations of the transformation are:

• Use the inverse trigonometric functions to get the values of the coordinates.

Conversion from Horizontal to Equatorial coordinates

• To convert horizontal to equatorial,

Ecliptic Coordinate System

• The ecliptic coordinate system is a celestial coordinate system that uses the ecliptic for its fundamental plane.

• The ecliptic is the path that the sun appears to follow across the sky over the course of a year.

• It is also the projection of the Earth's orbital plane onto the celestial sphere.

• The latitudinal angle is called the ecliptic latitude or celestial latitude (denoted β), measured positive towards the north.

• The longitudinal angle is called the ecliptic longitude or celestial longitiude (denoted λ), measured eastwards from 0° to 360°.

• Like right ascension in the equatorial coordinate system, the origin for ecliptic longitude is the vernal equinox.

• This choice makes the coordinates of the fixed stars subject to shifts due to the precession, so that always a reference epoch should be specified.

• Usually epoch J2000.0 is taken, but the instantaneous equinox of the day (called the epoch of date) is possible too.

Ecliptic Coordinate System

Ecliptic Coordinate System• This coordinate system can be particularly useful for charting

solar system objects. • Most planets (except Mercury), dwarf planets, and many

small solar system bodies have orbits with small inclinations to the ecliptic plane, and therefore their ecliptic latitude β is always small.

• Because of the planets' small deviation from the plane of the ecliptic, ecliptic coordinates were used historically to compute their positions.

• Conversion between celestial coordinate systems can be done as follows. In the formulas below : λ and β are the ecliptic longitude and ecliptic latitude, respectively; α and δ are the right ascension and declination, respectively; ε = 23.439 281° is the Earth's axial tilt.

• Conversion from Ecliptic coordinates to Equatorial coordinates• Declination δ and right ascension α are obtained from:

sin δ = sin ε sin λ cos β + cos ε sin β cos α cos δ = cos λ cos β

• Conversion from Equatorial Coordinates to Ecliptic Coordinatessin α cos δ = cos ε sin λ cos β - sin ε sin β

sin β = cos ε sin δ - sin α cos δ sin ε cos λ cos β = cos α cos δ

sin λ cos β = sin ε sin δ + sin α cos δ cos ε

Ecliptic Coordinate System – Precession & Nutation

• Due to the torques exerted primarily by the gravitational attraction of the moon and sun (and secondarily by other planets), the equator and the ecliptic of the Celestial Sphere precess and nutate.

• Precession, with a period of approximately 26000 years, consists of two components: luni-solar and planetary precession.

• The luni-solar effect causes a slow westerly circular motion of the pole of the equator relative to the pole of the ecliptic.

• The attraction of the planets causes an eastward motion of the equinox by about 12" (arcseconds) per century and a decrease of the obliquity by about 47" per century.

• Precession causes the equinox to move along with the equator by about 50" per century.

• Nutation is a short-period, irregular motion of the pole with a period ranging from 14 days to 18.6 years, and has a maximum amplitude of about 20".

• Both are described by the motion of the equator and equinox with respect to the fixed equator and equinox of a given epoch,

Ecliptic Coordinate System

Earth Centered Inertial (ECI) & Earth Centered Earth Fixed Coordinate Systems

ECI Coordinate System• The ECI coordinate system is typically defined as a Cartesian

coordinate system, where the coordinates (position) are defined as the distance from the origin along the three orthogonal (mutually perpendicular) axes.

• The z axis runs along the Earth's rotational axis pointing North, the x axis points in the direction of the vernal equinox (more on this in a moment), and the y axis completes the right-handed orthogonal system.

• The vernal equinox is an imaginary point in space which lies along the line representing the intersection of the Earth's equatorial plane and the plane of the Earth's orbit around the Sun or the ecliptic.

• Another way of thinking of the x axis is that it is the line segment pointing from the center of the Earth towards the center of the Sun at the beginning of Spring, when the Sun crosses the Earth's equator moving North.

• The x axis, therefore, lies in both the equatorial plane and the ecliptic.

• These three axes defining the Earth-Centered Inertial coordinate system are 'fixed' in space and do not rotate with the Earth.

Coordinate Systems

• Now to represent any point on the Physical Surface of the earth, you also need a suitable Coordinate System.

• An “Earth Centered Earth Fixed (ECEF) Coordinate System” is most popularly used. It is also sometimes known as “Conventional Terrestrial System”.

• In the figure below this coordinate system is shown. • Here the Physical Surface of the earth is simplified by an Ellipsoid.

The origin of the Coordinate System is taken as the centre of the Earth, which is also the centre of the Ellipsoid.

• The position of a location, P on the Physical Surface can be represented as an X, Y, and Z coordinate, which is known as Cartesian Coordinates of the Navigator.

• The position can be also represented as λ (Geodetic Longitude), Φ (Geodetic Latitude) and h (Height above Ellipsoid)

ECEF Coordinate System

ECEF Coordinate System

• The point (0,0,0) denotes the mass center of the earth, hence the name Earth- Centered.

• The Z-axis is defined as being parallel to the earth’s rotational axis, pointing towards north.

• The X-axis intersects the Ellipsoid at Semimajor axis of the Ellipsoid at 0° latitude, 0° longitude. This is actually the intersection of Greenwich Meridian with the semimajor axis of the Ellipsoid.

• This means the ECEF Coordinate system rotates with the earth around its Z-axis.

• Therefore, the coordinates of a point fixed on the surface of the earth do not change, hence the name earth-fixed. The Y axis is perpendicular to X axis along the semimajor axis.

• If you draw a normal from Point P to the Ellipsoid, it will not meet the origin of the Ellipsoid but somewhere else on the X-Y Plane, as shown.

• The Coordinates of Point P can be also represented as Φ, λ and h, which are known as Geodetic Latitude, Longitude and Height above the Ellipsoid respectively.

• These polar coordinates are more popular than the Cartesian Coordinates X,Y,Z in Satellite Navigation and satellite Geodesy.

ECEF Coordinate System

WGS84 is an ECEF coordinate system with:

• Its origin at the earth's centre of mass, the geocentre (for two reasons: the geocentre is the physical point about which the satellite orbits; and it is preferable to any local geodetic datum).

• Its "z-axis" is aligned parallel to the direction of the Conventional Terrestrial Pole (CTP) for polar motion, as originally defined by the Bureau International de l'Heure (BIH), and since 1989 by the International Earth Rotation Service (IERS).

• Its "x-axis" is the intersection of the WGS84 Reference Meridian Plane and the plane of the CTP Equator (the Reference Meridian being parallel to the Zero Meridian defined by BIH/IERS).

• Its "y-axis" completes a right-handed, earth-centered, earth-fixed (ECEF) orthogonal coordinate system, measured in the plane of the CTP Equator, 90 east of the x-axis.

WGS 84 Spheroid

The four defining parameters of the WGS84 ellipsoid are:

• Semi-major axis (a) : 6378137m.

• Ellipsoid flattening (f): 1/298.257223563 (derived from the value of the normalized second degree zonal harmonic coefficient of the gravitational field (J2) : -484.16685 x 10-6).

• Angular velocity of the earth (ωe) : 7292115 x 10-11 rad/sec.

• The earth's gravitational constant (atmosphere included) (GM) : 3986005 x 10-8 m3/sec2.

International Terrestrial Reference System (ITRS)

The ITRS definition fulfills the following conditions:

• It is geocentric, the center of mass being defined for the whole earth, including oceans and atmosphere.

• The unit of length is the meter (SI). This scale is consistent with the TCG time coordinate for a geocentric local frame, in agreement with IAU and IUGG (1991) resolutions. This is obtained by appropriate relativistic modeling.

• Its orientation was initially given by the BIH orientation at 1984.0.

• The time evolution of the orientation is ensured by using a no-net-rotation condition with regards to horizontal tectonic motions over the whole earth and corrected for Polar Motion and Earth Rotation.

International Terrestrial Reference Frame (ITRF)

• ITRF is the realization of International Terrestrial Reference System (ITRS).

• ITRS is a geocentric system. • The origin of the system is the centre of the mass of the Earth. • The unit of length is meter. • The orientation of the axes was established as consistent with that

of IERS’s predecessor, Bureau International de l’Heure, BIH, in 1984.

• The z-axis is the line from the center of Earth’s mass through the Conventional International Origin (CIO).

• Between 1900 and 1905, the mean position of Earth’s rotational pole was designated as the Conventional Terrestrial Pole (CTP).

• The x-axis is the line from the centre through the intersection of the zero meridian with the equator.

• The y-axis is the line from the centre to equator and perpendicular to x axis to make a right handed system.

International Terrestrial Reference Frame (ITRF)

• The ITRF is an alternative realization of WGS84 that is produced by the International Earth Rotation Service (IERS) based in Paris, France.

• It includes more than 500 stations at 290 sites all over the world.

• Four different space positioning methods contribute to the ITRF: Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR), GPS and Doppler Ranging Integrated on Satellite (DORIS).

• Each of these techniques has strengths and weaknesses but their combination produces a strong multipurpose Terrestrial Reference Frame (TRF).

• ITRF was created by the civil GPS community, quite independent of the US military organizations.

International Terrestrial Referrence Frame

• Each version of the ITRF is given a year code to identify it - the current version is ITRF2000.

• ITRF2000 is simply a list of coordinates (X, Y and Z in meters) and velocities (dX, dY and dZ in meters per year) of each station in the ITRF, together with the estimated level of error in those values.

• The coordinates relate to the epoch 2000.0 - to obtain the coordinates of a station at any other time. The station velocity is applied appropriately.

• This is to cope with the effects of tectonic plate motion.

• ITRF is of higher quality than the military WGS84

Tectonic Plates

ITRF

• ITRF has proven to be the most accurate global Terrestrial Reference Frame ever constructed.

• The ITRF is important to us for two reasons. • Firstly, we can use ITRF stations equipped with permanent GPS

receivers as reference points of known coordinates to precisely coordinate our own GPS stations, using GPS data downloaded from the Internet.

• This procedure is known as ‘fiducial GPS analysis’. • Secondly we can obtain precise satellite positions (known as

ephemeredes) in the ITRF2000. • Both these vital geodetic services are provided free on the Internet

by the International GPS Service.• ITRF not only takes care of coordinate variation due to Plate

Tectonics but also change of axes due to Polar motion and change in Earth’s rotation rate

Polar Motion

Earth’s Rotation Rate

Coordinate Transformation

• It is necessary to relate the coordinates determined using Satellite Geodesy (in ITRF or WGS 84) to the old coordinates, determined in local datums like Everest or Vice Versa.

• The most popular method is Bursa-Wolf Technique.

• The Bursa-Wolf method assumes a similarity three dimensioned relationship between two consistent sets of Cartesian coordinate through seven parameters

– Three Translations (DX, DY,DZ)– three rotations around X, Y, Z axis respectively ( ε, ψ, ω ) – a scale change ( DL )

Coordinate Transformation

Coordinate Transformation

• If U, V and W represent the Cartesian components of a station in reference frame 1 say Everest and X, Y, Z represent the Cartesian component of same stations in reference frame number 2 say WGS

– 84, the transformation can be expressed as :

• where R represents a 3 x 3 rotation matrix and defined a • R= R1 (ε) R2 (ψ) R3 (ω)

Rotation Matrices

1 0 0 R1 (ε) = [ 0 cos ε - sin ε ] 0 sin ε cos ε

cos ψ 0 - sin ψ R2 (ψ) = [ 0 1 0 ] sin ψ 0 cos ψ

cos ω - sin ω 0

R3 (ω) = [ sin ω cos ω 0 ] 0 0 1

• The Seven parameters can be determined by taking observations in both the coordinate systems at common Geodetic Points and using Least Square Technique

Everest to WGS 84 Transformation

• Mapwel Software available at http://www.embird.com/scripts/vysledokWGS84.php3

can be used to transfer coordinates from Everest to WGS 84 Spheroid.

• The results for Kalianpur is given below.

Everest WGS 84 Difference Latitude 240 07’ 11.26” N 240 07’ 8.56” N 2.7” (81 m)Longitude 770 39’ 17.63” E 770 39’ 13.0” E 15.72” (471 m)

• Mapwell software allows to transform 280+ local datum to WGS 84.

• Many of these transformations are performed with use of 7 parameter Helmert transformation

Everest to WGS 84 Transformation

Datum Ellipsoid dX dY dZ Region of Use εX εY εZ S Indian Everest (India 1830) 282 726 254 Bangladesh 10 8 12 6 Indian Everest (India 1956) 295 736 257 India, Nepal 12 10 15 7 Indian Everest(Pakistan) 283 682 231 Pakistan -1 -1 -1 0Indian 1954 Everest (India 1830) 217 823 299 Thailand 15 6 12 11Indian 1960 Everest (India 1830) 182 915 344 Vietnam 25 25 25 1 (Con SonIsland)Indian 1960 Everest (India 1830) 198 881 317 Vietnam 25 25 25 2 (Near 160 N)Indian 1975 Everest (India 1830) 210 814 289 Thailand 3 2 3 62

• This Table is published by Defense Mapping Agency, USA and taken from their web site. The units of dX, dY, dZ and εX, εY and εZ (error estimates) are in meters. The errors are less than about 25 m.

Position shift at a point Longitude = 970 44’ 25.19” W & Latitude = 300 16’ 28.82” N due to transformation from

WGS 84 Spheroid

Introduction to Map Projections

• What is a Map ? :

• A map is a graphical representation drawn to scale of natural and artificial features (objects) on the Earth's surface. Some of these features such as, roads, buildings or rivers, you would be able to see from a hill-top or aero plane. A map is a portrayal of the real world.

• Other features such as, names of places, boundaries or heights are added to the map because of the importance that they have for the map user. A map can tell us about things that are happening around us, close by and far away. It gives us this information without having to necessarily be at that place.

• The map maker has the task of bringing the real world to the map user. This is no easy task as the space available on the map is limited and the real world must be represented by symbols (points, lines and area fills).

Map Projection

• The process of making the map involves collecting data and making measurements (usually from aerial photographs now a days) of objects in the real world.

• This information is then translated into understandable symbols and names and other relevant information are added which the map user can interpret to get knowledge about the real world

• The Earth is round while the map is flat and so the map maker has to project the round surface on the Earth onto the flat surface on the map.

• This process is known as a map projection. There are different map projections, each with different properties of preserving true shape, area or distance.

Aerial Photo

The Corresponding Map

Uses of Maps

1. A map gives the location or position of places or features. The positions are usually given by the co-ordinates of the place, either as the Cartesian co-ordinates (x,y) in meters or as geographical co-ordinates (latitude and longitude) in degrees, minutes and seconds. The co-ordinates can be measured using the co-ordinate grid shown at set intervals along the borders of the map. The map user can, for example, find out that the position of Cape Town is 33º56' South latitude, 18º25' East longitude.

2. A map gives us the spatial relationship between features. For example: What province is the neighbor of another province? Which side of the road is the river on? Is there a dam on the farm? Where is the nearest railway station?

3. We can determine a lot of information from a map such as distances, directions and areas. We can measure the distance from Johannesburg to Durban, determine that Pretoria is to the north of Johannesburg, or calculate the size of the Gauteng province. In determining distances and areas the scale of the map has to be taken into consideration.

Types of Maps

• Being a representation of the real world on a limited size of paper means that a map is restricted as to what can be shown.

• The map maker (cartographer) has to select what to show and what to leave off.

• The map maker is guided by what the main purpose of the map is, such as a road map, a topographical map or a thematic map.

• A road map emphasizes the roads and towns but little else.

• A topographic map, also called a general map, shows as much of the landscape, elevations, roads, towns etc as possible.

• A thematic map is designed to depict a specific theme such as the population of various magisterial districts, the occurrence of crime in different districts, or annual rainfall.

Scale of a Map

• The scale of a map describes the relationship between a distance or size on the map and the corresponding distance or size on the ground.

• A map has a scale because it is not possible to show the whole area at its true size.

• A scale is normally given as a figure consisting of two parts, such as 1:50 000, the first part is a 1 and the second part 50 000.

• This means that 1 unit of measure on the map represents (equals) 50 000 units of measure on the ground - 1cm on the map represents (equals) 50 000cm on the ground.

Scale of a Map

• Scale can also be given on the map as a scale bar, which helps the user to measure distances from the map.

• Different maps have different scales. • A map with a scale of 1:50 000 is said to be a larger scale map than

a map with a scale of 1:250 000.

• This is because an object on the ground is shown bigger on a 1:50 000 scale map than on a 1:250 000 scale map.

• A 1:1 million scale map would be an even smaller scale map.

• However, the smaller the scale of the map the bigger the area that is covered on the map.

Map at a scale of 1 to 100000

Map at a scale of 1 to 250000

Map at a scale of 1 to 50,000

Map projection

• A map projection is any method used in cartography to represent the two-dimensional curved surface of the earth or other body on a plane.

• The term "projection" here refers to any function defined on the earth's surface and with values on the plane, and not necessarily a geometric projection.

• Flat maps could not exist without map projections, because a sphere cannot be laid flat over a plane without distortions. One can see this mathematically as a consequence of Gauss's Theorema Egregium.

• Flat maps can be more useful than globes in many situations: • (1) They are more compact and easier to store (2) They readily

accommodate an enormous range of scales (3) They are viewed easily on computer displays (4) They can facilitate measuring properties of the terrain being mapped (5) They can show larger portions of the earth's surface at once (6) They are cheaper to produce and transport.

• These useful traits of flat maps motivate the development of map projections.

Mercator Projection

Mercator Projection

Properties of Mercator Projection

• Type : cylindrical

• Properties :

1. Equator touches Cylinder,

2. Conformal true direction,

3. Reasonably true shapes and distances within 150 of Equator,

4. Rhumb lines (true directions between any two points) is a straight line),

5. Great distortion at high latitudes

• Regional Use : World, equatorial, east-west extent, large and medium scale

• General Use : Navigation, Large scale map series, USGS** maps

Transverse Mercator Projection

Properties of Transverse Mercator Projection

• Type : Cylindrical

• Properties :

1. Conformal,

2. Central Meridian selected by the Map maker touches Cylinder if the Cylinder is tangent.

3. Can show whole Earth, but the direction, distances and areas are reasonable accurate only within 150 of the Central Meridian.

4. No Straight Rhumb lines

• Regional Use : Continents/ Oceans, Equatorial/ Mid-latitude, North-South extent, large and medium scale, topographic large scale

• General Use : Map series, N.T.S. and USGS maps

Lambert Conformal Conic Projection

Properties of Lambert Conformal Conic Projection

• Type : A Lambert conformal conic projection (LCC) is a conic map projection, which is often used for aeronautical charts.

• In essence, the projection superimposes a cone over the sphere of the Earth, with two reference parallels secant to the globe and intersecting it.

• Properties : 1. Conformal true direction 2. Minimizes distortion from projecting a three dimensional surface to

a two-dimensional surface. 3. Distortion is least along the standard parallels, and increases

further from the chosen parallels. 4. Pilots favor these charts because a straight line drawn on a

Lambert conformal conic projection approximates a great-circle route between endpoints.

• Regional Use : Continents/ oceans, Equatorial/ mid-latitude, East-west extent, large and medium scale

• General Use : Mapping countries of Canada and USA, National Atlas of Canada 5th edition, IMW

Alber’s Projection

Properties of Alber’s Projection

• Type : The Albers equal-area conic projection, or Albers projection, is a conic, equal area map projection that uses two standard parallels.

• Properties :

1. Although scale and shape are not preserved, distortion is minimal between the standard parallels.

2. Scale & Distortion are constant any given Parallel• Use :

1. Frequently used in the ellipsoidal form for maps of the United States in the National Atlas of the United States, for thematic maps, and for world atlases. Also used and recommended for equal-area maps of regions that are predominantly east-west in

extent. • The Albers projection is the standard projection for

British Columbia.

Miller Cylindrical Projection

Miller Cylindrical Projection

Properties of Miller Cylindrical Projection

• Type : • The Miller cylindrical projection is a modified Mercator

projection, proposed by Osborn Maitland Miller.

• The parallels of latitude are scaled by a factor of 0.8, projected according to Mercator, and then the result is divided by 0.8 to retain scale along the equator.

• Only the Equator is true to scale

• Regional Use : World

• General Use : Thematic, Reference maps, USGS maps

Space Oblique Mercator Projection

Properties of Space Oblique Mercator Projection

• The Space Oblique Mercator is a projection designed to show the curved ground-track of IRS images.

• There is little distortion along the ground-track but only within the narrow band (about 15 degrees) of the IRS image.

• Modified Cylindrical Projection with map surface defined by Satellite Orbit.

• Designed especially for continuous mapping of Satellite Imagery.• Groundtrack of Satellite, a curved line on the globe, is shown as a

curved line on the map and is continuously true to scale as orbiting continues.

• All meridians and parallels are curved lines, except the meridian at each polar approach.

• Recommended only for a relatively narrow band along the ground track.