3. georeferencing projection

8/19/2019 3. Georeferencing Projection

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3.Georeferencing and Projection

Understanding Earth

Earth’s shape

The earth is generally viewed as a sphere; however its shape is not as perfect as a sphere in reality. Given beloware the models that have attempted to describe the shape of the earth:

Spherical model

Based on a circle, it treats earth as a sphere to make mathematical calculations easier.

Ellipsoid/ Oblate spheroid model

Based on an ellipse, rotating an ellipse around the semi-minor axis creates an ellipsoid.Latitude, longitude and planar coordinate systems are determined with respect to the ellipsoid.

Earth is flattened at poles with a bulge at equator and this is attributed to the earth’s rotation. Rotation of

earth has centrifugal force associated with it, which causes an object to move away from the centre ofgravity. The force is greatest at equator causing an outward bulge and thus giving that region a largercircumference

Geoid model

Describes unique and irregular shape of the earth. The variation in the density of different rock types andirregularities caused by mountain ranges and ocean depths affect the gravity of earth.

Geoid can be perceived as a sea level surface (where dynamic effects such as tides and waves areexcluded) whose irregular shape is attributed to the earth’s gravity

No simple surface such as sphere or spheroid/ellipsoid can model the sea level surface completely so bestfit of the spheroid/ellipsoid to the sea level surface is performed.


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Figure 1: Representation of geoid model

The geoid differs from the shape of ellipsoid by upto ± 100 m and this difference is known as geoid

separation or geoid undulation

Elevations and contour lines depicted on maps are measured with respect to the geoid

Datum

A datum is a reference point or surface against which measurements are made using models of the shape of theearth.

Vertical Datum: A vertical datum is a reference surface used to measure elevations of the point on earth’ssurface. It is tidal, based on sea level, or geodetic, based on ellipsoid

The tidal vertical datum takes local mean sea level as reference for height measurement. Mean sea level is thearithmetic mean of the hourly water elevation taken over a specific 19 years cycle which is defined as zeroelevation for local area and is close approximation to the geoid (geoid and local mean sea level differ by notmore than a couple of meters). As zero elevation defined for one country is not necessarily same for othercountries, therefore a number of local vertical datums are defined.


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Fi gure 2 : Vertical Datum

The mean sea level height is also known as orthometric height or geoid height.

The geodetic vertical datum uses ellipsoid as the reference surface. The surface of the ellipsoid is considered torepresent zero altitude. Points above the ellipsoid represent positive altitude and points below the surfacerepresent negative altitude. The altitude is also known as ellipsoidal or geodetic height. GPS devices furnishellipsoidal heights.

The relationship between ellipsoidal height H and geoid height h is given as

where N refers to the geoid ellipsoid separation.

Horizontal datum : A horizontal or geodetic datum is defined as an ellipsoid which is used as a referencesurface for the planimetric measurements on the Earth surface usually expressed in latitudes and longitudes. Itcan be of two types:

a. Local geodetic datum: The one which best approximates the size and shape of a particular part of earth’ssea level surface. The centre of this spheroid doesn’t coincide with centre of mass of the earth


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Fi gure 3: L ocal geodetic datum

b. Global/Geocentric datum: The one that best approximates the size and shape of the whole earth. Thecentre of this spheroid coincides with centre of mass of the earth. The US Global Positioning System usesgeocentric datum

F igur e 4: Geocentr ic datum


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The use of local datums results in uneven connectivity of longitudinal and latitudinal lines between differentcountries/regions. These mismatches were common over hundred meters and created confusion about locatingan area correctly. With the advent of Global Positioning System (GPS) technology this disagreement was nolonger acceptable. World-wide datums which are now used in all countries/regions began to be developed.

The datum presently used for GPS is called WGS 84 (World Geodetic System 1984). It consists of a three-dimensional Cartesian coordinate system and an associated ellipsoid. The positions can either be described asXYZ Cartesian coordinates or latitude, longitude and ellipsoid height coordinates. The origin of the datum is thecentre of mass of the Earth and it is designed for positioning anywhere on Earth.

Coordinate System

A coordinate system is a reference system used for locating objects in a two or three dimensional space

Geographic Coordinate System

A geographic coordinate system, also known as global or spherical coordinate system is a reference system thatuses a three-dimensional spherical surface to determine locations on the earth. Any location on earth can bereferenced by a point with longitude and latitude.

We must familiarize ourselves with the geographic terms with respect to the Earth coordinate system in order touse the GIS technologies effectively.

Pole: The geographic pole of earth is defined as either of the two points where the axis of rotation of the earth


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meets its surface. The North Pole lies 90º north of the equator and the South Pole lies 90º south of the equator

Latitude : Imaginary lines that run horizontally around the globe and are measured from 90º north to 90º south.Also known as parallels, latitudes are equidistant from each other.

Equator : An imaginary line on the earth with zero degree latitude, divides the earth into two halves–Northernand Southern Hemisphere. This parallel has the widest circumference.

Figure 5: Division of earth into hemispheres

Longitude : Imaginary lines that run vertically around the globe. Also known as meridians, longitudes aremeasured from 180º east to 180º west. Longitudes meet at the poles and are widest apart at the equator

Prime meridian : Zero degree longitude which divides the earth into two halves–Eastern and Western

hemisphere. As it runs through the Royal Greenwich Observatory in Greenwich, England it is also known asGreenwich meridian


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Figure 6: Latitude and longitude measurements

Equator (0º) is the reference for the measurement of latitude. Latitude is measured north or south of theequator. For measurement of longitude, prime meridian (0º) is used as a reference. Longitude is measured eastor west of prime meridian. The grid of latitude and longitude over the globe is known as graticule. Theintersection point of the equator and the prime meridian is the origin (0, 0) of the graticule.


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Coordinate measurement

The geographic coordinates are measured in angles. The angle measurement can be understood as per following:

A full circle has 360 degrees 1 circle = 360º

A degree is further divided into 60 minutes 1º = 60′

A minute is further divided into 60 seconds 1′ = 60″

An angle is expressed in Degree Minute Second.

While writing coordinates of a location, latitude is followed by longitude. For example, coordinates of Delhi iswritten as 28° 36′ 50″ N, 77° 12′ 32″ E.

Decimal Degree is another format of expressing the coordinates of a location. To convert a coordinate pair from

degree minute second to decimal degree following method is adopted:

We have 28 full degrees, 36 minutes - each 1/60 of a degree, and 50 seconds - each 1/60 of 1/60 of a degree

While writing coordinates of a location, latitude is followed by longitude. For example, coordinates of Delhi iswritten as

Similarly 77° 12′ 32″ can be written as 77.2088. So, we can write coordinates of Delhi in decimal degree formatas: 28.6138 N, 77.2088 E

Local Time and Time Zones

With rotation of earth on its axis, at any moment one of the longitudes faces the Sun (noon meridian), and atthat moment, it is noon everywhere on it. After 24 hours the earth completes one full rotation with respect to theSun, and the same meridian again faces the noon. Thus each hour the Earth rotates by 360/24 = 15 degrees.


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This implies that with every 15º of longitude change a new time zone is created which is marked by a differenceof one hour from the neighboring longitudes specified at 15º gap. The earth's time zones are measured from theprime meridian (0º) and the time at Prime meridian is called Greenwich Mean Time. Thus, there are 24 timezones created around the globe.

Date

The International Date Line is the imaginary line on the Earth that separates two consecutive calendar days.Generally, it is said to be lying exactly opposite to the prime meridian having a measurement of 180º meridianbut it is not so. It zigs and zags the 180º meridian following the political jurisdiction of the states but for sake ofsimplicity it is taken as 180º meridian. Starting at midnight and going east to the International Date Line, thedate is one day ahead of the date on the rest of the Earth.

Projected Coordinate system

A projected coordinate system is defined as two dimensional representation of the Earth. It is based on a

spheroid geographic coordinate system, but it uses linear units of measure for coordinates. It is also known asCartesian coordinate system.

In such a coordinate system the location of a point on the grid is identified by (x, y) coordinate pair and theorigin lies at the centre of grid. The x coordinate determines the horizontal position and y coordinate determinesthe vertical position of the point.

Figure 7: Cartesian coordinate system


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In such a coordinate system the location of a point on the grid is identified by (x, y) coordinate pair and theorigin lies at the centre of grid. The x coordinate determines the horizontal position and y coordinate determinesthe vertical position of the point.

Map Projection

Map projection is a mathematical expression using which the three-dimensional surface of earth is represented in atwo dimensional plane. The process of projection results in distortion of one or more map properties such as shape,size, area or direction.

A single projection system can never account for the correct representation of all map properties for all the regions ofthe world. Therefore, hundreds of projection systems have been defined for accurate representation of a particularmap element for a particular region of the world.

Classification of Map Projections

Map projections are classified on the following criteria:

Method of construction

Development surface used Projection properties Position of light source


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Method of Construction

The term map projection implies projecting the graticule of the earth onto a flat surface with the help of shadow cast.However, not all of the map projections are developed in this manner. Some projections are developed usingmathematical calculations only. Given below are the projections that are based on the method of construction:

Perspective Projections : These projections are made with the help of shadow cast from an illuminated globe onto a developable surface

Non Perspective Projections :These projections do not use shadow cast from an illuminated globe on to adevelopable surface. A developable surface is only assumed to be covering the globe and the construction ofprojections is done using mathematical calculations.

Development Surface Projection transforms the coordinates of earth on to a surface that can be flattened to a plane without distortion(shearing or stretching). Such a surface is called a developable surface. The three basic projections are based on thetypes of developable surface and are introduced below:

1. Cylindrical Projection

It can be visualized as a cylinder wrapped around the globe.

Once the graticule is projected onto the cylinder, the cylinder is opened to get a grid like pattern of latitudesand longitudes.

The longitudes (meridians) and latitudes (parallels) appear as straight lines

Length of equator on the cylinder is equal to the length of the equator therefore is suitable for showingequatorial regions.


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Aspects of cylindrical projection:

(a) (b) (c)


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(a) Normal: when cylinder has line of tangency to the equator. It includes Equirectangular Projection, the Mercatorprojection, Lambert's Cylindrical Equal Area, Gall's Stereographic Cylindrical, and Miller cylindrical projection.

(b) Transverse: when cylinder has line of tangency to the meridian. It includes the Cassini Projection, TransverseMercator, Transverse cylindrical Equal Area Projection, and Modified Transverse Mercator.

(c) Oblique: when cylinder has line of tangency to another point on the globe. It only consists of the Oblique Mercatorprojection.

2. Conic Projection

It can be visualized as a cone placed on the globe, tangent to it at some parallel.

After projecting the graticule on to the cone, the cone is cut along one of the meridian and unfolded. Parallelsappear as arcs with a pole and meridians as straight lines that converge to the same point.

It can represent only one hemisphere, at a time, northern or southern.

Suitable for representing middle latitudes.


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Aspects of conic projection:

(a) Tangent: when the cone is tangent to only one of the parallel.

(b) Secant: when the cone is not big enough to cover the curvature of earth, it intersects the earth twice at twoparallels.

3. Azimuthal/Zenithal Projection

It can be visualized as a flat sheet of paper tangent to any point on the globe The sheet will have the tangent point as the centre of the circular map, where meridians passing through the

centre are straight line and the parallels are seen as concentric circle.

Suitable for showing polar areas


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Aspects of zenithal projection:


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(a) Equatorial zenithal: When the plane is tangent to a point on the equator.

(b) Oblique zenithal: when the plane is tangent to a point between a pole and the equator.

(c) Polar zenithal: when the plane is tangent to one of the poles.

Projection Properties

According to properties map projections can be classified as:

Equal area projection: Also known as homolographic projections. The areas of different parts of earth are correctlyrepresented by such projections.

True shape projection: Also known as orthomorphic projections. The shapes of different parts of earth arecorrectly represented on these projections.

True scale or equidistant projections: Projections that maintain correct scale are called true scale projections.However, no projection can maintain the correct scale throughout. Correct scale can only be maintained along someparallel or meridian.


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Position of light source

Placing light source illuminating the globe at different positions results in the development of different projections.These projections are:

Gnomonic projection: when the source of light is placed at the centre of the globe

Stereographic Projection: when the source of light is placed at the periphery of the globe, diametrically oppositeto the point at which developable surface touches the globe

Orthographic Projection: when the source of l ight is placed at infinity from the globe opposite to the point atwhich developable surface touches the globe

Figure 8: Projections and position of light source

(Imagesource:http://www.geo.hunter.cuny.edu/~jochen/GTECH201/Lectures/Lec6concepts/Map%20coordinate%20systems/Perspective.htm)

http://www.geo.hunter.cuny.edu/~jochen/GTECH201/Lectures/Lec6concepts/Map%20coordinate%20systems/Perspective.htmhttp://www.geo.hunter.cuny.edu/~jochen/GTECH201/Lectures/Lec6concepts/Map%20coordinate%20systems/Perspective.htmhttp://www.geo.hunter.cuny.edu/~jochen/GTECH201/Lectures/Lec6concepts/Map%20coordinate%20systems/Perspective.htmhttp://www.geo.hunter.cuny.edu/~jochen/GTECH201/Lectures/Lec6concepts/Map%20coordinate%20systems/Perspective.htm


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Constructing Map Projections ( Adapted from Singh, G 2004, Map work and practical geography )

Cylindrical Projection Let us draw a network of Simple cylindrical Projection for the whole globe on the scale of 1: 400,000,000 spacingmeridians and parallels at 30º interval

Calculations:


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Figure 9 : Simple cylindrical projection graticule

Steps of constrution:

Draw a line AB, 9.975 cm long to represent the equator. The equator is a circle on the globe and is subtended

by 360º. Since the meridians are to be drawn at an interval of 30º divide AB into 360/30 or 12 equal parts.

The length of a meridian is equal to half the length of the equator i.e. 9.975/2 or 4.987 cm.

To draw meridians, erect perpendiculars on the points of divisions of AB. Take these perpendiculars equal tothe length specified for a meridian and keep half of their length on either side of the equator.

A meridian on a globe is subtended by 180º. Since the parallels are to be drawn at an interval of 30º, dividethe central meridian into 180/30 i.e. 6 parts.

Through these points of divisions draw lines parallel to the equator. These lines will be parallels of latitude.Mark the equator and the central meridian with 0º and the parallels and other meridians. EFGH is the required

graticule.

Conical Projection

Let us draw a graticule on simple conical projection with one standard parallel on the scale of 1: 180,000,000 for thearea extending from the equator to 90º N latitude and from 60º W longitude to 100º E longitude with parallels spacedat 15º interval, meridians at 20º, and standard parallel 45º N.

Calculations:


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Steps of construction:

1. Draw a circle with a radius of 3.527 cm that represents the globe. Let NS be the polar diameter and WE be theequatorial diameter which intersect each other at right angles at O.

2. To draw the standard parallel 45º N, draw OP making an angle of 45º with OE.3. Draw QP tangent to OP and extend ON to meet PQ at point Q.4. Draw OA making an angle equal to the parallel interval i.e. 15º with OE.

Draw line LM, it represents the central meridian


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With L as the centre and QP as the radius, draw an arc intersecting LM at n. This arc describes the standard parallel

45º N.

The distance between the successive parallels is 15º. The length of the arc subtended by 15º is calculated as under:

From point n, mark off distances nr, rs, st, nu, uv and vM , each distance being equal to 0.923 cm. With L as centre,

draw arcs passing through the points t, s, r, u, v and M. These arcs represent the parallels.

5. Draw OB making an angle of 20° with OW Length of the arc subtended by 20° is calculated as under:


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1. With O as centre and radius equal to the arc WB (1.231 cm) draw arc abc.2. From point b, drop perpendicular bd on line ON. Now db is the distance between the meridians.

Keeping in view the number of meridians to be drawn, mark off distances along the standard parallel toward the east

and west of the point n, each distance being equal to db.

Join point L with the points of divisions marked on the standard parallel and produce them to meet the equator.

Fi gure 10 : Simple conic projection


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Azimuthal Projection

Let us draw Polar zenithal equal area projection for the northern hemisphere on the scale of 1: 200,000,000 spacingparallels at 15° interval and meridians at 30° interval.

Calculations:


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Steps of construction:

Draw a circle with radius equal to 3.175 cm representing a globe. Let NS and WE be the polar and equatorialdiameter respectively which intersect each other at right angles at O, the centre of the circle.

Draw radii Oa, Ob, Oc, Od, and Oe making angles of 15°, 30°, 45°, 60° and 75° respectively with OE. Join Ne,

Nd, Nc, Nb, Na and NE by straight lines. With radius equal to Ne, and N’ as centre draw a circle. This circle represents 75° parallel. Similarly with centre

N’ and radii equal to Nd, Nc, Nb, Na and NE draw circles to represent the parallels of 60°, 45°, 30°, 15° and 0°

respectively. Draw straight lines AB and CD intersecting each other at the centre i.e. point N. Radius N’B represents 0° meridian, N’A 180° meridian, N’D 90° E meridian and N’C 90° W meridian. Using protractor, draw other radii at 30° interval to represent other meridians

Fi gure 11 : Polar zenithal equal area projection


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Selection of Map Projection

Choosing a correct map projection for an area depends on the following:

Map Purpose

Considering the purpose of the map is important while choosing the map projection. If a map has a specific purpose,one may need to preserve a certain property such as shape, area or direction

On the basis of the property preserved, maps can be categorized as following

a. Maps that preserve shapes.

Used for showing local directions and representing the shapes of the features. Such maps include:

Topographic and cadastral maps. Navigation charts (for plotting course bearings and wind direction).

Civil engineering maps and military maps. Weather maps (for showing the local direction in which weather systems are moving).

b. Maps that preserve area

The size of any area on the map is in true proportion to its size on the earth. Such projections can be used to show

Density of an attribute e.g. population density with dots Spatial extent of a categorical attribute e.g. land use maps Quantitative attributes by area e.g. Gross Domestic Product by country World political maps to correct popular misconceptions about the relative sizes of countries.

c. Maps that preserve scale

Preserves true scale from a single point to all other points on the map. The maps that use this property include:

Maps of airline distances from a single city to several other cities Seismic maps showing distances from the epicenter of an earthquake Maps used to calculate ranges; for example, the cruising ranges of airplanes or the habitats of animal species


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d. Maps that preserve direction

On any Azimuthal projection, all azimuths, or directions, are true from a single specified point to all other points onthe map. On a conformal projection, directions are locally true, but are distorted with distance.

General purpose maps

There are many projections which show the world with a balanced distortion of shape and area. Few of these areWinkel Tripel, Robinson and Miller Cylindrical.For larger-scale maps, from continents to large countries, equidistantprojections are good at balancing shape and area distortion. Depending on the area of interest, one might useAzimuthal Equidistant,Equidistant Conic and Plate Carrée.

Study area

Geographical location

The line of zero distortion for a cylindrical projection is equator. For conical projections it is parallels and for Azimuthalit is one of the poles. If the study area is in tropics use cylindrical projection, for middle latitudes use conical and forPolar Regions use Azimuthal projections.

Shape of the area

Young in 1920 described a way of selecting the map projection which is known as Young’s rule. According to this rule,if the ratio of maximum extent (z) (measured from the centre of the country to its most distant boundary) and thewidth (δ) of the country comes out to be less than 1.41, Azimuthal projection is preferable. If the ratio is greater than

1.41 a conical or cylindrical projection should be used.Z/δ < 1.41 Azimuthal Projection Z/δ >1.41 Conical or Cylindrical projections


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Projection Systems

Given below is the description of the projection systems that are mostly used:

Cylindrical Projection I. Equirectangular projection This is a Projection on to a cylinder which is tangent to the equator. It is believed to be invented by Marinus of Tyre,about A.D. 100.

(Image source: http://en.wikipedia.org/wiki/File:Equirectangular_projection_SW.jpg )

1. Poles are straight lines equal in length to the equator

2. Meridians are straight parallel lines, equally spaced and are half as long as the equator. All meridians are ofsame length therefore scale is true along all meridians.

3. Parallels are straight, equally spaced lines which are perpendicular to the meridians and are equal to thelength of the equator.

4. Length of the equator on the map is the same as that on the globe but the length of other parallels on map ismore than the length of corresponding parallels on the globe. So the scale is true only along the Equator andnot along other parallels.

5. Distance between the parallels and meridians remain same throughout the map.

6. Since the projection is neither equal area nor orthomorphic, maps on this projection are used for generalpurposes only.

http://en.wikipedia.org/wiki/File:Equirectangular_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Equirectangular_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Equirectangular_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Equirectangular_projection_SW.jpg


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II. Lambert's cylindrical equal-area projection

It is devised by JH Lambert in 1772. It is a normal perspective projection onto a cylinder tangent at the equator

(Image source: http://en.wikipedia.org/wiki/File:Lambert_cylindrical_equal-area_projection_SW.jpg )

1. Parallels and meridians are straight lines

2.

The meridians intersect parallels at right angles3. The distance between parallels decrease toward the poles but meridians are equally spaced

4. The length of the equator on this projection is same as that on globe but other parallels are longer thancorresponding parallels on globe. So, the scale is true along the equator but is exaggerated along otherparallels

5. Shape and scale distortions increase near points 90 degrees from the central line resulting in verticalexaggeration of Equatorial regions with compression of regions in middle latitudes

6. Despite the shape distortion in some portions of a world map, this projection is well suited for equal-areamapping of regions which are predominantly north-south in extent, which have an oblique central line, orwhich lie near the Equator.

http://en.wikipedia.org/wiki/File:Lambert_cylindrical_equal-area_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Lambert_cylindrical_equal-area_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Lambert_cylindrical_equal-area_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Lambert_cylindrical_equal-area_projection_SW.jpg


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III. Gall's stereographic cylindrical projection Invented by James Gall in 1855, this projection is a cylindrical projection with two standard parallels at 45ºN and45ºS.

(Image source: http://en.wikipedia.org/wiki/File:Gall%E2%80%93Peters_projection_SW.jpg )

Poles are straight lines.

Meridians are straight lines and are equally spaced.

Parallels are straight lines but the distance between them increases away from the equator.

Shapes are true at the standard parallels. Distortion increases on moving away from these latitudes and ishighest at the poles.

Scale is true in all directions along 45ºN and 45ºS.

Used for world maps in British atlases.

http://en.wikipedia.org/wiki/File:Gall%E2%80%93Peters_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Gall%E2%80%93Peters_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Gall%E2%80%93Peters_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Gall%E2%80%93Peters_projection_SW.jpg


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IV. Mercator projection

Gerardus Mercator in 1569 invented this projection.

(Image source: http://en.wikipedia.org/wiki/File:Mercator_projection_SW.jpg )

Parallels and meridians are straight lines

Meridians intersect parallels at right angle

Distance between the meridians remains the same but distance between the parallels increases towards thepole

The length of equator on the projection is equal to the length of the equator on the globe whereas otherparallels are drawn longer than what they are on the globe, therefore the scale along the equator is correct butis incorrect for other parallels

As scale varies from parallel to parallel and is exaggerated towards the pole, the shapes of large sized

countries are distorted more towards pole and less towards equator. However, shapes of small countries arepreserved

The image of the poles are at infinity

Commonly used for navigational purposes, ocean currents and wind direction are shown on this projection

http://en.wikipedia.org/wiki/File:Mercator_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Mercator_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Mercator_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Mercator_projection_SW.jpg


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V.Transverse Mercator

This projection results from projecting the sphere onto a cylinder tangent to a central meridian.

(Image source: http://en.wikipedia.org/wiki/Transverse_Mercator_projection )

Only centre meridian and equator are projected as straight lines. The other parallels and meridians areprojected as curves.

The meridians and the parallels intersect at right angles

Small shapes are maintained but larger shapes distort away from the central meridian.

The area distortion increases with distance from the central meridian

Used to portray areas with larger north-south extent. British National Grid is based on this projection only.

http://en.wikipedia.org/wiki/Transverse_Mercator_projectionhttp://en.wikipedia.org/wiki/Transverse_Mercator_projectionhttp://en.wikipedia.org/wiki/Transverse_Mercator_projectionhttp://en.wikipedia.org/wiki/Transverse_Mercator_projection


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Pseudo-cylindrical Projections A pseudo cylindrical projection is that projection in which latitudes are parallel straight lines but meridians are curved.

I. Mollweide Projection

(Image source: http://www.mapthematics.com/ProjectionsList.php?Projection=69#Mollweide )

The poles are points and the central meridian is a straight line

The meridians 90º away from central meridians are circular arcs and all other meridians are elliptical arcs.

The parallels are straight but unequally spaced.

Scale is true along 40º 44' North and 40º 44' South.

Equal –area projection

Used for preparing world maps

http://www.mapthematics.com/ProjectionsList.php?Projection=69#Mollweidehttp://www.mapthematics.com/ProjectionsList.php?Projection=69#Mollweidehttp://www.mapthematics.com/ProjectionsList.php?Projection=69#Mollweidehttp://www.mapthematics.com/ProjectionsList.php?Projection=69#Mollweide


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II. Sinusoidal Projection

(Image source: http://www.mapthematics.com/ProjectionsList.php?Projection=110#sinusoidal )

The central meridian is a straight line and all other meridians are equally spaced sinusoidal curves.

The parallels are straight lines that intersect centre meridian at right angles.

Shape and angles are correct along the central meridian and equator

The distortion of shape and angles increases away from the central meridian and is high near the edges

Equal area projection

Used for world maps illustrating area characteristics. Used for continental maps of South America, Africa, andoccasionally other land masses, where each has its own central meridian.

http://www.mapthematics.com/ProjectionsList.php?Projection=110#sinusoidalhttp://www.mapthematics.com/ProjectionsList.php?Projection=110#sinusoidalhttp://www.mapthematics.com/ProjectionsList.php?Projection=110#sinusoidalhttp://www.mapthematics.com/ProjectionsList.php?Projection=110#sinusoidal


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III. Eckert VI

(Image source: http://www.mapthematics.com/ProjectionsList.php?Projection=54#Eckert%20VI )

Parallels are unequally spaced straight lines.

Meridians are equally spaced sinusoidal curves.

The poles and the central meridians are straight lines and half as long as equator.

It stretches shapes and scale by 29% in the north-south direction, along the equator. This stretching reducesto zero at 49º 16' N and 49º 16' S.

The areas near the poles are compressed in north-south direction.

Suitable for thematic mapping of the world.

http://www.mapthematics.com/ProjectionsList.php?Projection=54#Eckert%20VIhttp://www.mapthematics.com/ProjectionsList.php?Projection=54#Eckert%20VIhttp://www.mapthematics.com/ProjectionsList.php?Projection=54#Eckert%20VIhttp://www.mapthematics.com/ProjectionsList.php?Projection=54#Eckert%20VI


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Conical Projection

I. Bonne’s Projection

(Image source: http://en.wikipedia.org/wiki/File:Bonne_projection_SW.jpg )

Pole is represented as a point and parallels as concentric arcs of circles

Scale along all the parallels is correct

Central meridian is a straight line along which the scale is correct.

Other meridians are curved and longer than corresponding meridians on the globe. Scale along meridiansincreases away from the central meridian

Central meridian intersects all parallels at right angle. Other meridians intersect standard parallel at right anglebut other parallels obliquely. Shape is only preserved along central meridian and standard parallel

The distance and scale between two parallels are correct. Area between projected parallels is equal to the areabetween the same parallels on the globe. Therefore, is an equal area projection

Maps of European countries are shown in this projection. It is also used for preparing topographical sheets ofsmall countries of middle latitudes.

http://en.wikipedia.org/wiki/File:Bonne_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Bonne_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Bonne_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Bonne_projection_SW.jpg


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II. Polyonic Projection

(Image source: http://en.wikipedia.org/wiki/File:Polyconic_projection_SW.jpg )

The parallels are arcs of circles with different centers

Each parallel is a standard parallel i.e. each parallel is developed from a different cone

Equator is represented as a straight line and the pole as a point

Parallels are equally spaced along central meridian but the distance between them increases away from thecentral meridian.

Scale is correct along every parallel.

Central meridian intersects all parallels at right angle so the scale along it, is correct. Other meridians arecurved and longer than corresponding meridians on the globe and so scale along meridians increases awayfrom the central meridian.

It is used for preparing topographical sheets of small areas.

http://en.wikipedia.org/wiki/File:Polyconic_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Polyconic_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Polyconic_projection_SW.jpghttp://en.wikipedia.org/wiki/File:Polyconic_projection_SW.jpg


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Azimuthal/Zenithal Projection

I. Polar Zenithal Equal area projectionThis projection is invented by J.H Lambert in the year 1772. It is also known as Lambert’s Equal Area Projection.

(Image source: http://www.mapthematics.com/ProjectionsList.php?Projection=167#zenithal%20equal-area )

The pole is a point forming the centre of the projection and the parallels are concentric circles.

The meridians are straight lines radiating from pole having correct angular distance between them.

The meridians intersect the parallels at right angles.

The scale along the parallels increases away from the centre of the projection. The decrease in the scale along meridians is in the same proportion in which there is an increase in the scale

along the parallels away from the centre of the projection. Thus the projection is an equal area projection.

Shapes are distorted away from the centre of the projection. Scale along the meridians is small and along theparallels is large so the shapes are compressed along the meridians but stretched along the parallels.

Used for preparing political and distribution maps of polar regions. It can also be used for preparing generalpurpose maps of large areas in Northern Hemisphere.

http://www.mapthematics.com/ProjectionsList.php?Projection=167#zenithal%20equal-areahttp://www.mapthematics.com/ProjectionsList.php?Projection=167#zenithal%20equal-areahttp://www.mapthematics.com/ProjectionsList.php?Projection=167#zenithal%20equal-areahttp://www.mapthematics.com/ProjectionsList.php?Projection=167#zenithal%20equal-area


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II. Polar Zenithal Equidistant Projection

(Image source: http://www.mapthematics.com/ProjectionsList.php?Projection=168#zenithal%20equidistant )




The spacing between the parallels represent true distances, therefore the scale along the meridians is correct.

The scale along the parallels increases away from the centre of the projection.

The exaggeration and distortion of shapes increases away from the centre of the projection.

The projection is neither equal area nor orthomorphic.

It is used for preparing maps of polar areas for general purposes.

http://www.mapthematics.com/ProjectionsList.php?Projection=168#zenithal%20equidistanthttp://www.mapthematics.com/ProjectionsList.php?Projection=168#zenithal%20equidistanthttp://www.mapthematics.com/ProjectionsList.php?Projection=168#zenithal%20equidistanthttp://www.mapthematics.com/ProjectionsList.php?Projection=168#zenithal%20equidistant


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III. Gnomonic ProjectionIt is also known as great-circle sailing chart.

(Image source: http://www.mapthematics.com/ProjectionsList.php?Projection=156#gnomonic )




The parallels are unequally spaced. The distances between the parallels increase rapidly toward the margin ofthe projection. This causes exaggeration of the scale along the meridians.

The scale along the parallels increases away from the centre of the projection.

The exaggeration and distortion of shapes increases away from the centre of the projection. The exaggerationin the meridian scale is greater than that in any other zenithal projection.

It is neither equal area nor orthomorphic.

An arc on the globe which is a part of a great circle is represented as a straight line on this projection. This isbecause the radii from the centre of the globe are produced to meet the plane placed tangentially at the pole.

It is used to show great-circle paths as straight lines and thus to assist navigators and aviators in determiningappropriate courses.

http://www.mapthematics.com/ProjectionsList.php?Projection=156#gnomonichttp://www.mapthematics.com/ProjectionsList.php?Projection=156#gnomonichttp://www.mapthematics.com/ProjectionsList.php?Projection=156#gnomonichttp://www.mapthematics.com/ProjectionsList.php?Projection=156#gnomonic


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IV. Sereographic Projection

(Image source: http://www.mapthematics.com/ProjectionsList.php?Projection=162#stereographic )




The parallels are unequally spaced. The distances between the parallels increase toward the margin of theprojection. The exaggeration in the meridian scale is less than that in the case of Gnomonic projection.

The scale along the parallels also increases away from the meridian and in the same proportion in which itincreases along the meridians. At any point scale along the parallel is equal to the scale along the meridian.

The areas are exaggerated on this projection and the exaggeration increases away from the centre of theprojection.

A circle drawn on the globe is represented by a circle on this projection.

It is used to show world in hemispheres. Also used for preparing aeronautical charts and daily weather maps ofthe polar areas.

http://www.mapthematics.com/ProjectionsList.php?Projection=162#stereographichttp://www.mapthematics.com/ProjectionsList.php?Projection=162#stereographichttp://www.mapthematics.com/ProjectionsList.php?Projection=162#stereographichttp://www.mapthematics.com/ProjectionsList.php?Projection=162#stereographic


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V. Orthographic Projection

(Image source: http://www.mapthematics.com/ProjectionsList.php?Projection=159#orthographic )




The parallels are not equally spaced. The distances between them decrease rapidly towards the margin of theprojection. So, the scale along the meridians decreases away from the centre of the projection.

The scale along the parallel is correct.

The distortion of the shapes increases away from the centre of the projection.

It is neither equal area nor orthomorphic.

The projection is used to prepare charts for showing the celestial bodies such as moon and other planets.

http://www.mapthematics.com/ProjectionsList.php?Projection=159#orthographichttp://www.mapthematics.com/ProjectionsList.php?Projection=159#orthographichttp://www.mapthematics.com/ProjectionsList.php?Projection=159#orthographichttp://www.mapthematics.com/ProjectionsList.php?Projection=159#orthographic


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Universal Transverse Mercator (UTM) UTM projection divides the surface of the Earth into a number of zones, each zone having a 6 degree longitudinalextent, Transverse Mercator projection with a central meridian in the center of the zone. UTM zones extend from 80degrees South latitude to 84 degrees North latitude. The zones are numbered from west to east. The first zone beginsat the International Date Line (180°).

(Image source: http://www.resurgentsoftware.com/GeoMag/utm_coordinates.htm )

The particular transverse Mercator map that is used to represent each zone has its central meridian running north-to-south down the center of the zone. This means that no portion of any particular zone is very far from the centralmeridian of the transverse Mercator map that is used to depict the zone. Since a Universal Transverse Mercator zoneis 6° of longitude wide, no portion of a UTM zone is more than 3° of longitude from the zone's central meridian. Sincethe distortion in a transverse Mercator map is relatively low near the map's central meridian, the result of this closeproximity to the map's central meridian is that the transverse Mercator map used to depict each zone within thecoordinate system contains relatively little distortion.

Adapted from the report Map Projections of Europe (2001), the table gives an account of the commonly usedprojection systems.

http://www.resurgentsoftware.com/GeoMag/utm_coordinates.htmhttp://www.resurgentsoftware.com/GeoMag/utm_coordinates.htmhttp://www.resurgentsoftware.com/GeoMag/utm_coordinates.htmhttp://www.resurgentsoftware.com/GeoMag/utm_coordinates.htm


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Property Developable

surface

Aspect Projections Extent of use

Conformal

(True shape)

Cylinder

Normal Mercator Equatorial regions (east-west extent)

Transverse UTM (Universal Transverse

Mercator)

Whole world except polar areas

Oblique Rosenmund ObliqueMercator

Small regions, oblique & east - west extent

Cone Normal Lambert Conformal Conic Small regions, oblique & east - west extent (1

or 2 standard parallels)

Plane Any Stereographic Small regions upto the hemisphere

Polar UPS (Universal Polar

Stereographic)

Polar regions

Homolographic

(Equal area)

Cylinder Normal Lambert Equal Area Equatorial areas (east-west extent)

Cone Normal Albers Equal Area Smaller regions & continents with east-west

extent

Plane

Any Lambert Azimuthal Equal

Area

Smaller regions about same north-south , east-

west extent

Equatorial Hammer-Aitoff World

Equidistant

Cylinder Normal Plate Caree World

Transverse Cassini Soldner Locally used for large scale mapping

Cone Normal Equidistant Conic Smaller regions & continents with (1 or 2

standard parallels) east-west extent

Plane Any Azimuthal Equidistant Smaller regions about same north-south , east-

west extent


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Transformation

The process of transformation, maps every point from one coordinate space to another. Using transforms, one canrotate, translate and scale content freely in two-dimensional space.

Rotation

A rotation is a transformation that is performed by spinning the object around a fixed point known as the center ofrotation. It can be performed clockwise as well as counterclockwise. The angle by which the object is turned is calledthe angle of rotation.

F igur e 12 :Rotation (a) Clockwise (b) Counterclockwise


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Translation

It is the process of moving each point of an object by the same distance in the same direction. It involves shifting theorigin of the current coordinate system horizontally and vertically by a specific amount.

Fi gure 13: I ll ustrating translation of (2, 1).

Scaling

Scaling changes the size of the grid. It lets the stretching and shrinking of the grid along the x and y axesindependently.


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Fi gure 14: Scalin g

(Image Source: http://commons.oreilly.com/wiki/index.php/Image:SVG_Essentials_I_5_tt74.png )

It does not change the origin of the grid but makes the grid multiplied by a given value. The figure above shows scalingtransformation, which doubles the scale of both axes. The square maintains its origin at (10, 10) even after scaling.Because the scale is made twice, therefore origin i.e. (10, 10) of the new grid lies at (20, 20) of the original grid.

Changing Map Projections

To transform data from one coordinate system to another, forward and inverse equations are used. The inverseequation of the source projection transforms the coordinates of source projection into geographic coordinates. Then,the forward equation of the target projection is used to transform the geographic coordinates into the target projectioncoordinates.

http://commons.oreilly.com/wiki/index.php/Image:SVG_Essentials_I_5_tt74.pnghttp://commons.oreilly.com/wiki/index.php/Image:SVG_Essentials_I_5_tt74.pnghttp://commons.oreilly.com/wiki/index.php/Image:SVG_Essentials_I_5_tt74.pnghttp://commons.oreilly.com/wiki/index.php/Image:SVG_Essentials_I_5_tt74.png


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Cartesian coordinate Transformation

Two dimensional Cartesian coordinates (x, y) can be transformed from one coordinate system to another coordinatesystem using three primary transformation methods which are:

Conformal Transformation : It is a linear transformation which changes one coordinate system into another by theprocess of rotation, uniform scale change followed by translation

Affine Transformation : It is a linear transformation which changes one coordinate system into another by the

process of rotation , scale change in x and y direction, followed by translation.

The difference between the conformal and affine transformations could be understood by the figure given below.

Figure 15 : Transformation (a) Conformal (b) Affine

(Image Source: http://kartoweb.itc.nl/geometrics/Coordinate%20transformations/coordtrans.html )

http://kartoweb.itc.nl/geometrics/Coordinate%20transformations/coordtrans.htmlhttp://kartoweb.itc.nl/geometrics/Coordinate%20transformations/coordtrans.htmlhttp://kartoweb.itc.nl/geometrics/Coordinate%20transformations/coordtrans.htmlhttp://kartoweb.itc.nl/geometrics/Coordinate%20transformations/coordtrans.html


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The shape of the original rectangular grid is maintained due to the uniform scale change in the case of conformaltransformation whereas in affine transformation the shape of the rectangular grid is changed due to the difference inscale in x and y direction.

Polynomial Transformation : It is a non-linear transformation which relates two 2D Cartesian coordinate systems

through a translation, a rotation and a variable scale change

Datum transformation

Datum transformation is a transformation of a three dimensional coordinate system into another three dimensionalcoordinate system. It is done when the source projection is based on a different datum than the target projection. Thetransformation parameters are estimated on the basis of a set of selected points whose coordinates are known in bothdatum systems.

Mathematically a datum transformation can be realized directly by relating the geographic coordinates of both datum

systems or indirectly by relating the geocentric coordinates of the datums.

Datum transformation via geographic coordinates

1. Geographic Offset2. Molodensky and Abridged Molodensky transformation3. Multiple regression transformation

3. georeferencing projection

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