# Map Projection Transformation Principles And Applications Pdf

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The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Mercator. When paired with a suitable geodetic datum , the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

Mercator projection , type of map projection introduced in by Gerardus Mercator. It is often described as a cylindrical projection , but it must be derived mathematically.

## Map Projections - types and distortion patterns

The shape of the Earth is represented as a sphere. It is also modeled more accurately as an oblate spheroid or an ellipsoid. A globe is a scaled down model of the Earth. Although they can represent size, shape, distance and directions of the Earth features with reasonable accuracy, globes are not practical or suitable for many applications. Globes are not suitable for use at large scales, such as finding directions in a city or following a hiking route, where a more detailed image is essential.

They are expensive to produce, especially in varying sizes scales. On a curved surface, measuring terrain properties is difficult, and it is not possible to see large portions of the Earth at once.

Maps do not suffer from the above shortcomings and are more practical than globes in most applications. Historically cartographers have tried to address the challenge of representing the curved surface of the Earth on a map plane, and to this end have devised map projections. During such transformation, the angular geographic coordinates latitude, longitude referencing positions on the surface of the Earth are converted to Cartesian coordinates x, y representing position of points on a flat map.

One way of classifying map projections is by the type of the developable surface onto which the reference sphere is projected. A developable surface is a geometric shape that can be laid out into a flat surface without stretching or tearing. The three types of developable surfaces are cylinder, cone and plane, and their corresponding projections are called cylindrical , conical and planar. Projections can be further categorized based on their point s of contact tangent or secant with the reference surface of the Earth and their orientation aspect.

Keep in mind that while some projections use a geometric process, in reality most projections use mathematical equations to transform the coordinates from a globe to a flat surface. The resulting map plane in most instances can be rolled around the globe in the form of cylinder, cone or placed to the side of the globe in the case of the plane.

The developable surface serves as a good illustrative analogy of the process of flattening out a spherical object onto a plane. In cylindrical projections , the reference spherical surface is projected onto a cylinder wrapped around the globe. The cylinder is then cut lengthwise and unwrapped to form a flat map. The cylinder may be either tangent or secant to the reference surface of the Earth. The diameter of the cylinder is equal to the diameter of the globe.

The tangent line is the equator for the equatorial or normal aspect; while in the transverse aspect, the cylinder is tangent along a chosen meridian i. At the place where the cylinder cuts through the globe two secant lines are formed. Such lines of true scale are called standard lines. These are lines of equidistance.

Distortion increases by moving away from standard lines. In normal aspect of cylindrical projection, the secant or standard lines are along two parallels of latitude equally spaced from equator, and are called standard parallels. In transverse aspect, the two standard lines run north-south parallel to meridians. Secant case provides a more even distribution of distortion throughout the map. Features appear smaller between secant lines scale 1.

The aspect of the map projection refers to the orientation of the developable surface relative to the reference globe. The graticule layout is affected by the choice of the aspect. The meridians are vertical and equally spaced; the parallels of latitude are horizontal straight lines parallel to the equator with their spacing increasing toward the poles.

Therefore the distortion increases towards the poles. Meridians and parallels are perpendicular to each other. The meridian that lies along the projection center is called the central meridian. In conical or conic projections , the reference spherical surface is projected onto a cone placed over the globe.

The cone is cut lengthwise and unwrapped to form a flat map. For the polar or normal aspect , the cone is tangent along a parallel of latitude or is secant at two parallels. These parallels are called standard parallels. Distortion increases by moving away from standard parallels. Features appear smaller between secant parallels and appear larger outside these parallels. Secant projections lead to less overall map distortion. The polar aspect is the normal aspect of the conic projection.

The cone can be situated over the North or South Pole. The polar conic projections are most suitable for maps of mid-latitude temperate zones regions with an east-west orientation such as the United States. Oblique aspect has an orientation between transverse and polar aspects. Transverse and oblique aspects are seldom used. In planar also known as azimuthal or zenithal projections, the reference spherical surface is projected onto a plane. The plane in planar projections may be tangent to the globe at a single point or may be secant.

In the secant case the plane intersects the globe along a small circle forming a standard parallel which has true scale. The normal polar aspect yields parallels as concentric circles, and meridians projecting as straight lines from the center of the map.

The distortion is minimal around the point of tangency in the tangent case, and close to the standard parallel in the secant case. The polar aspect is the normal aspect of the planar projection. The plane is tangent to North or South Pole at a single point or is secant along a parallel of latitude standard parallel.

The polar aspect yields parallels of latitude as concentric circles around the center of the map, and meridians projecting as straight lines from this center. Azimuthal projections are used often for mapping Polar Regions, the polar aspect of these projections are also referred to as polar azimuthal projections.

In transverse aspect of planar projections, the plane is oriented perpendicular to the equatorial plane. And for the oblique aspect, the plane surface has an orientation between polar and transverse aspects.

These projections are named azimuthal due to the fact that they preserve direction property from the center point of the projection. Great circles passing through the center point are drawn as straight lines.

Some classic azimuthal projections are perspective projections and can be produced geometrically. They can be visualized as projection of points on the sphere to the plane by shining rays of light from a light source or point of perspective.

Three projections, namely gnomonic, stereographic and orthographic can be defined based on the location of the perspective point or the light source. The point of perspective or the light source is located at the center of the globe in gnomonic projections. Great circles are the shortest distance between two points on the surface of the sphere known as great circle route.

Gnomonic projections map all great circles as straight lines, and such property makes these projections suitable for use in navigation charts. Distance and shape distortion increase sharply by moving away from the center of the projection. In stereographic projections, the perspective point is located on the surface of globe directly opposite from the point of tangency of the plane.

Points close to center point show great distortion on the map. Stereographic projection is a conformal projection , that is over small areas angles and therefore shapes are preserved. It is often used for mapping Polar Regions with the source located at the opposite pole. In orthographic projections, the point of perspective is at infinite distance on the opposite direction from the point of tangency.

The light rays travel as parallel lines. The resulting map from this projection looks like a globe similar to seeing Earth from deep space. There is great distortion towards the borders of the map. As stated above spherical bodies such as globes can represent size, shape, distance and directions of the Earth features with reasonable accuracy. It is impossible to flatten any spherical surface e. Similarly, when trying to project a spherical surface of the Earth onto a map plane, the curved surface will get deformed, causing distortions in shape angle , area, direction or distance of features.

All projections cause distortions in varying degrees; there is no one perfect projection preserving all of the above properties, rather each projection is a compromise best suited for a particular purpose. Different projections are developed for different purposes. Some projections minimize distortion or preserve some properties at the expense of increasing distortion of others.

As mentioned above, a reference globe reference surface of the Earth is a scaled down model of the Earth. This scale can be measured as the ratio of distance on the globe to the corresponding distance on the Earth. Throughout the globe this scale is constant. For example, a representative fraction scale indicates that 1 unit e.

The principal scale or nominal scale of a flat map the stated map scale refers to this scale of its generating globe. However the projection of the curved surface on the plane and the resulting distortions from the deformation of the surface will result in variation of scale throughout a flat map.

In other words the actual map scale is different for different locations on the map plane and it is impossible to have a constant scale throughout the map. This variation of scale can be visualized by Tissot's indicatrix explained in detail below. Measure of scale distortion on map plane can also be quantified by the use of scale factor. This can be alternatively stated as ratio of distance on the map to the corresponding distance on the reference globe.

A scale factor of 1 indicates actual scale is equal to nominal scale, or no scale distortion at that point on the map. Scale factors of less than or greater than one are indicative of scale distortion. The actual scale at a point on map can be obtained by multiplying the nominal map scale by the scale factor. As an example, the actual scale at a given point on map with scale factor of 0. A scale factor of 0. As mentioned above, there is no distortion along standard lines as evident in following figures.

On a tangent surface to the reference globe, there is no scale distortion at the point or along the line of tangency and therefore scale factor is 1.

## Mercator projection

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Many types of map projections are being used for map making. They are basically classified into four groups in accordance with the Map Projection Theory or the types of surfaces that are tangent with the globe. The four categories are: - Planar, Azimuthal or Zenithal projection - Conic projection - Cylindrical projection - Mathematical or Conventional projection obtained from mathematical calculation. The Gnomonic projection has its origin of light at the center of the globe. Less than half of the sphere can be projected onto a finite map.

The shape of the Earth is represented as a sphere. It is also modeled more accurately as an oblate spheroid or an ellipsoid. A globe is a scaled down model of the Earth. Although they can represent size, shape, distance and directions of the Earth features with reasonable accuracy, globes are not practical or suitable for many applications. Globes are not suitable for use at large scales, such as finding directions in a city or following a hiking route, where a more detailed image is essential. They are expensive to produce, especially in varying sizes scales. On a curved surface, measuring terrain properties is difficult, and it is not possible to see large portions of the Earth at once.

## Transverse Mercator projection

The conversion from geographical to plane coordinates is called forward transformation. The inverse transformation, which yields geographical coordinates captured from paper maps, is a more recent development, due to the need for transformation between different map projections especially in Geographic Information Systems GIS. Deriving the invers equations is sometimes not easy for the projections that have complicated forward functions including parametric variables. This paper describes an iteration algorithm using jacobian matrix for the inverse transformation of the pseudo-cylindrical map projections with non-linear forward projection equations. The method has been tested for ten pseudocylindrical world map projection.

With the advance of science and technology, there have been breakthroughs in the field of classical research and methods of map projection. Among these, computer science and space science have had the greater influence upon the field of research and the formation of a working body of map projection, developing them in breadth and depth. This book reflects several aspects of the development of modern mathematical cartography, especially the theory and methods of map projection transformation.

### Mercator projection

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