Craster Parabolic Projections

Figure 1. The Craster parabolic projection in its uninterrupted form.

Along with the Mollweide, the Craster parabolic is the projection of choice for the aficionado of interrupted maps. It is widely used to build both parallel (Figure 2) and sinusoidal interrupted maps (Figure 3).

Figure 2. A parallel interrupted version of the Craster parabolic projection.

The earliest form of what is now known as the Craster parabolic projection was developed by John Evelyn Edmond Craster in 1929; it was refined into its modern form by Charles H. Deetz and O.S. Adams in 1934. Also in 1934, another geodetic scientist named Reinholds V. Putnins independently introduced a new projection that he called the P₄. It turns out that the P₄ was identical to the Craster parabolic, and since Craster introduced his projection first, he is typically given the credit for developing (and the right of naming) this projection.

Figure 3. A sinusoidal interrupted version of the Craster parabolic projection.

• Form: The Craster parabolic projection is what is termed pseudocylindrical.

• Case: The original cylindrical form upon which the Craster parabolic projection is based is secant, with lines of tangency running along the 36° 46' 0"N and 36° 46' 0"S lines of latitude. However, since in its final form the Craster parabolic is pseudocylindrical and hence is not based on a developable surface with an understandable physical shape, most geodetic scientists say that the Craster parabolic projection has no meaningful case.

• Aspect: Craster parabolic projections have normal aspects.

• Variation Within Craster Parabolic Projections: Craster parabolic projections differ from one another in the locations of their central meridians.

• Distortions
  • Shearing: The Craster parabolic projection is not conformal; shapes are distorted more than they would be in a truly conformal projection. However, shapes are not distorted along either the lines of tangency (which run along the 36° 46' 0"N and 36° 46' 0"S lines of latitude) or along the map's central meridian, but shape distortion increases rapidly as you move away from these lines. By interrupting the projection (which in effect places many more central meridians into the map), this shape distortion can be reduced dramatically. It is this quality of Craster parabolic projections that make them so attractive for use in interrupted maps.
  • Tearing: Craster parabolic maps show lines of latitude as parallel straight lines and lines of longitude as nonparallel lines that become increasingly curved as you move farther away from the map's central meridian. The edges of the map are highly curved lines of longitude 180 degrees from the map's central meridian. Tearing occurs along these edges. As mentioned previously, the Craster parabolic projection is extremely well suited to making both parallel (Figure 2) and sinusoidal (Figure 3) interrupted maps, and it is at least as common to see the Craster parabolic projection used to make interrupted maps as it is to see uninterrupted Craster parabolic maps.
  • Compression: Craster parabolic projections are equivalent; they do not suffer from compression.
  • Equivalence: Craster parabolic projections are equivalent; they do not suffer from compression.
  • Conformality: The Craster parabolic projection is not conformal; shapes are distorted more than they would be in a truly conformal projection. However, shapes are not distorted along either the lines of tangency (which run along the 36° 46' 0"N and 36° 46' 0"S lines of latitude) or along the map's central meridian, but shape distortion increases rapidly as you move away from these lines. By interrupting the projection (which in effect places many more central meridians into the map), this shape distortion can be reduced dramatically. It is this quality of Craster parabolic projections that make them so attractive for use in interrupted maps.
  • Equidistance: The Craster parabolic projection is not equidistant; there is no point or points from which all distances are shown accurately.
  • Azimuthality: The Craster parabolic projection is not azimuthal; there is no point or points from which all directions are shown accurately.
  
  
• Uses: The Craster parabolic projection is widely used to make world maps; it is not very useful for mapping smaller areas. It is widely used in atlases, in both its interrupted and uninterrupted forms. In this capacity, perhaps its only competition is the Mollweide projection, to which it is fairly similar.