A projected coordinate system is based on map projections and designed for a flat surface. A mathematical transformation is carried out to convert spherical coordinates on a globe to the planar coordinates on a flat surface.
2. Three-dimensional coordinate system: In a three-dimensional coordinate system, the location of a point is given by three real values (coordinates) that represent the point’s distance from perpendicular projections on fixed perpendicular lines, called axes, intersecting at the origin. The common three-dimensional coordinate systems used in GIS are as follows.
Three-dimensional Cartesian coordinates
Geographical coordinate system (latitude and longitude)
A geographical coordinate system (GCS) uses three-dimensional spherical surfaces to define the location on the earth’s surface. A GCS uses a datum, an angular unit of measure, and a meridian to define the locations. A point on the earth’s surface is referenced by its lat–long values. Latitude and longitude are the angles measured, usually in degree or grad, from the earth’s centre to the point on the earth’s surface.
These are global or specific coordinate systems.
A coordinate system, either GCS or projected, provides a framework to define the location on the earth’s surface. All spatial information must be referenced by a coordinate system having an associated coordinate value.
Figure 2.14 Local datum and geocentric datum Source Robinson, Morrison, Muehrcke, et al. (1995)
Figure 2.13 Representation of earth as sphere or a spheroid
a particular spheroid for modelling the earth’s surface. This is where datum comes into play. A datum specifies the spheroid that should be used for modelling the earth’s surface and the exact location (a point) at which the spheroid needs to be aligned with the earth’s surface. A datum defines the origin for the GCS. An origin is the point on the surface at which the spheroid perfectly matches with the earth’s surface and the lat–long coordinates of the sphere are true and accurate. All other points in the coordinate system are referenced using the origin.
Usually, there are two basic types of datum—geocentric datum and local datum (Figure 2.14).
Geocentric datum uses the earth’s centre of mass as the origin, while the local datum aligns a spheroid to closely match the earth’s surface at a particular region. The point at which the spheroid meets the earth’s surface is known as the origin and all other points are calculated accordingly.
With a defined origin, a datum assigns lat–long values for a feature location on the earth’s surface defined by the GCS. A GCS (Figure 2.15) uses a network of intersecting lines, called graticules (Figure 2.16), to represent locations and features on the curved surface of the earth.
The intersecting lines are latitudes and longitudes. The horizontal lines or lines of latitude are called parallels, while vertical lines or lines of longitude are called meridians. The measures of lines of latitude start at the equator with 0° and range from 0° to 90° towards the North Pole and 0° to –90° towards the South Pole of the earth’s surface. Vertical lines or lines of longitude start with a prime meridian (with zero value) and range from 0° to 180° towards the east and 0° to –180° towards the west. Latitude and longitude values are the angles measured from the centre of the earth to the point on the earth’s surface.
The globe is the best representation of the earth’s surface with accurate location, shape, and proportions, but it may have size constraints. If a globe is cut into pieces to make flat images, the shape and other information may get distorted; hence, a map is required. So the question is how to convert a three-dimensional spheroid with location information given in latitude and longitude values to a two-dimensional flat map?
The answer is map projection.
Map projection is a blended technique of mathematical transformation and geometrics to convert a three-dimensional spheroid into a two- dimensional flat map. In other words, map projection is used to convert a GCS (lat–long system) to a projected (planar) coordinate system. It is not easy to flatten a spheroid onto a flat surface such as a map; it always
Figure 2.15 Geographical coordinate system
creates some distortion to the actual spatial information regarding location. There are different types of projections based on the spatial information (shape, size, or direction) they preserve. Two basic criteria to classify different types of projection are as follows.
1. Based on the spatial attributes (shape, size, or direction) preserved by projection: basic type.
2. Based on the techniques used to project the spheroid onto the flat surface: basic technique.
The most common approach of projecting a spheroid onto a flat surface is using a developable surface, which may be a cylinder, cone, or plane. There are different projection methods, and each aims to follow both the basic criteria of projection—minimize distortion and preserve spatial characteristics. The selection of a projection type depends on the purpose of the map.
Basic Types of Projections
• Conformal: The conformal projection preserves the shape for small areas or angles for large areas. It is often used for navigation maps, topographic maps, and weather maps. A conformal projection shows 90° graticule lines intersecting at 90° angles on the map;
hence, it preserves shape (Eklundh, Arnberg, Arnborg, et al. 1999).
There are four conformal projections in use—Mercator, transverse Mercator, Lambert’s conformal conic with two standard parallels, and stereographic azimuthal Mercator’s projection (Robinson, Morrison, Muehrcke, et al. 1995).
Figure 2.16 Graticules
• Equal area: The equal area projection preserves the area of the projected region. It is often used for dot density maps and thematic maps.
• Equidistant: The equidistant projection preserves distances between features on the projected region. It is specially used for airline route maps and seismic maps. Most equidistant projections have one or more lines that have the same length on the map as that on the globe.
• True direction: Projections that preserve the direction between features are called true direction projections. True direction or azimuthal maps can be combined with equidistant, equal area, and conformal projections. These maps are used for navigation routes.
Each of these properties (shape, size, and direction) is preserved, one at the expense of others. Selection of a projection type depends on application as there is no perfect projection that can preserve all the properties.
Basic Techniques of Projection
The most common technique of projection is using a developing surface such as a plane, cylinder, or cone. A developing surface either touches (tangent) or intersects (secant) the spheroid. There are three ways to project a map.
1. Planar projection: In a planar projection (Figure 2.17), the earth’s surface is projected on a plane surface. Planar projection is the most accurate at the centre where it touches the spheroid. The tangential point at which the plane meets with the spheroid determines the type of planar projection—polar, equatorial, or oblique planar projection.
2. Conical projection: A conical projection uses a cone as the developing surface to project a region on the earth’s surface (Figure 2.18). An elementary method of developing a conic projection is to
Figure 2.17 Planar projection
place a cone shape over the globe. There are two types of conical projections—normal conical and secant normal conical.
A normal conical projection is tangent to the globe along the line of latitude. This line is called a standard parallel with no distortion.
The lines of longitude are projected on the conic surface meeting at the apex, and the lines of latitude are projected as rings. This type of projection is preferably used for polar sections.
A secant normal conical projection intersects the globe at two locations; hence, it is defined by two “standard parallel” lines (Figure 2.19). A secant conical has less distortion than a normal conical projection.
3. Cylindrical projection: Another widely used and important technique to project the globe on a flat surface is by using a cylinder as the developing surface. In cylindrical projection, the globe is projected by inserting it inside a cylinder prepared by rolling a plane paper. The cylinder touches the earth along the equator. When the cylinder is open and flattened, the regions near the equator are most accurate and the regions near the poles are most distorted.
Figure 2.18 Normal conical projection
Figure 2.19 Secant normal conical projection
Cylindrical projections are also of three types—normal cylindrical, transverse cylindrical, and oblique cylindrical (Figure 2.20). A normal cylindrical projection has a cylinder in which the equator is the line of tangency. A transverse cylindrical projection has its tangency at a meridian, and an oblique cylinder is rotated around a great circle line, located anywhere between the equator and the meridian.
Figure 2.21 depicts the steps of projection of a geoid on a planar map with coordinate system.