Charts and Projections (Part One)

There are several basic terms and ideas relative to charts and projections that the reader should be familiar with before discussing the various projections used in the creation of aeronautical charts.

  1. A map or chart is a small scale representation on a plane of the surface of the earth or some portion of it.
  2. The chart projection forms the basic structure on which a chart is built and determines the fundamental characteristics of the finished chart.
  3. There are many difficulties that must be resolved when representing a portion of the surface of a sphere upon a plane. Two of these are distortion and perspective.
    • Distortion cannot be entirely avoided, but it can be controlled and systematized to some extent in the drawing of a chart. If a chart is drawn for a particular purpose, it can be drawn in such a way as to minimize the type of distortion that is most detrimental to the purpose. Surfaces that can be spread out in a plane without stretching or tearing, such as a cone or cylinder, are called developable surfaces, and those like the sphere or spheroid that cannot be formed into a plane without distortion are called non-developable. [Figure 1-19] The problem of creating a projection lies in developing a method for transferring the meridians and parallels to the chart in a manner that preserves certain desired characteristics as nearly as possible. The methods of projection are either mathematical or perspective.

      Figure 1-19. Developable and nondevelopable surfaces.

      Figure 1-19. Developable and nondevelopable surfaces.

    • The perspective or geometric projection consists of projecting a coordinate system based on the earth-sphere from a given point directly onto a developable surface. The properties and appearance of the resultant map depends upon two factors: the type of developable surface and the position of the point of projection.
  4. The mathematical projection is derived analytically to provide certain properties or characteristics that cannot be arrived at geometrically. Consider some of the choices available for selecting projections that best accommodate these properties and characteristics.
 

Choice of Projection

The ideal chart projection would portray the features of the earth in their true relationship to each other; that is, directions would be true and distances would be represented at a constant scale over the entire chart. This would result in equality of area and true shape throughout the chart. Such a relationship can only be represented on a globe. On a flat chart, it is impossible to preserve constant scale and true direction in all directions at all points, nor can both relative size and shape of the geographic features be accurately portrayed throughout the chart. The characteristics most commonly desired in a chart projection are conformality, constant scale, great circles as straight lines, rhumb lines as straight lines, true azimuth, and geographic position easily located.

Conformality

Conformality is very important for air navigation charts. For any projection to be conformal, the scale at any point must be independent of azimuth. This does not imply, however, that the scales at two points at different latitudes are equal. It means the scale at any given point is, for a short distance, equal in all directions. For conformality, the outline of areas on the chart must conform in shape to the feature being portrayed. This condition applies only to small and relatively small areas; large land masses must necessarily reflect any distortion inherent in the projection. Finally, since the meridians and parallels of earth intersect at right angles, the longitude and latitude lines on all conformal projections must exhibit this same perpendicularity. This characteristic facilitates the plotting of points by geographic coordinates.

Constant Scale

The property of constant scale throughout the entire chart is highly desirable, but impossible to obtain as it would require the scale to be the same at all points and in all directions throughout the chart.

Straight Line

The rhumb line and the great circle are the two curves that a navigator might wish to have represented on a map as straight lines. The only projection that shows all rhumb lines as straight lines is the Mercator. The only projection that shows all great circles as straight lines is the gnomonic projection. However, this is not a conformal projection and cannot be used directly for obtaining direction or distance. No conformal chart represents all great circles as straight lines.

 

True Azimuth

It would be extremely desirable to have a projection that showed directions or azimuths as true throughout the chart. This would be particularly important to the navigator, who must determine from the chart the heading to be flown. There is no chart projection representing true great circle direction along a straight line from all points to all other points.

Coordinates Easy to Locate

The geographic latitudes and longitudes of places should be easily found or plotted on the map when the latitudes and longitudes are known.

Chart Projections

Chart projections may be classified in many ways. In this book, the various projections are divided into three classes according to the type of developable surface to which the projections are related. These classes are azimuthal, cylindrical, and conical.

Azimuthal Projections

An azimuthal, or zenithal projection, is one in which points on the earth are transferred directly to a plane tangent to the earth. According to the positioning of the plane and the point of projection, various geometric projections may be derived. If the origin of the projecting rays (point of projection) is the center of the sphere, a gnomonic projection results. If it is located on the surface of the earth opposite the point of the tangent plane, the projection is a stereographic, and if it is at infinity, an orthographic projection results. Figure 1-20 shows these various points of projection.

Figure 1-20. Azimuthal projections.

Figure 1-20. Azimuthal projections. [click image to enlarge]

Gnomonic Projection

All gnomonic projections are direct perspective projections. Since the plane of every great circle cuts through the center of the sphere, the point of projection is in the plane of every great circle. This property then becomes the most important and useful characteristic of the gnomonic projection. Each and every great circle is represented by a straight line on the projection. A complete hemisphere cannot be projected onto this plane because points 90° from the center of the map project lines parallel to the plane of projection. Because the gnomonic is nonconformal, shapes or land masses are distorted, and measured angles are not true. At only one point, the center of the projection, are the azimuths of lines true. At this point, the projection is said to be azimuthal. Gnomonic projections are classified according to the point of tangency of the plane of projection. A gnomonic projection is polar gnomonic when the point of tangency is one of the poles, equatorial gnomonic when the point of tangency is at the equator and any selected meridian. [Figure 1-21]

Figure 1-21. Polar gnomonic and stereographic projections.

Figure 1-21. Polar gnomonic and stereographic projections. [click image to enlarge]

Stereographic Projection

The stereographic projection is a perspective conformal projection of the sphere. The term oblique stereographic is applied to any stereographic projection where the center of the projection is positioned at any point other than the geographic poles or the equator. If the center is coincident with one of the poles of the reference surface, the projection is called polar stereographic. The illustration in Figure 1-21 shows both gnomonic and stereographic projections. If the center lies on the equator, the primitive circle is a meridian, which gives the name meridian stereographic or equatorial stereographic.

 

Cylindrical Projections

The only cylindrical projection used for navigation is the Mercator, named after its originator, Gerhard Mercator (Kramer), who first devised this type of chart in the year 1569. The Mercator is the only projection ever constructed that is conformal and, at the same time, displays the rhumb line as a straight line. It is used for navigation, for nearly all atlases (a word coined by Mercator), and for many wall maps.

Imagine a cylinder tangent to the equator, with the source of projection at the center of the earth. It would appear much like the illustration in Figure 1-22, with the meridians being straight lines and the parallels being unequally spaced circles around the cylinder. It is obvious from Figure 1-22 that those parts of the terrestrial surface close to the poles could not be projected unless the cylinder was tremendously long, and the poles could not be projected at all.

Figure 1-22. Cylindrical projection.

Figure 1-22. Cylindrical projection.

On the earth, the parallels of latitude are perpendicular to the meridians, forming circles of progressively smaller diameters as the latitude increases. On the cylinder, the parallels of latitude are shown perpendicular to the projected meridians but, since the diameter of a cylinder is the same at any point along the longitudinal axis, the projected parallels are all the same length. If the cylinder is cut along a vertical line (a meridian) and spread flat, the meridians appear as equally spaced, vertical lines, and the parallels as horizontal lines, with distance between the horizontal lines increasing with distance away from the false (arbitrary) meridian.

The cylinder may be tangent at some great circle other than the equator, forming other types of cylindrical projections. If the cylinder is tangent at a meridian, it is a transverse cylindrical projection; if it is tangent at any point other than the equator or a meridian, it is called an oblique cylindrical projection. The patterns of latitude and longitude appear quite different on these projections because the line of tangency and the equator no longer coincide.