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 purposes1.
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’ = a11x, y’ = y/a11). 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) dφ.
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 = ( φβ sinαφ )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 = ( λα λβ cosβφ ) 1/(α +β)
one has
φ
y = ∫ cos φ dφ / (∂x/∂λ)
0
which, if α = β = 1, yields
x = λ (cos φ )½
φ
y = ∫ (cos φ)½ dφ
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 ∫ (γk - yk)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 (γk - yk)1/k ] λ
and y = f(φ, α, γ, k) is obtained from
y
sin φ - αy + (1 - α)/γ ∫ (γk - 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 a11 = 0.8679 and a22 = 1/a11 (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 a11). On a modern computing machine such a computation should take approximately
(CM)p 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 2ω 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.
REFERENCES
Adams, 0., General Theory of Equivalent Projections, Spec. Pub.. 236, pp. 46-56, U.S. Coast
and Geodetic Survery, Washington, D. C., 1945.
Airy, G., Explanation of a projection by balance of errors for maps applying to a very large extent of the earth’s surface, and comparison of this projection with other projections, Phil. Mag.,
22, 409-421, 1861.
Baar, E., The manipulation of projections for world maps, Geogr. Rev., 32, 112-120, 1947.
Behrmann, W., Zur Kritik der flächentreuen Projektionen der ganzen Erde und einer Halbkugel,
Bericht d..Königlichen Bayer. Akad. Wiss., 13, 1-48, 1909.
Biernacki, F., Theory of Representation of Surfaces for Surveyors and Cartographers, translated
from Polish, pp. 53-147, U.S. Department of Commerce, Washington, D.C., 1965.
Boggs, S., A new equal area projection for world maps, Geogr. J., 73, 241-245, 1929.
Collignon, E., Recherches sur la representation plane de la surface du globe terrestre, J. Ecole
Polytech., 24, 125-132, 1865.
Craster, J., Some equal area projections of the sphere, Geogr. J., 74, 471-474, 1929.
Eckert, M., Neue Entwürfe für Erdkarten, Petermann’s Geogr. Mitt., 52, 97-109, 1906.
Foucaut, P., Notice sur la construction de nouvelles mappemondes et de nouveaux atlas, pp.
5-10, Arras, France, 1862.
Frolov, Y., Issledovanie iskajenii ravnoelikich pseudo-tzilundaricheskich projektzii, Vestn. Leningrad. Univ. Geol. Geogr., no. 12, 148-157, 1961.
Gardner, M., The superellipse: A curve that lies between the ellipse and the rectangle, Sci.
Amer.,
113, 222-232, 1965.
Graf, U., Über die Aequideformaten der fläehentreuen Zylinderentwürfe, Petermann’s Geogr.
Mitt., 87, 281-289, 1941.
Hammer, E., Die Netzentwürfe geographischer Karten, pp. 25-82, Metzler, Stuttgart, Germany,
1887.
Hammer, E., Unechtcylindrische und unechtkonische flächentreue Abbildungen, Petermann’s
Geogr. Mitt., 46, 42-46, 1890.
Jordan, W., Zur Vergleichung der soldnerschen Koordinaten, Z. Vermess., 4, 175-190, 1875.
Kavraisky, V., Mathematischeskaya Kartografiya, pp. 229-236, Moscow, 1934.
Lambert, J., Notes and Comments on the Construction of Terrestrial and Celestial Maps, 1772.
(Introduced and translated by W. Tobler, Geography Department, University of Michigan, Ann
Arbor, 1972.)
McBryde, F., and P. Thomas, Equal Area Projections for World Statistical Maps, Spec. Pub.
245, pp. 11-44, U.S. Coast and Geod. Surv., Washington, D.C., 1949.
Meshcheryakov, G., The problem of choosing the most advantageous projections, Geod.
Aerophotogr., no. 4, 263-268, 1965.
Mollweide, C., Über die vom Prof. Schmidt in Giessen in der zweyten Abtheilung seines Handbuchs der Naturlehre S. 595 angegebene Projection der Halbkugelfläche, Mon. Korresp., 13,
152-163, 1805.
Nell, A., Aequivalente Kartenprojektionen, Petermann’s Geogr. Mitt., 45, 93-98, 1890.
Robinson, A., The use of deformational data in evaluating world map projections, Ann. Ass.
Amer. Geogr., 41, 58-74, 1951.
Tissot, M., Mémoire sur la representation des surfaces et les projections des cartes géographiques, 260 pp., Paris, 1881.
Wagner, K., Kartographische Netzentwurfe, pp. 222, 241-242, Bibliographisches Institut,
Leipzig, 1949.
Werenskiold, W., A class of equal area map projections, Norske Vidensk. Akad. Oslo, 11, 3-23,
1945.
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.