Gnomonic Projections

Figure 1. All great circles appear as straight lines on a gnomonic projection.

The gnomonic projection (the g is silent -- this is pronounced "nomonic") is a bit unfortunately named, because geodetic scientists use the term "gnomonic" in two different ways -- first, to refer to the particular family of projections described here, and second, to describe more generally any projection whose light source is located at the center of the model of the Earth. The onus is on you to understand the context in which the term gnomonic is being used and to determine which of these two meanings is appropriate.

The gnomonic projection has been around for at least 2,500 years -- the ancient Greeks are credited with using this projection to map stars. The most attractive feature of the gnomonic projection is that all great circles appear as straight lines -- a highly desirable feature for navigation. Unfortunately, neither the lengths nor the directions of these great circles are accurate (with the exception of those passing through the map's point of tangency: The directions -- put not the lengths -- of these great circles are accurate). These inaccuracies obviously detract from the projection's usefulness. Despite these drawbacks, its advantages ensure that gnomonic projections will continue to be widely used long into the future.

• Form: Gnomonic projections are planner.

• Case: Gnomonic projections are tangent.

• Aspect: Gnomonic projections can use any possible aspect; normal, transverse, or oblique.

• Variation Within Gnomonic Projections: Gnomonic projections differ from one another in their aspect. Usually, a cartographer will specify the projection's aspect by specifying the latitude and longitude of the point of tangency.

• Distortions
  • Shearing: Gnomonic projections do distort shapes, and this distortion can become very sever as one moves farther from the point of tangency. For this reason, most geodetic scientists recommend that gnomonic projections not be used to cover areas more than about 30 degrees of latitude and/or longitude from the projection's point of tangency.
  • Tearing: Gnomonic maps tend to have rounded edges, and are typically used to cover areas the size of continents. Tearing occurs along these edges. Mathematically, a gnomonic projection could be used to map just less than half the Earth (i.e., just less than a single hemisphere), but the amount of distortion along the edges of a gnomonic map that large would be enormous. For this reason, geodetic scientists recommend that gnomonic projections not be used to cover areas more than about 30 degrees of latitude and/or longitude from the projection's point of tangency. This edge distortion also makes the gnomonic projection a poor choice upon which to build an interrupted map.
  • Compression: Gnomonic projections do distort areas; they are not equivalent. Like shearing, the amount of compression in a gnomonic map increases rapidly as you move away from the point of tangency.
  • Equivalence: Gnomonic projections do distort areas; they are not equivalent. Like shearing, the amount of compression in a gnomonic map increases rapidly as you move away from the point of tangency.
  • Conformality: Like shearing and compression, gnomonic maps do distort shapes and this distortion increases rapidly as you move away from the map's point of tangency.
  • Equidistance: Like shearing and compression, gnomonic maps do distort distances and this distortion increases rapidly as you move away from the map's point of tangency.
  • Azimuthality: Gnomonic projections are azimuthal; directions from the map's point of tangency are shown accurately. Directions from all other points on the map are not shown accurately.
  
  
• Uses: The great strength of gnomonic projections is their ability to show great circles as straight lines. The projection's azimuthality is largely an afterthought.

  Shearing and compression are severe and get progressively worse toward the edges of the map; for this reason, gnomonic projections are rarely used for general purpose mapping. The gnomonic projection is an excellent choice for maps intended to show the shortest routes between points no more than about 60 degrees of latitude and/or longitude apart from one another. When used in this fashion, one must be careful not to misinterpret the map by measuring the distances and directions of these shortest routes; gnomonic maps do not show these qualities accurately.