Polyconic Projections

Figure 1. The Polyconic Projection. This map's central meridian runs along the prime meridian.

The origins of the polyconic projection are a little obscure, but most researchers credit Ferdinand Rudolph Hassler with developing the first comprehensive description of this projection around 1820. Hassler was a Swiss scientist who became famous for conducting the first complete and accurate survey of the coast of the United States. He also founded what is now known as the U.S. Coast Survey Office, and in general instilled a tradition of scientific rigor into the U.S. government's efforts to measure quantities and conduct surveys.

The polyconic projection is neither conformal nor equivalent; it is a compromise projection that attempts to minimize all distortions while not eliminating any particular type of distortion. It was an extremely popular projection from the mid 1800s through the mid 1900s; its popularity has waned in the last 50 years. This loss in popularity is vividly demonstrated by how the polyconic projection has been handled by the U.S. Geological Survey. From 1886 through approximately 1957, the USGS used the polyconic projection as its standard for the production of all of its 7½ minute quad maps. However, since 1957, new quad maps have used the Lambert projection instead (Note that since 1957, many 7½ minute quads incorrectly state that they were produced using a polyconic projection; someday the USGS will get it right).

• Form: Polyconic means "many cones," and the developable surface in a polyconic projection is exactly that. A classic tangent, conic projection consists of a single cone draped over a model of the Earth (Figure 2). However, there is no reason why you can't extend this idea and have two cones with differing angles draped over your model of the Earth (Figure 3). But why stop at two? Figure 4 shows three cones, one of which has its apex pointed in the opposite direction from the other two.
  

  Figure 2. A classic tangent, conic projection with the cone-shaped developable surface (in red) draped over the model of the Earth (in blue).	Figure 3. A multiconic projection with a developable surface made up of two cones. In this case, both cones (shown in red and green) have their apexes above the North Pole of the model of the Earth.
  Figure 4. A multiconic projection with a developable surface made up of three cones. In this case, two cones (shown in red and green) have their apexes above the North Pole and the third cone (shown in yellow) has its apex over the South Pole of the model of the Earth.

  A true polyconic projection carries this idea to the extreme, and has an infinite number of cones draped over the model of the Earth. Half of these cones have their apexes over the North Pole, and the other half have their apexes over the South Pole. Thus, the form of the polyconic projection could accurately be described as involving an infinite number of cones, or a little less specifically, as involving many cones -- but that's exactly what polyconic means. Thus, the form of the polyconic projection is usually described as being "polyconic."

• Case: The polyconic projection is tangent, with each of its infinite number of cones having a single line of tangency. Thus, the projection as a whole has an infinite number of lines of tangency.

• Aspect: The polyconic projection uses a normal aspect.

• Variation Within Polyconic Projections: Polyconic projections differ in the location of their central meridians.

• Distortions
  • Shearing: The polyconic projection is not conformal. There is little shape distortion along the map's central meridian; but the distortion increases as you move to the east or west of the central meridian.
  • Tearing: Polyconic projections produce maps with an unusual, butterfly-like shape. The outer edges of the map are curved lines of longitude 180 degrees from the map's central meridian. Tearing occurs along these edges. The polyconic projection is not used to build interrupted maps.
  • Compression: The polyconic projection is not equivalent. There is little area distortion along the map's central meridian; but the distortion increases as you move to the east or west of the central meridian.
  • Conformality: The polyconic projection is not conformal. There is little shape distortion along the map's central meridian; but the distortion increases as you move to the east or west of the central meridian.
  • Equidistance: The polyconic projection is not equidistance; there is no set of point(s) from which distances are accurately depicted. In general, east-to-west distances are not badly distorted in the polyconic projection, but north-to-south distances become badly distorted as you move farther from the map's central meridian.
  • Azimuthality: The polyconic projection is not azimuthal; the only place where directions are shown with any degree of accuracy at all is along the map's central meridian.
  
  
• Uses: The polyconic is a somewhat unusual projection that produces maps with a unique set of qualities. In the past, the projection was more highly regarded than it is today. Currently, the polyconic is considered suitable only for mapping relatively small areas near the projection's central meridian. Other projections are considered more suitable for mapping large areas.