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.
spatial data in one coordinate system need to be transformed to another coordinate system. Here, the term geographical transformation comes into play. There are various definitions for transformation. In general terms, transformation is a technique to register data layers to a common coordinate scheme with which a standard data layer is already registered for further processing. ESRI defines transformation as “the process of converting coordinates of a map or an image from one system to another system by shifting, rotating, scaling, skewing, or projecting them”. Based on the types of coordinate systems to be transformed, transformations are categorized into two classes: projection transformation and datum transformation.
Projection Transformation
Projection transformation is the process of transforming a two- dimensional coordinate system of one map projection into another two-dimensional coordinate (x, y) system of a specified map projection.
For example, a reference system is projected with a Mercator projection.
To add a new data layer, which is projected with a Universal Transverse Mercator (UTM) projection system, the specialist has to convert the coordinates of the UTM projection into coordinates of the Mercator projection. In projection transformation, coordinates (x, y) of source projection system are converted into geographical coordinates (latitude, longitude), which is called inverse transformation. Converting geographical coordinates into coordinate values (x, y) of the target projection system is called forward transformation (Figure 2.22). The transformation process is carried out by two equations.
Projection transformation is an equation-based transformation. It is the simplest form of transformation where Cartesian values of one two-dimensional system need to be converted into Cartesian values of another two-dimensional system using mathematical equations. There are two types of transformations.
Figure 2.21 Projection system
1. Affine transformation: Affine is a Latin word that means “connected with”. Affine transformation between two two-dimensional Cartesian systems is a linear transformation in which a rotation along the x-axis and y-axis is followed by a translation. The transformation function is expressed as follows.
x¢ = ax – by + xorigin y¢ = cx + dy + yorigin
where a, b, c, d, xorigin, and yorigin are transformation parameters.
Affine transformation rotates and enlarges a map, preserving the primary shape of the original map.
2. Polynomial transformation: A polynomial transformation is a non- linear transformation. It ascertains the relationship between two two-dimensional Cartesian systems through a translation (slides an object to a fixed distance in a given direction), rotation (turns an image about a fixed point called the centre of rotation), and a variable scale change. The transformation function is represented as follows.
x¢ = xorigin + a1x + a2y + a3xy + a4x 2 +a5y 2 + a6x2y + … y¢ = yorigin + b1x + b2y + b3xy + b4x 2 + b5y2 + b6x2y + …
Polynomial transformation is used to geo-reference aerial photographs or satellite imagery.
Datum Transformation
The earth is not an even surface. Because of its complex structure and uneven surface, specific datum is required to represent distinct zones. For a comprehensive analysis of any geographical phenomena,
Figure 2.22 Projection transformation
a specialist may require to combine spatial information from different sources about different zones. Now if different data had been used to represent this spatial information, how this information can be combined? The solution is to transform one projection into another one as well as the underlying datum of one projection into another.
Datum transformation is a mathematical procedure that transforms source datum into the target datum using three-dimensional analysis (three-dimensional coordinate system). A mathematical function can be used to map the geographical coordinates (e, l, h) of one datum into another or can map the geocentric coordinates (x, y, z) of source datum into target datum (Figure 2.23). There are several methods for datum transformation such as geocentric transformation, 7-parameter transformation, and Helmert 7-parameter methods.
A question arises here. If the coordinates of a projection system for input data are not known prior to the transformation, how the transformation can be performed? The answer is ground control point (GCP), which is a point on the earth’s surface with known locations used to geo-reference (defining coordinate values for geographical data) satellite images and aerial photographs. GCPs are used to establish the relationships between a known set of coordinates and an unknown set of coordinates.
Many GIS software provides the functionality to transform projection systems and integrate data from different sources. Integration of map sheets with the same projection allows a single data layer of spatial data
Figure 2.23 Datum transformation
to be generated from different sources for a GIS application. If there is some distortion in any map sheet to be joined with an already registered map, it should be rubber sheeted. The layers that are supposed to be superimposed should be registered to a similar coordinate system.
Selected layers are registered to a common reference system. Geometric transformations are used to assign ground coordinates to a map or data layer within the GIS or to adjust one data layer so that it can be correctly superimposed on another of the same area. The procedure to correct geographical data is called registration. Two approaches are used in registration—the adjustment of absolute positions and the adjustment of relative positions. Relative position refers to the location of features in relation to a geographic coordinate system. Rubber sheeting is a relative position registration technique. It is the process of transforming geometric properties of a raster map or edge matching of one map sheet of data layer with another map sheet depicting the same location using a common control point. Primarily, it is a GIS functionality to manipulate geospatial data. Rubber sheeting is necessary because the imagery and the vector data rarely match up correctly due to various reasons such as the angle at which the image was taken, the curvature of the surface of the earth, minor movements in the imaging platform (such as a satellite or aircraft), and other errors in imagery. Rubber sheeting starches one data layer to meet the predefined ground point.