Where is distortion on azimuthal projections least and greatest

The cone is then cut along one of the lines from its apex to the edge resulting in a projection plane with the meridians converging in the apex and curved parallels. All lines intersect at right angles as in the azimuthal and cylindrical projections. Conic projections are mainly used for medium to local scale areas.

They are especially suitable for mid-latitude regions with a large east-west extension e. As shown in Figure 6. For example, such a pseudo projection may look like a cylindrical one with its lines at their ends curved inwards. Pseudo projections are mostly used for mapping the entire surface of the Earth planisphere. Given that most GIS applications operate in larger scale areas, the pseudo projections play a less important role in the context of GIS. The map projection aspect refers to the point or line s of tangency on the projection surface.

In theory it can touch or intersect anywhere on the projection surface. The quality of a projection is best where the projection surface touches the Earth surface. For example, an azimuthal projection where the projection plane touches the Earth at or near the north pole results in significantly better projections for the close-to-polar areas than the equatorial regions.

In contrast, a cylindrical projection tangential to the equator is better for the equatorial areas than for a polar region. How is it that highly accurate, large-scale maps are based on a cylindrical projection? To project any small part of the Earth surface in high quality we change the position of the geometrical projection body relative to the Earth.

For example, with a horizontal cylinder, even areas outside the equatorial zone are represented with sufficient accuracy. We distinguish between the following three aspects of projection:. The distortion properties of any given projection surface, however, remain unchanged when the aspect is changed.

As you already know, there are three types of projection surfaces: a cone, cylinder, or plane. Imagine the Earth wrapped up in one of these projection developable surfaces. The resulting standard parallel is the line of latitude where the projection surface touches the globe. The standard parallel has no distortion. Distortion, however, increases towards the middle and the edges. Compared to the tangent approach, the secant approach allows almost doubling the area to be projected while at the same time preserving the quality of projection.

So far, we assumed the light source used for projection to be at the centre of the Earth. Changing the position of this light source can result in different perspectives in projection, i.

The light source can be positioned in three locations: the centre of the Earth, opposite sides of the Earth, and infinite location in space Figure 6. Conical and cylindrical projections usually use a light source emanating from the centre of the Earth.

In azimuthal planar projections the effects of using light sources in different positions can be seen clearly, especially on polar projections. Orthographic projection, on the other hand, almost resembles the view that a geographer would expect when seeing Earth from space. The analogy to positioning a light source applies well to the example in Figure 6. In those cases it is necessary to allow existence of curved projection rays and account for them in the underlying mathematical algorithms.

Projections are used to map the entire world as well as to map a specific area such as a continent, a war zone, etc. In any of these cases you want to have a projection that is just right exact ; that is, you want to select a projections for which distortions are known and kept to a minimum. This can be achieved by selection of appropriate parameters.

Projection parameters are a series of values that define a particular projection, that tell how the projection is related to the Earth. Projection parameters may indicate the point of tangency, or the lines where a secant surface intersects the Earth. They also define the spheroid used to create the projection, and any other information necessary to identify the projection. Projection parameters are of two types: angular and linear Kimerling et al.

Such a meridian lies in the centre of the resulting map sheet. The origin of the projection is also the origin for the x and y coordinates in the projection.

In the Figure 6. The intersection of the latitude and longitude of origin is the origin for map projections x, y. A change of latitude of origin does not affect the mapped content; only the origin of the y-coordinate values changes accordingly. An exception here are azimuthal projections: the change of the latitude of origin changes the location of the projection surface and so significantly influences the resulting map content map body. A tangent conic or cylindrical projection has one standard parallel, while a secant conic or cylindrical projection has two.

At the standard parallel, the projection shows no distortion. See Figure 6. Its intersection with the central meridian is the origin of map projection point. It is used mainly with projections that have single points of zero distortion like Gnomonic and Orthographic projections.

Note: while the latitude of origin does not have to lie in the centre, the central parallel does. Projected coordinates i. This depends on where the x- and y-axes intersect. On published maps that use x, y coordinates as reference, it is common practice to have all coordinate values positive.

For example, if your area of interest is favourably located, no action is required at your end. You can influence it by the choice of central meridian and latitude of origin. A convenient way is the use of false easting and false northing values: these are two big numbers — constants — that are added to each x- and y-coordinate, respectively.

The selected constants are big enough to make sure that all the coordinate values in your area of interest have a positive value. The y values in Austria range from ca. In order to speed up writing, the false northing value of is introduced; so from each y the value m is subtracted resulting in values with considerably fewer digits. A map scale is a ratio between a distance on a map and the respective distance in the reality. Due to stretching and shrinking that occurs while transforming ellipsoidal Earth surface onto a plane, the map scale will vary across the mapped area i.

We therefore distinguish between 2 types of scales Kimerling et al. The actual true scale is the one we can measure at any point on the map. It varies from location to location and is a direct consequence of the geometrical distortion from flattening 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. If the azimuth value from a to b is the same on a map as on the earth, then the map preserves direction from a to b. An azimuthal projection is one that preserves direction for all straight lines passing through a single, specified point.

No map has true direction everywhere. A few projections with different properties. The Lambert Conformal Conic preserves shape. The Mollweide preserves area. Compare the relative sizes of Greenland and South America in one and then the other. The Orthographic projection preserves direction. The Azimuthal Equidistant preserves both distance and direction.

The Winkel Tripel is a compromise projection. More about scale. Scale is the relationship between distance on a map or globe and distance on the earth. Suppose you have a globe that is 40 million times smaller than the earth. Its scale is ,, These two are examples of Azimuthal Projection maps which are not centred on a Pole. Most commonly, the tip of the cone is positioned over a Pole and the line where the cone touches the earth is a line of latitude; but this is not essential.

The line of latitude where the cone touches the Earth is called a Standard Parallel. This is a typical example of a world map based on the Conic Projection technique. Note how the shapes of land masses near the Standard Parallel are fairly close to the true shape when viewed from space — see the images at the beginning of this section.

Also note how land masses furthest away from the Standard Parallel are very distorted when compared to the views from space. Particularly note how massively large northern Canada and the Arctic icecaps look. Because of the distortions away from the Standard Parallel, Conic Projections are usually used to map regions of the Earth — particularly in mid-latitude areas. This map uses the same settings as the previous World Map, but it is more typical of a Conic Projection map. Distortions are greatest to the north and south — away from the Standard Parallel.

But, because the Standard Parallel runs east-west, distortions are minimal through the middle of the map. The cylinder is usually positioned over the Equator, but this is not essential.

This is an example of a cylindrical map projection and it is one of the most famous projections ever developed. It was created by a Flemish cartographer and geographer — Geradus Mercator in It is famous because it was used for centuries for marine navigation. The sole reason for this is that any line drawn on the map was a true direction.

However, shapes and distances were distorted. The first Cylindrical Projections developed had the lines of latitude and lines of longitude shown as straight lines — see the section on the Mercator projection. With advances in computers it became possible to calculate the lines of longitude as curves — thereby reducing distortions near the Poles — see the section on the Robinson projection. To distinguish between these two projections the first continued to be called a Cylindrical Projection, but the second with the curving lines of longitude was called Pseudo—Cylindrical Projection.

Firstly Projections are often named after their creator — famous names include Albers, Lambert, Mercator and Robinson.

However, without inside knowledge, this gives no indication of the properties of a projection. 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.