General Projections Concepts

For almost 500 years, it has been conclusively established that the Earth is essentially a sphere, although there were a number of intellectuals nearly 2,000 years earlier who were convinced of this. Even to the scholars who considered the Earth flat, the skies appeared hemispherical, however. It was established at an early date that attempts to prepare a flat map of a surface curving in all directions leads to distortion of one form or another.


A map projection is a device for reproducing all or part of a round body on a flat sheet. Since this cannot be done without distortion, the cartographer must choose the characteristic that is to be shown accurately at the expense of others, or a compromise of several characteristics. There is literally an infinite number of ways in which this can be done, and several hundred projections have been published, most of which are rarely used novelties. Most projections may be infinitely varied by choosing different points on the Earth as the center or as a starting point. Manifold, for example, allows the setting of the projection’s center point for most projections.


It cannot be said that there is one "best" projection for mapping. It is even risky to claim that one has found the "best" projection for a given application, unless the parameters chosen are artificially constricting. Even a carefully constructed globe is not the best map for most applications because its scale is by necessity too small, a straightedge cannot be satisfactorily used on it for measurement of distance, and it is awkward to use in general.


The characteristics normally considered in choosing a map projection are as follows:


Area - Many map projections are designed to be equal-area, so that a coin, for example, on one part of the map covers exactly the same area of the actual Earth as the same coin on any other part of the map. Shapes, angles, and scale must be distorted on most parts of such a map, but there are usually some parts of an equal-area map which are designed to retain these characteristics correctly, or very nearly so. Less common terms used for equal-area projections are equivalent, homolographic, authalic, and equiareal.


Shape - Many of the most common and most important projections are conformal or orthomorphic, in that normally the shape of every small feature of the map is shown correctly. On a conformal map of the entire Earth there are usually one or more "singular" points at which shape is still distorted. A large landmass must still be shown distorted in shape, even though its small features are shaped correctly. An important result of conformality is that relative angles at each point are correct, and the local scale in every direction around any one point is constant. Consequently, meridians intersect parallels at right (90 degree) angles on a conformal projection, just as they do on the Earth. Areas are generally enlarged or reduced throughout the map, but they are relatively correct along certain lines, depending on the projection. Nearly all large-scale maps of the Geological Survey and other mapping agencies throughout the world are now prepared on a conformal projection.


Scale - No map projection shows scale correctly throughout the map, but there are usually one or more lines on the map along which the scale remains true. By choosing the locations of these lines properly, the scale errors elsewhere may be minimized, although some errors may still be large, depending on the size of the area being mapped and the projection. Some projections show true scale between one or two points and every other point on the map, or along every meridian. They are called equidistant projections.


Direction - While conformal maps give the relative local directions correctly at any given point, there is one frequently used group of map projections, called azimuthal or zenithal, on which the directions or azimuths of all points on the map are shown correctly with respect to the center. One of these projections is also equal-area, another is conformal, and another is equidistant. There are also projections on which directions from two points are correct, or on which directions from all points to one or two selected points are correct, but these are rarely used.


Special Characteristics - Several map projections provide special characteristics that no other projection provides. On the Mercator projection, all rhumb lines, or lines of constant direction, are shown as straight lines. On the Gnomonic projection, all great circle paths - the shortest routes between points on a sphere - are shown as straight lines. On the Stereographic, all small circles, as well as great circles, are shown as circles on the map. Some newer projections are specially designed for satellite mapping. Less useful but mathematically intriguing projections have been designed to fit the sphere conformally into a square, an ellipse, a triangle, or some other geometric figure.


Method of Construction - In the days before ready access to computers, ease of construction was of greater importance. Some projections have become popular simply because they are easy to compute. With the advent of computers, very complicated formulas can be handled as routinely as simple projections in the past.


While the above features should ordinarily be considered in choosing a map projection, they are not so obvious in recognizing a projection. In fact, if the region shown on a map is no much larger than the United States, for example, even a trained eye cannot often distinguish whether the map is equal-area or conformal. It is necessary to make measurements to detect small differences in spacing or location of meridians and parallels, or to make other tests. The type of construction of the map projection is more easily recognized with experience, if the projection falls into one of the common categories.


Categories of Projections


A developable surface is one that can be transformed to a plane without distortion. There are three types of developable surfaces onto which most of the map projections used by USGS and other agencies are at least partially geometrically projected. They are the cylinder, the cone, and the plane. Actually all three are variations of the cone. A cylinder is a limiting form of a cone with an increasingly sharp point or apex (i.e., drawn out to infinity). As the cone becomes flatter, its limit is a plane.


If a cylinder is wrapped around the globe representing the Earth, so that its surface touches the Equator throughout its circumference, the meridians of longitude may be projected onto the cylinder as equidistant straight lines perpendicular to the Equator, and the parallels of latitude marked as lines parallels to the Equator, around the circumference of the cylinder and mathematically spaced for certain characteristics. When the cylinder is cut along some meridian and unrolled, a cylindrical projection with straight meridians and straight parallels results. The Mercator projection is the best-known example.


images\ill_reg_cyl_proj.gif Regular Cylindrical Projection


If a cone is placed over the globe, with its peak or apex along the polar axis of the Earth and with the surface of the cone touching the globe along some particular parallel of latitude, a conic (or conical) projection can be produced. This time the meridians are projected onto the cone as equidistant straight lines radiating from the apex, and the parallels are marked as lines around the circumference of the cone in planes perpendicular to the Earth’s axis, spaced for the desired characteristics.


images\ill_reg_conic_proj.gif Regular Conic Projection


When the cone is cut along a meridian, unrolled, and laid flat, the meridians remain straight radiating lines, but the parallels are now circular arcs centered on the apex. The angles between meridians are shown smaller than the true angles.


A plane tangent to one of the Earth’s poles is the basis for polar azimuthal projections. In this case, the group of projections is named for the function, not the plane, since all common tangent-plane projections of the sphere are azimuthal. The meridians are projected as straight lines radiating from a point, but they are spaced at their true angles instead of the smaller angles of the conic projections. The parallels of latitude are complete circles, centered on the pole.


images\ill_polar_azi_proj.gif Polar Azimuthal Projection


On some important azimuthal projections, such as the Stereographic (for the sphere) the parallels are geometrically projected from a common point of perspective; on others, such as the Azimuthal Equidistant, they are non-perspective.


The concepts outlined above may be modified in two ways, which still provide cylindrical, conic, or azimuthal projections (although the azimuthals retain this property precisely only for the sphere, not for ellipsoidal Earth models):


·      The cylinder or cone may be secant to or cut the globe at two parallels instead of being tangent to just one. This conceptually provides two standard parallels (as settable in some Manifold projections); but for most conic projections this construction is not geometrically correct. The plane may likewise cut through the globe at any parallel instead of touching a pole. Those Manifold projections which allow secant projection surfaces will allow the setting of additional standard parallels beyond what is required for the simple tangent form of the projection.

·      The axis of the cylinder or cone can have a direction different from that of the Earth’s axis, while the plane may be tangent to a point other than a pole. This type of modification leads to important oblique, transverse and Equatorial projections, in which most meridians and parallels are no longer straight lines or arcs of circles. What were standard parallels in the normal orientation now become standard lines not following parallels of latitude.


images\ill_trans_cyl_proj.gif Transverse Cylindrical Projection


images\ill_oblq_cyl_proj.gif Oblique Cylindrical Projection


images\ill_oblq_azim_proj.gif Oblique Azimuthal


Some other projections in common use resemble one or another of these categories only in some respects. The Sinusoidal projection is called pseudocylindrical because its latitude lines are parallel and straight, but its meridians are curved. The Polyconic projection is projected onto cones tangent to each parallel of latitude, so the meridians are curved, not straight. Still others are more remotely related to cylindrical, conic, or azimuthal projections, if at all.


Projection Names


Manifold System includes a vast array of different projections that are named mostly in accordance as described in US Geological Survey bulletins which tend to follow international cartographic practise for the names of the most common projections.


Many of the "standard" projections allow the use of various projection parameters as described above. Some countries have standardized on the use of a particular projection for mapping their countries that has acquired a local name when used with the locally-preferred set of parameters. Manifold includes these "projections" as choices for many of the more well known national projections.


Datum names used within Manifold originate with the U.S. National Imagery and Mapping Agency the official keeper of such data for the U.S. government.