Projection System Demystified

Before beginning to explain something as important as Projection System, let me have a small discussion about how people came across measuring distances in the first place.

In the past geomatics, or the science of quantifying the earth, was known as land surveying. Famous people have started their career as land surveyors. Even US presidents, at some time in their life worked as land surveyors and has been commemorated by having their face carved into the rock at Mount Rushmore National Monument, South Dakota. Carl Friedrich Gauss (1777–1855), one of the most notable mathematician and astronomer of all time was the first to take interest in geodesy, and successfully established a geodetic coordinate system in northern Germany, which in those days belonged to England and was known as the Kingdom of Hannover ruled by King George IV during the period 1828 and 1847. To honor his contribution, the German Central Bank put his picture in a 10-mark note in 1991.

The end of the eighteenth century and the beginning of the nineteenth century was an era full of exploration, discoveries and pioneering of land surveying. It was also the time when the East India Company required accurate maps of the Indian subcontinent for constructing roads, establish visual telegraphy and later on railways and electric telegraph. And like everyone else, East India Company used a geodetic reference system, comprising a network of known points, points of which the coordinates have been accurately determined.

Dive into Geomatics
The term geomatics was born in Canada in 1980s. Geomatics is discipline for gathering, storing, processing, and delivering of Geographic Information or spatially referenced information. This broad term applies both to science and technology. Geomatics is a new term incorporating the older version of ‘land surveying’ and other possible aspects of spatial data management. Under this ‘broad’ term we discuss Geodesy.

Geodesy is a discipline which studies the position and nature of a point on or above the surface of a planet, by critically examining the size and shape of that planets, within a terrestrial reference system. But the surface of a planet like Earth is not as simple and smooth as a circle or ellipse drawn on a sheet of paper. So we need an approximate shape of the Earth before we start calculating. Geomatic Engineers around the world uses a simpler ellipsoid to model the world. But one ellipsoid cannot represent the whole world, so we need many different ellipsoids to fit best with the shape and size of a particular country or other parts of the Earth. For example, the ellipsoid used for mapping Britain, the Airy 1830 ellipsoid is designed to best-fit Britain and is not useful in Africa, Australia or other parts of the world. The World Geodetic Reference System in 1980s used WGS84 ellipsoid which is so far the best to fit the shape and the size of the Earth as a whole.

Within an ellipsoid we use three exis, or reference coordinates to pin-point a location on the surface. It is called latitude (y), longitude (x) and height above the ellipsoid (h). Latitude and longitude are also expressed as phi (φ) and lambda (λ). This triplet (x, y, h) is called the geodetic coordinate system or geographic coordinate system.

Geodetic or Geographic Coordinate System
This system divides the earth into hemispheres and calculates positions by measuring angles in degrees. If the equator is considered as the center most line, this huge x latitude is referred as 0 degree. This line divides the word into northern and southern hemisphere. If you travel towards any pole from 0 degree, you will have a value measured with reference to the geodetic center. We use N and S to refer the subsequent hemisphere.

Just like the equator, let’s imagine another axis perpendicular to the x. This time we have y or latitude. This center latitude divides the earth into eastern and western hemisphere. If you travel east or west from this line, you will have your distance in degrees just like before. Here we will use E and W. These imaginary lines are called meridians. The two meridians (x and y) with 0 values are called prime-meridians. We can draw more meridians all over the world parallel to the prime-meridians for our convenience.

Notice that the latitudes start count from 0 and ends at 90 degree at poles. But the longitudes start from 0 and ends at 180 degree. So we have a valid range for GCS coordinates, for latitudes it is either 0-90 degree N or 0-90 degree S. For Longitude the range is 0-180 degree E or 0-180 degree W. Besides degrees, grads and radians are also used. Depending on the precision required, the degrees (360° for a full circle) can be subdivided into 60 min of arc, and each minute of arc can be further subdivided into 60 sec of arc. Seconds can be subsequently divided into decimals of seconds. 92 degree 23 minute 34.56 second east longitude is expressed as 92° 23ʹ 34.56ʺ E. This system is known as sexagesimal.

Datum
A datum is an origin or starting point for calculating a location or plane. Without datum we cannot answer questions like “height above what?”, “what is the origin?” and “on what surface do they lie?”. In a coordinate system the reference point or datum is the very center of the world. In real world, a datum is considered as errorless, at least in the abstract. It has the most accurate coordinates and is unique. In GCS system, the datum is called geographic datum.

Map Projection and Projected Coordinate System
There are two kind of coordinate operation: coordinate conversion and coordinate transformation. Coordinate transformation are the basic operations in geodesy. It refers to transformation between different coordinate system. It is usually achieved by translation of datum, rotation and change in scale. Coordinate conversion includes change of coordinate system from one datum to another. In this case, mathematical rules are applied. Map projection is an example of coordinate conversion.

In GIS, cartography and surveying two-dimensional plane is more applicable than three-dimensional ellipsoid. The first problem of map projection is that the surface of an ellipsoid is like the orange peel, non-developable. In other words, flattening its surface leads to distortion. It may be possible to translate coordinates between dimensions, but it is never possible to avoid distortions in area, distance and direction. For example, conformal (or orthomorphic) projection preserves angles and shape of any area, but the scale does not.

There are dozens or more choices for projection system. If we want to classify map projection systems by their nature of preserving different items, we may end up with this list

• Preserving direction (azimuthal or zenithal), a trait possible only from one or two points to every other point
• Preserving shape locally (conformal or orthomorphic)
• Preserving area (equal-area or equiareal or equivalent or authalic)
• Preserving distance (equidistant), a trait possible only between one or two points and every other point
• Preserving shortest route, a trait preserved only by the gnomonic projection.

To preserve the physical properties of the actual surface, we must consider more local solutions. The idea of a self-consistent local map projection based on small flat planes tangent to the Earth is convenient, but only for small projects. As long as there is no need to venture outside the bounds of a particular local system, this method can be entirely adequate.

Azimuthal Projection to preserve direction
This projection creates maps which are generated on a large tangent plane touching the globe at a single point. At pole, the parallels of latitudes are usually concentric circles. The scale is correct at the center, but just as in the smaller local systems mentioned earlier, the farther you get from the center of the map, the more distorted it becomes. An imaginary light source is used to device in imaging the projection onto a developable surface.

Cylindrical and Conic projection
Cylindrical System, refers any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal lines. Cylindrical projections stretch distances to east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the secant of the latitude as a multiple of the equator’s scale.

The term “conic projection” is used to refer to any projection in which meridians are mapped to equally spaced lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centering the apex. When making a conic map, the map maker arbitrarily picks two standard parallels. Those standard parallels may be visualized as secant lines where the cone intersects the globe—or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe.

Conic projection is best suited for polar or higher latitude mapping.

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