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Equations of the Mayr projection.

Article Excerpt
Introduction

Distinguished from cylindrical projections by curved meridians, but sharing the pattern of latitude, the pseudocylindrical projection emerged as a favorite design concept for new projections as the 20th century began (Snyder 1993; Delmelle 2991). Equivalency was the common characteristic among examples of these types of projections. On equal-area projections, all areas on the map are represented in their correct proportion, which is an essential criterion for the mapping of thematic variables (Hsu 1981 ; Delmelle 9001). As Tissot demonstrated, the use of the equal area property (equivalency) generally implies a high distortion of shape (Delmelle 2001).

Of the pseudo-cylindrical projections available today, Franz Mayr's (1964) seems to be the least used. The Mayr projection is a pointed-polar, equal-area, pseudo-cylindrical, world map projection. It is constructed by spacing meridians in proportion to the square-root of the cosine of the latitude and requires numerical integration for values of y (Figure 1).

The computation of one of Mayr's projection equations depends on the solution of an elliptical integral. Mayr used the Legendre tables for the elliptical functions E and F and gave the plane coordinates within one-degree latitude intervals on the 90[degrees] meridian. It is this characteristic of the projection that likely contributes to its minimal use today. The research reported here derives analytical expressions instead of using the elliptical integral and the interpolation between the table values. Four different solutions have been introduced for mapping applications. In the sections that follow, a short background is provided to orient the reader, then each solution is described, and related distortion quantities are presented and discussed.

[FIGURE 1 OMITTED]

Designing an Equivalent Pseudo-cylindrical Projection

Geometrically, cylindrical projections can be developed by unrolling a cylinder which has been wrapped around the Earth and touches at the equator. Then, meridians and parallels have been projected from the center of the globe (Snyder 1993). The construction of this graticule can be realized either graphically or mathematically with partial similarity to the geometric projections and they are called pseudo-cylindrical. Many of them are designed to be equal-area with horizontal straight lines for parallels and curved lines for meridians. The equivalency means that areas are shown correctly, i.e., the map covers exactly the same area of the corresponding region on the actual Earth.

Assuming a sphere for the geometric model of the Earth (Figure 2), the area dF of an infinitely narrow zone with an infinitely short height d[phi] at an arbitrary geographic latitude [phi] can be expressed as:

dF = 2[pi] R cos[phi] Rd[phi] = 2[pi] [R.sup.2] cos[phi] d[phi] (1)

This equation takes the form below for a segment of an arbitrary geographic longitude [lambda]:

dF - [DELTA][lambda] [R.sup[2] cos[phi] d[phi] (2a)

where:

[DELTA][lambda] = [lambda] - [lambda].sub.0] and

[[lambda].sub.0] = the central meridian in radians.

In the case of equivalency, the infinitely small area dF in Equation (2a) on the sphere equals to an infinitely small area df on the projection plane and for the cylindrical projections, this may be expressed as:

df = x dy (2b)

The multipliers x and dy in the Equation (2b) can be variously defined using different combinations of the components of Equation (2a). If the multipliers are selected as follows:

x = [lambda] R cos[phi] dy = Rd[phi] (3)

then the length x of a zone at the latitude [phi] becomes (cos [phi]) times shorter than the length ([lamdba],R) at the equator, where the distances dy remain the same. Therefore, scale is true along the parallel circles and along the central meridian. In other words, the distances on the sphere are preserved through those corresponding directions on the projection plane. This gives the equations of the pseudo-cylindrical, equal-area sinusoidal projection, known also as the Mercator-Sanson projection (Mayr 1964; Snyder and Voxland 1989; Richardus and Adler 1972), in Figure 3a:

x = [lambda] R cos [phi]

y = R[phi] (4)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

If, on the other hand, the multipliers are taken as:

x = [lambda] R

dy = R cos[phi] d[phi] (5)

then the length x is constant as ([lambda],R) and the distances dy become (cos [phi])) times shorter. The equations

x= [lambda] R y = R sin[phi] (6)

then give the equal-area, orthographic cylindrical projection of...

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