Where is distortion on cylindrical projections least and greatest

Directions along a Rhumb line are true between any two points on a map. Distances are true only along the Equator. Although it has a conformal property, areas are greatly distorted increasing size at poles. Lambert introduced the Transverse Mercator in It uses a horizontally oriented cylinder tangent to a Meridian. This is particularly useful for mapping large areas that are mainly north-south in extent.

The whole UTM grid system uses 60 horizontally oriented cylinders secant to the globe. While horizontal and vertical cylinders make up a Mercator and Transverse Mercator, an oblique aspect projection uses neither.

The USGS uses the Transverse Mercator in their , to , quadrangle maps because they can be joined at their edges. Further to this, State Plane Coordinate Systems use a Transverse Mercator when its orientation is a north-east extent.

The Transverse Mercator m projection is conformal with shapes being true in small areas. While the equator is a straight line, other parallels are complex curves concave toward nearest pole.

The Miller Projection was developed by O. 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.

Distortion increases with distance from the point or line of tangency. Scale distortion on a tangent surface to the globe. Between the secant lines where the surface is inside the globe, features appear smaller than in reality and scale factor is less than 1.

At places on map where the surface is outside the globe, features appear larger than in reality and scale factor is greater than 1. A map derived from a secant projection surface has less overall distortion than a map from a tangent surface.

Scale distortion on a secant surface to the globe. A common method of classification of map projections is according to distortion characteristics - identifying properties that are preserved or distorted by a projection. The distortion pattern of a projection can be visualized by distortion ellipses , which are known as Tissot's indicatrices.

Each indicatrix ellipse represents the distortion at the point it is centered on. The two axes of the ellipse indicate the directions along which the scale is maximal and minimal at that point on the map.

Since scale distortion varies across the map, distortion ellipses are drawn on the projected map in an array of regular intervals to show the spatial distortion pattern across the map. The ellipses are usually centered at the intersection of meridians and parallels. Their shape represents the distortion of an imaginary circle on the spherical surface after being projected on the map plane. The size, shape and orientation of the ellipses are changed as the result of projection. Map projections and distortion.

Converting a sphere to a flat surface results in distortion. This is the most profound single fact about map projections—they distort the world—a fact that you will investigate in more detail in Module 4, Understanding and Controlling Distortion. Imagine a map projection as an attempt to reconstruct your face in two dimensions. Some maps will get the shapes of all your features just right, but not the sizes—your forehead and chin, for instance, may come out huge. Other maps will get the sizes right, but the shapes will be stretched—maybe your full, round mouth will appear wide, thin, and rather mean.

Some maps preserve distances. Measurements from the tip of your nose to your chin, ears, and eyes will be right, even though the size and shape of your features is wrong.

Other maps preserve direction. Your features may look weird, and they may be scrunched up or set too far apart, but their relative positions will be correct.

Finally, some maps are compromises—they get nothing exactly right but nothing too far wrong. In particular, compromise projections try to balance shape and area distortion. So the four spatial properties subject to distortion in a projection are:. Shape If a map preserves shape, then feature outlines like country boundaries look the same on the map as they do on the earth. A map that preserves shape is conformal. Even on a conformal map, shapes are a bit distorted for very large areas, like continents.

A conformal map distorts area—most features are depicted too large or too small. The amount of distortion, however, is regular along some lines in the map. For example, it may be constant along any given parallel. This would mean that features lying on the 20th parallel are equally distorted, features on the 40th parallel are equally distorted but differently from those on the 20th parallel , and so on. Area If a map preserves area, then the size of a feature on a map is the same relative to its size on the earth.

For example, on an equal-area world map, Norway takes up the same percentage of map space that actual Norway takes up on the earth. To look at it another way, a coin moved to different spots on the map represents the same amount of actual ground no matter where you put it. In an equal-area map, the shapes of most features are distorted. No map can preserve both shape and area for the whole world, although some come close over sizeable regions. Distance If a line from a to b on a map is the same distance accounting for scale that it is on the earth, then the map line has true scale.

No map has true scale everywhere, but most maps have at least one or two lines of true scale. An equidistant map is one that preserves true scale for all straight lines passing through a single, specified point. For example, in an equidistant map centered on Redlands , California , a linear measurement from Redlands to any other point on the map would be correct.

Direction Direction, or azimuth , is measured in degrees of angle from north. On the earth, this means that the direction from a to b is the angle between the meridian on which a lies and the great circle arc connecting a to b.

The azimuth of a to b is 22 degrees.