Map Projection PDF

Chapter- Three Map Projection 1 Map Projection Map projection is the method of transferring the graticule of latitude and longitude on a plane surface. It can also be defined as the transformation of spherical network of parallels and meridians on a plane surface....

Chapter- Three Map Projection 1 Map Projection Map projection is the method of transferring the graticule of latitude and longitude on a plane surface. It can also be defined as the transformation of spherical network of parallels and meridians on a plane surface. The earth on which we live in is not flat. It is geoid in shape like a sphere. A globe is the best model of the earth. Due to this property of the globe, the shape and sizes of the continents and oceans are accurately shown on it. It also shows the directions and distances very accurately. The globe is divided into various segments by the lines of latitude and longitude.  The horizontal lines represent the parallels of latitude and the vertical lines represent the meridians of the longitude. The network of parallels and meridians is called graticule. This network facilitates drawing of maps. Drawing of the graticule on a flat surface is called projection. Besides, on the globe the meridians are semi-circles and the parallels are circles. When they are transferred on a plane surface, they become intersecting straight lines or curved lines. Creation of a Map Projection The creation of a map projection involves three steps : 1. Selection of a model for the shape of the earth or round body (choosing between a sphere or ellipsoid) 2. Transform geographic coordinates (longitude and latitude) to plane coordinates (eastings and northings). 3. Reduce the scale (in manual cartography this step came second, in digital cartography it comes last ELEMENTS OF MAP PROJECTION A. Reduced Earth: A model of the earth is represented by the help of a reduced scale on a flat sheet of paper. This model is called the “reduced earth”. This model should be more or less spheroid having the length of polar diameter lesser than equatorial and on this model the network of graticule can be transferred. B. Parallels of Latitude: These are the circles running round the globe parallel to the equator and maintaining uniform distance from the poles. Each parallel lies wholly in its plane which is at right angle to the axis of the earth. They are not of equal length. They range from a point at each pole to the circumference of the globe at the equator. They are demarcated as 0º to 90º North and South latitudes. C. Meridians of Longitude: These are semi-circles drawn in north south direction from one pole to the other, and the two opposite meridians make a complete circle, i.e. circumference of the globe. Each meridian lies wholly in its plane, but all intersect at right angle along the axis of the globe. There is no obvious central meridian but for convenience, an arbitrary choice is made, namely the meridian of Greenwich, which is demarcated as 0° longitudes. It is used as reference longitudes to draw all other longitudes D. Global Property: In preparing a map projection the following basic properties of the global surface are to be preserved by using one or the other methods: (i) Distance between any given points of a region; (ii) Shape of the region; (iii) Size or area of the region in accuracy; (iv) Direction of any one point of the region bearing to another point. Properties of Map Projection A map projection property is an alteration of area, shape, distance, and direction on a map projection. These map projection properties exist because the conversion from a three dimensional object, such as the earth, to a two-dimensional representation, such as a flat paper map, requires the deformation of the three-dimensional object to fit onto a flat map. The three-dimensional spherical surface is torn, sheared, or compressed to flatten it onto a flat developable surface.  Of the four projection properties, area and shape are considered major properties and are mutually exclusive. That means, that if area is held to its true form on a map, shape must be distorted, and vice versa. Distance and direction, on the other hand, are minor properties, and can coexist with any of the other projection properties. However, distance and direction cannot be true everywhere on a map.  A map can show one or more, but never all, of the following map projection properties at the same time: true direction, true distance, truth area, and true shape. Area and shape are two major map projection properties, and they still remain mutually exclusive. Distortions are unavoidable when making flat maps of the earth. Distortion is not constant across the map, as distortion may take different forms in different parts of the map. There are few points were distortions are going to be zero, however, distortion is usually less near the points or lines of intersection where the developable surface intersects the globe.  By determining where the standard points and lines are placed will directly affect where the map will have the least and most amount of distortion. Classification of map projections Map Projections may be classified on the following bases: A. Drawing Techniques: On the basis of method of construction, projections are generally classified into perspective, non-perspective and conventional or mathematical. Perspective projections can be drawn taking the help of a source of light by projecting the image of a network of parallels and meridians of a globe on developable surface.  Non–perspective projections are developed without the help of a source of light or casting shadow on surfaces, which can be flattened. Mathematical or conventional projections are those, which are derived by mathematical computation, and formulae and have little relations with the projected image. B. Developable Surface: A developable surface is one, which can be flattened, and on which, a network of latitude and longitude can be projected. A non-developable surface is one, which cannot be flattened without shrinking, breaking or creasing. A globe or spherical surface has the property of non-developable surface whereas a cylinder, a cone and a plane have the property of developable surface.  On the basis of nature of developable surface (Based on the projection surface used) , the projections are classified as cylindrical, conical and zenithal projections. Cylindrical projections are made through the use of cylindrical developable surface. A paper-made cylinder covers the globe, and the parallels and meridians are projected on it. When the cylinder is cut open, it provides a cylindrical projection on the plane sheet. A Conical projection is drawn by wrapping a cone round the globe and the shadow of graticule network is projected on it. When the cone is cut open, a projection is obtained on a flat sheet.  Zenithal projection is directly obtained on a plane surface when plane touches the globe at a point and the graticule is projected on it. Generally, the plane is so placed on the globe that it touches the globe at one of the poles. These projections are further subdivided into normal, oblique or polar as per the position of the plane touching the globe. If the developable surface touches the globe at the equator, it is called the equatorial or normal projection. If it is tangential to a point between the pole and the equator, it is called the oblique projection; and if it is tangential to the pole, it is called the polar projection. C. Global Properties: The correctness of area, shape, direction and distances are the four major global properties to be preserved in a map. But none of the projections can maintain all these properties simultaneously. Therefore, according to specific need, a projection can be drawn so that the desired quality may be retained. Thus, on the basis of global properties, projections are classified into equal area, orthomorphic, azimuthal and equi-distant projections. Equal Area Projection is also called homolographic projection. It is that projection in which areas of various parts of the earth are represented correctly.  Orthomorphic or True-Shape projection is one in which shapes of various areas are portrayed correctly. The shape is generally maintained at the cost of the correctness of area. Azimuthal or True-Bearing projection is one on which the direction of all points from the centre is correctly represented. Equi-distant or True Scale projection is that where the distance or scale is correctly maintained. However, there is no such projection, which maintains the scale correctly throughout. It can be maintained correctly only along some selected parallels and meridians as per the requirement. D. Source of Light: On the basis of location of source of light, projections may be classified as gnomonic, stereographic and orthographic. Gnomonic projection is obtained by putting the light at the centre of the globe. Stereographic projection is drawn when the source of light is placed at the periphery of the globe at a point diametrically opposite to the point at which the plane surface touches the globe. Orthographic projection is drawn when the source of light is placed at infinity from the globe, opposite to the point at which the plane surface touches the globe. 1. Cylindrical Projections Cylindrical projections are those that provide the appearance of a rectangle. The rectangle can be seen as a developed cylindrical surface that can be rolled into a cylinder. Whereas these projections are created mathematically rather than from the cylinder, the final appearance may suggest a cylindrical construction. A cylindrical map projection can have one line or two lines of no scale distortion. Classic examples of cylindrical projections include the conformal Mercator and Lambert's original cylindrical equal area. Cylindrical projections are often used for world maps with the latitude limited to a reasonable range of degrees south and north to avoid the great distortion of the polar areas by this projection method.  The normal aspect Mercator projection is used for nautical charts throughout the world, while its transverse aspect is regularly used for topographic maps and is the projection used for the UTM coordinate system. 1.1 Cylindrical Equal Area Projection The cylindrical equal area projection, also known as the Lambert’s projection, has been derived by projecting the surface of the globe with parallel rays on a cylinder touching it at the equator. Both the parallels and meridians are projected as straight lines intersecting one another at right angles. The pole is shown with a parallel equal to the equator; hence, the shape of the area gets highly distorted at the higher latitude. Properties 1. All parallels and meridians are straight lines intersecting each other at right angle. 2. Polar parallel is also equal to the equator. 3. Scale is true only along the equator. Limitations 1. Distortion increases as we move towards the pole. 2. The projection is non-orthomorphic (non-True-Shape). 3. Equality of area is maintained at the cost of distortion in shape. Uses 1. The projection is most suitable for the area lying between 45º N and S latitudes. 2. It is suitable to show the distribution of tropical crops like rice, tea, coffee, rubber and sugarcane. 1.2 Mercator’s Projection A Dutch cartographer Mercator Gerardus Karmer developed this projection in 1569. The projection is based on mathematical formulae. So, it is an orthomorphic projection in which the correct shape is maintained. The distance between parallels increases towards the pole. Like cylindrical projection, the parallels and meridians intersect each other at right angle. It has the characteristics of showing correct directions. A straight line joining any two points on this projection gives a constant bearing, which is called a Laxodrome or Rhumb line. Universal Transverse Mercator (UTM) Coordinate System UTM system is transverse-secant cylindrical projection, dividing the surface of the Earth into 6 degree zones with a central meridian in the center of the zone. Each zone is a different Transverse Mercator projection that is slightly rotated to use a different meridian. UTM zone numbers designate 6 degree longitudinal strips extending from 80 degrees South latitude to 84 degrees North latitude. UTM is a conformal projection, so small features appear with the correct shape. scale is same in all directions. ( distances, directions, shapes, and areas are reasonably accurate ). Scale factor is 0.9996 at the central meridian and at most 1.0004 at the edges of the zones.  UTM coordinates are in meters, making it easy to make accurate calculations of short distances between points (error is less than 0.04%). Although the distortions of the UTM system are small, they are too great for some accurate surveying. zone boundaries are also a problem in many applications, because they follow arbitrary lines of longitude rather than boundaries between jurisdictions. o It is best for maps large/ small scale covering small areas with predominantly N – S trending extends. o UTM is the most prevalent plane grid system used in GIS, metric measurements, remote sensing work, topographic map preparation and natural resource database development operations. Properties 1. All parallels and meridians are straight lines and they intersect each other at right angles. 2. All parallels have the same length which is equal to the length of equator. 3. All meridians have the same length and equal spacing. But they are longer than the corresponding meridian on the globe. 4. Spacing between parallels increases towards the pole. 5. Scale along the equator is correct as it is equal to the length of the equator on the globe; but other parallels are longer than the corresponding parallel on the globe; hence the scale is not correct along them. For example, the 30º parallel is 1.154 times longer than the corresponding parallel on the globe. 6. Shape of the area is maintained, but at the higher latitudes distortion takes place. 7. The shape of small countries near the equator is truly preserved while it increases towards poles. 8. It is an azimuthal projection (True-Bearing/directions). 9. This is an orthomorphic projection as scale along the meridian is equal to the scale along the parallel. Limitations 1. There is greater exaggeration of scale along the parallels and meridians in high latitudes. 2. As a result, size of the countries near the pole is highly exaggerated.  For example, the size of Greenland equals to the size of USA, whereas it is 1/10th of USA. 2. Poles in this projection cannot be shown as 90º parallel and meridian touching them are infinite. Uses 1. More suitable for a world map and widely used in preparing atlas maps. 2. Very useful for navigation purposes showing sea routes and air routes. 3. Drainage pattern, ocean currents, temperature, winds and their directions, distribution of worldwide rainfall and other weather elements are appropriately shown on this map Conic projections 2. Conic projections Conical projections, the developmental surface used for projecting the graticule of parallels and meridians is a cone which is placed over a sphere. The cone touches the globe along a parallel which is called as standard parallel. This is because along this parallel the scale is correct. Properties Common to Conical Projections 1. All parallels are arcs of concentric circles or concentric curves. 2. Meridians in general are straight lines (except for example in Bonne’s Projection meridians are smooth curves and only central meridian is a straight line). 3. Scale is true along the standard parallel(s). 4. It can be orthomorphic or homolographic. 5. The pole is represented as an arc or a point. Simple Conical Projection With One Standard Parallel A conical projection is one, which is drawn by projecting the image of the graticule of a globe on a developable cone, which touches the globe along a parallel of latitude called the standard parallel. As the cone touches the globe located along AB, the position of this parallel on the globe coinciding with that on the cone is taken as the standard parallel. The length of other parallels on either side of this parallel are distorted. Properties of simple conical projection with one standard parallel 1. All the parallels are arcs of concentric circle and are equally spaced. 2. All meridians are straight lines merging at the pole. The meridians intersect the parallels at right angles. 3. The scale along all meridians is true, i.e. distances along the meridians are accurate. 4. An arc of a circle represents the pole. 5. The scale is true along the standard parallel but exaggerated away from the standard parallel. 6. Meridians become closer to each other towards the pole. 7. This projection is neither equal area nor orthomorphic. Limitations 1. It is not suitable for a world map due to extreme distortions in the hemisphere opposite the one in which the standard parallel is selected. 2. Even within the hemisphere, it is not suitable for representing larger areas as the distortion along the pole and near the equator is larger. Uses 1. This projection is commonly used for showing areas of mid-latitudes with limited latitudinal and larger longitudinal extent. 2. A long narrow strip of land running parallel to the standard parallel and having east-west stretch is correctly shown on this projection. 3. Direction along standard parallel is used to show railways, roads, narrow river valleys and international boundaries. 4. This projection is suitable for showing the Canadian Pacific Railways, Trans-Siberian Railways, international boundaries between USA and Canada and the Narmada Valley. Conical projection with two standard parallels This projection is an improvement over simple conical projection. The scale is correct along the two standard parallels, so a greater longitudinal area or north-south extent is correctly represented in this projection. Properties of Conical projection with two Standard Parallels. 1. All parallels are arcs of concentric circles. 2. All meridians are straight lines radiating from the pole as radii of concentric curves. 3. The scale is true along both the standard parallels. 4. The scale is true along all the meridians. 5. The distances between the standard parallels are shorter than their actual distances while beyond them they are longer than their actual distances. 6. This projection is an improvement over the conical projection with one standard parallel as it is suitable for comparatively large areas of mid- latitudes like that of Canada, Russia etc. This is because of the two standard parallels which reduces the north-south distortion. 1. Two properties of conical projections are: i) All parallels are arcs of concentric circles or concentric curves. ii) Scale is true along standard parallel(s). 2. The two differences between simple conical projection with one standard parallel and simple conical projection with two standard parallel is that the former is having only one standard parallel which is true to scale while in the latter there are two standard parallels along which the scale is correct. The second difference is that since the scale is correct along two standard parallels in the latter one, so scale is correct along greater latitudinal extent and so, it is suitable for showing countries in mid-latitudinal areas having greater north-south extent like Russia, Canada etc. while the former one is suitable for areas in mid-latitudes having only a small north-south extent. 3. Azimuthal Projection: The azimuthal projection plots the surface of Earth using a flat plane. Some of the common perspective azimuthal projections include gnomonic, stereographic, and orthographic.  The Orthographic projection geometrically projects the globe onto a plane with the point of projection as infinity. All the projection lines are orthogonal to the projection plane. Orthographic Map Distortion The orthographic projection distorts shape and area near edges due to perspective. Directions are true from the point of projection, with scale defeating away from its radiating lines. The orthographic projection isn’t conformal nor equal area. Stereographic Projection This map projection is commonly used for polar aspects and navigation maps because of how it preserves shapes (conformal). Despite how the scale is greatly stretched by perspective, it’s been used to map large continents or oceans including the Arctic and Antarctic. Stereographic Map Properties In terms of map distortion, the stereographic projection is conformal but the distortion of area and distance increases away from the center point of projection. Direction is true from the center point with each straight line representing a great circle. The stereographic projection isn’t equal in area nor equidistant. Gnomonic Projection Unlike the stereographic projection, the Gnomonic projection light source is located at the sphere center. This means that it can only present less than a hemisphere at a time.  Each great circle (geodesic) including meridians is mapped to a straight line. This makes the gnomonic projection easiest to plot out the shortest route. This is why navigators have used the gnomonic projection along with Mercator maps (cylindrical projections) for finding the shortest route between two points. Furthermore, seismologists use this map projection because seismic waves often travel down great circles. Gnomonic Projection Distortion The Gnomonic projection isn’t equal area, equidistant, or conformal as the distortion of these two properties increases away from the center point. You should avoid using the Gnomonic projection for measuring distances. However, it’s particularly useful for navigation as a straight line drawn on the map is a great circle (geodesic). 4. Conventional projections Conventional projections are pure-mathematical constructions designed to map the entire sphere with minimal distortion. 1. Sanson-Flamsteed (sinusoidal) projection Parallels are equispaced and their length is chosen to make an equal area projection. It is, in fact, the equatorial case of Bonne's projection. 2. Aitov's projection This projection is developed from the equatorial case of the zenithal equal area projection for a hemisphere. The equatorial scale and the longitude coverage are both doubled, thereby mapping the whole sphere while preserving the equal area property. 3. Mollweide's projection Meridians are projected as ellipses, parallels as straight lines spaced so as to make the projection equal area. Suitability of Map Projection When you choose a projection, the first thing to consider is the purpose of your map. For general reference and atlas maps, you usually want to balance shape and area distortion. If your map has a specific purpose, you may need to preserve a certain spatial property—most commonly shape or area—to achieve that purpose. Choosing the Right Projection When selecting a map projection, consider the following factors: 1. Purpose of the Map: Determine whether area, shape, distance, or direction is most important for your application. 2. Geographic Area: The location being mapped can influence which projection minimizes distortion. 3. Scale of the Map: Larger areas may require different projections than smaller, localized maps to maintain accuracy. Aspects of Map Projection The projection aspect is the position of the projection axis in relation to the geographic sphere parameterization axis. Projections can also be described in terms of the direction of the projection plane's orientation (whether cylinder, plane or cone) with respect to the globe. This is called the aspect of a map projection. The three possible aspects are normal, transverse and oblique. In a normal projection, the main orientation of the projection surface is parallel to the Earth's axis. A transverse projection has its main orientation perpendicular to the Earth's axis. Oblique projections are all other, non-parallel and non-perpendicular, cases.