The Hyperelliptical and Other New Pseudo Cylindrical
Equal Area Map Projections †
W. R. TOBLER
Department of Geography, University of Michigan, Ann Arbor, Michigan 48104
Abstract : Several, mostly new, finite parameter instances of a special class of map projections are derived, illustrated, and evaluated. It is demonstrated that most of the usual evaluation measures are ordinally equivalent and thus that any one of them suffices. The new projections are shown to be at least as good as existing projections and appear otherwise advantageous.
My colleagues, Professors Kolars and Nystuen, recently asked me to suggest a map projection on which they could plot geographical information. A priori requirements were that the parallels should be horizontal on the page and that the map be equal area. Two well-known projections that satisfy these criteria are Lambert’s [1772] cylindrical equal area projection and Mercator’s
(1538) sinusoidal projection. But the former squashes the polar regions excessively, and the latter has too much shearing at the polar edges. This suggests that a compromise between the two might be more acceptable. The consequences of this idea follow, with a few examples drawn by computer for illustrative purposes 1 .
In the usual notation the requirement that the parallels be horizontal lines requires that ∂ y/ ∂λ =
0; the equal area condition is then, for a unit sphere, ∂ y/ ∂φ ∂ x/ ∂λ = cos φ . Two procedures are available to solve this equation, with the usual requirement for symmetry about the equator and about the mid-meridian. The simpler approach is to arbitrarily choose the spacing of the parallels on the central meridian and then to solve for the shape of the remaining meridians. Conversely, by choosing the shape of the meridians, one can solve for the spacing of the parallels. Meridian shapes chosen or obtained to date include trigonometric polynomials (sine and tangent curves) and algebraic polynomials (straight lines, circles, ellipses, parabolas, hyperbolas, and quartics).
With these elementary facts in mind suppose, for example, that an average is desired between
Lambert’s cylindrical projection and the sinusoidal projection. Let α be the weight of the
Lambert projection and β = 1 - α the weight of the sinusoidal. Then there are two possible combinations. In the first instance, taking the weighted average of the parallels y = β φ + α sin φ and solving for the meridian gives x = [cos φ / ( β + α cos φ )] λ . This projection, apparently first described by Foucaut [1862], is illustrated in Figure 1 for the weights α = β , further modified to make the equatorial axis twice the length of the polar axis by an equal area affine transformation
(x’ = a
11 x, y’ = y/a
11
).
Boggs [1929] combined the sinusoidal with Mollweide’s [1805] projection in this same manner.
The second weighted arithmetical average is obtained from the equation of the meridians x =
( α + β cos φ ) λ , and the spacing of the parallels is then given by
φ
y = ∫ cos φ /( α + β cos φ ) d φ .
0
The evaluation of this integral can be effected by using standard tables but depends on the ratio of α to β , which also governs the length of the line representing the poles on the map.
The simplest case, in which α = β , yields y = 2 ( φ - tan φ /2), as illustrated in Figure 2. This projection was apparently first derived in a paper by Hammer [1890] following a suggestion by

Nell [1890]. It is also briefly mentioned by Wagner [1949]. A map using α = 1/3 resembles
Adams’ [1945] quartic projection as modified by Thomas [McBryde and Thomas, 1949] to have the poles represented by lines.
A range of pseudo conical equal area projections can similarly be obtained by pairwise combination from the Werner-Bonne sinusoidal series with the corresponding projection from
Lambert’s sequence of azimuthal, conical, and cylindrical equal area projections. It would be of interest to know the corresponding polar to equatorial sequence that has Mollweide’s projection as its limit.
One may also wish to consider the geometrical averages. For example, beginning with the spacing of the parallels, y = ( φ β sin α φ ) 1/( α + β ) to obtain x = [cos φ /( ∂ y/ ∂φ )] λ ; or, again taking the simplest case a = β = 1, y = ( φ sin φ ) ½ x = λ [2 cos φ ( φ sin φ ) ½ ] / (sin φ + φ cos φ ) as illustrated in Figure 3. Alternately, starting with a weighted geometrical mean of the meridians from the Lambert and sinusoidal projections x = ( λ α λ β cos β φ ) 1/( α + β ) one has
φ
y = ∫ cos φ d φ / ( ∂ x/ ∂λ )
0 which, if α = β = 1, yields x = λ (cos φ ) ½
φ y = ∫ (cos φ ) ½ d φ
0 as shown in Figure 4, after evaluating the integral by numerical methods.
Another approach is to propose some function, not yet employed, for either the spacing of the parallels or the shape of the meridians. One such curve, recently popularized by Hein as cited by
Gardner [1965], is the hyperelliptical. This yields as the equation for the meridians
(x/ λ ) k + (y/ γ ) k = 1 from which the spacing of the parallels is deduced to be contained in
y
1/ γ ∫ ( γ k - y k ) 1/k dy = sin φ
0
The constant γ must have the value of y when φ = π /2 if the pole is to be a point. This new projection can now also immediately be averaged with, for example, Lambert’s cylindrical equal area projection with weight α ; thus x = [ α + (1 - α )/ γ ( γ k - y k ) 1/k ] λ and y = f( φ , α , γ , k) is obtained from y sin ( γ k - yk)1/k dy = 0
0
Since y cannot be computed until γ is known and γ is not available until the integral has been evaluated, it is necessary to use an iterative numerical technique, which, of course, is easily done on a digital computer. Depending on the choice of the constants α , γ , and k , this generalized hyperelliptical projection takes several different forms. The value of γ controls the length of the

pole line, and k controls the amount of bending of the meridians. If α = 0, γ = 6, and k = ½ , one obtains an equal area hypocycloid (Figure 5); if α = 0 and k = 1, the meridians are straight lines,
Collignon’s [1865] projection; using α = 0 and k = 2 gives Mollweide’s projection; α = ¼ , γ =
1.194, and k = 2 gives an average between Mollweide’s and Lambert’s projections, with emphasis on Mollweide’s (Figure 6); α = ½, γ = π /2, and k = 2 is similar but gives equal weight to Lambert and Mollweide and has a pole line (Figure 7); α = 0, γ = 1.461, and k = 1.5 has the appearance of an American football (Figure 8). Lambert’s projection, of course, is obtained as α
= 1. The map finally chosen uses α = 0, γ = 1.183136, and k = 2.5, as modified by the affine coefficients a
11
= 0.8679 and a
22
= 1/a
11
(Figure 9), and has an attractive overall shape. The hyperelliptical equation is obviously of great generality and contains many special cases.
Tables 1, 2, 3, and 4 allow some of the new projections to be compared with those already mentioned and with those of Eckert [1906], Craster [1929], Kavraislcy [1934], Werenskiold
[1945], Baar [1947], and McBryde and Thomas [1949]. The tables show the measures of distortion as computed before multiplication of the maps by the equal area affine transformation often used to adjust the ratio of the midmeridian to the equator or to obtain a pair of standard parallels. The use of such affine coefficients may reduce the distortion even further. The most extensive computation of distortion for equal area projections is that by Behrmann [1909], based on the work of Tissot [1881] and Hammer [1887]. More recent results have been provided by
Graf [1941] and by Robinson [1951]. Further measures of distortion were suggested by Airy
[1861] and by Jordan [1875] and are described in works by Biernacki [1965], Mescheryakov
[1965], and Frolov [1961]. If it is considered necessary to reduce the distortion to an absolute minimum, it seems possible to do so by numerical exploration of the topography of the function space, since the projections in question depend on only a finite small number of parameters ( α , γ , k , and a
11
(CM) p
). On a modern computing machine such a computation should take approximately
seconds, where M is the multiplication time of the computer, p is the number of parameters, and C depends on the density of the values of the parameters and the resolution of the finite difference scheme employed for the evaluation of the distortion. In all cases it is necessary to decide on geographical grounds which properties of the projection are most important and then to give these properties a precise definition as a mathematical statement capable of being computed. What then is the mathematical equivalent to the intuitive objective of reducing the squashing of the polar regions on Lambert’s projection or of the shearing of the sinusoidal projection? My candidate is Jordan’s measure, not Behrmann’s, but, for equal area projections, they all tend to measure departure from conformality. Careful examination of an expanded version of the tables presented here in fact demonstrates that the measures are all monotone relative to each other and thus that any one of these measures is sufficient to rank order these pseudo cylindrical equal area map projections. The projections analyzed in the tables given here can be evaluated by averaging the maximum angular distortion 2 ω along each parallel, multiplying this average by the cosine of the latitude, summing these values, and then dividing by the sum of the cosines. This provides a rapid finite difference estimate of
Behrmann’s mean maximum angular distortion. When tables giving twice the detail of those published here are used, this computation in each case yields 30 degrees, approximately, and demonstrates that all three of these projections compare very favorably with the ones currently in use.
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1 The computer programs used to calculate these projections and the world outline tape can be obtained at cost from the Geography Program Exchange, Computer Institute, Michigan State
University, East Lansing, Michigan 48823.
† JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 78, NO. 11, APRIL 10, 1973: 1753-1759
(Received September 12, 1972; revised December 18, 1972.)
Copyright © 1973 by the American Geophysical Union.
I have added some (unpublished) supplementary notes to the end of this scanned version.