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A "slice-and-dice" approach to area equivalence in polyhedral map projections.

Article Excerpt
Introduction

Polyhedral globes (Figure 1) go as far back as to at least the times when the artist Albrecht Durer (1525) suggested the projection of a sphere upon polyhedra. They serve two primary purposes: they approximate spheres in three dimensions, thus eliminating curvature problematic to printing and manufacture. They also serve as a route to interrupted projections, whose benefits are low distortion in full-world maps. Often, polyhedral maps are created (usually as novelties) with the intent of serving both functions; the outcome is a low-distortion flat map that may also be folded into a sphere. Because polyhedral projections tessellate the sphere in a regular way, they relate to the problem of partitioning the sphere, which is important to the representation and structure of data in geographical data bases (Huang 1998).

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In 1943, the Life magazine published a map projection on a cuboctahedron, This projection was created by R. Buckminster Fuller and is one of the polyhedra Durer proposed for projections. An interesting aspect of Fuller's world map is that its transformation equations project exactly the polyhedron's edges with constant scale (Gray 1995). Then, in 1946, Bradley published a cylindrical equal-area projection on the icosahedron, which made him the first to publish an equal-area projection for polyhedral globes. But instead of using 20 equilateral spherical triangles, Bradley divided the globe into four equal-area zones that were in turn mapped onto triangles. (1) All projections mentioned so far introduce cusps (discontinuities in the first derivative) along the edges of the polyhedron's faces. The unpublished Fisher projection (see Bradley 1946) also introduces cusps along the symmetry lines of the polyhedron's faces. Cusps project as angles in parallels and meridians (Figure 2).

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Lee (1965) documents conformal projections on polyhedral faces using elliptic functions. A conformal approach eliminates angle distortion in the graticule everywhere--something that is not possible in an equal-area projection. This characteristic ensures there will never be cusps; neither between the polyhedron's faces, nor along their symmetry lines. Lee addresses an interesting advantage of the tetrahedron over all the other regular polyhedra: when unfolded repeatedly, a tetrahedron produces a map with each face enclosed by all neighboring faces.

When Fisher's approach (see Bradley 1946) is applied to non-Platonic polyhedra, discontinuities are introduced along the edges between non-congruent adjacent faces. Discontinuities are projected as actual jumps in parallels and meridians (Figure 3).

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In his modified Lambert azimuthal equal-area projection for polyhedral globes, Snyder (1992) corrects Fisher's approach for discontinuities between non-congruent adjacent faces. For Platonic polyhedra, where no non-congruent adjacent faces exist, Snyder's projection is equivalent to Fisher's projection. The drawbacks of Snyder's equal-area projection are the variations in scale along the polyhedron's edges and, more disturbingly, the serious cusps between the polyhedron's faces and along their symmetry lines.

In Song et al. (2002), a projection is proposed that uses small circles to recursively partition the polyhedron's faces into sub-triangles equal in area. Song's projection eliminates Snyder's largest cusps, but not those between adjacent faces. Song states that computational complexity appears to be the major drawback to the widespread use of small circle subdivision.

In this paper, two equal-area projections with some useful properties are presented: one with a constant scale along the polyhedron's edges and one with only very small cusps, both without discontinuities. The projections are implemented using a simple "slice-and-dice" method to obtain area equivalence. First, this method is explained. Subsequently, the two implementations are presented in terms of exact equations and compared with Snyder's projection. The objective here is to demonstrate the method and characterize its mathematical properties. Distortion analyses help the reader compare the method to other projections with the same purpose.

The Slice-and-Dice Method

The slice-and-dice method helps to obtain area equivalence in the projection of regular spherical polygons onto their regular plane counterparts. Regular polygons can be composed from a set of equal isosceles triangles, each defined by one of the edges of the polygon and the center of the polygon. Each of the isosceles triangles can, in turn, be composed from two symmetrical right triangles (Figure 4).

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Because of symmetry, analysis is required for only one right triangle. This means that area equivalence is obtained when spherical right triangle T, and any shape contained in T, are mapped onto plane right triangle T with constant areal scale. Triangles T and T' can be broken up into a large number of small cells. In the limit of infinitely small cells, the area of any shape equals the sum of the areas of all small cells that together construct that shape. So, if infinitesimal cells are mapped with constant areal scale everywhere, then any shape will also be mapped with constant areal scale.

In other words, to prove area equivalence between T and T', it is sufficient to show that infinitely small cells anywhere on T are mapped onto T' with constant areal scale. This proof is...

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