NEW P R O J E C T I O N S F O R W O R L D MAPS / A Q U A N T I T A T I V E -P E R C E P T I V E A P P R O A C H

FRANK CANTERS

Geografisch InstituutlVrije UniversiteitlBrussel, Belgium

ABSTRACT The increasing use of maps in mass communication influences to a large extent our concept of geographic space. This is particularly so for world maps. Therefore, the preservation of continental shape should be regarded as an important property of such maps. This paper provides a quantitative measure of continental shape distortion. An operational tool is proposed for the develop­ment of world maps that are acceptable from a perceptive point of view but which, at the same time, result from the minimization of an objective distortion criterion.

INTRODUCTION

In the past the development of new map projections was very often seen as a challenging mathematical exercise, which leaves us today with hundreds of pro­jections of which only a small fraction is actually used. Most projections for the mapping of the entire world were not developed for specific purposes and the mathematical complexity of many of these projections is not necessarily a guaran­tee for good maps.

The selection of a projection for a small-scale map should normally involve a qualitative, followed by a quantitative analysis (Canters and Decleir 1989). During the qualitative analysis a number of constraints are set up, which may already define to a large extent the final appearance of the map. These constraints are related to the intended purpose of the map and may impose the special properties which are desired (equivalence, conformality, rectilinear great circles etc.), the class of projection (which defines the nature of meridians and parallels, e.g., a pseudocylindrical projection), the way of representing the pole (point, straight line, curved line), the aspect (normal, transverse, oblique). After formulation of these requirements the spectrum of possible map projections is already consider­ably narrowed. A quantitative analysis, based on distortion characteristics, can then help to select the most suitable one. If the set of constraints is very restrictive it may even happen that none of the existing projections meets the given require­ments. In those cases the development of an entirely new projection is justified.

Until recently, the equal-area property was regarded as the most important property for world maps, especially when it concerned the mapping of statistical data. Today, projections which are neither conformai nor equal-area are becom­ing increasingly popular in small-scale cartography. Their associated favourable distortion patterns result from balancing the angular and areal distortions and lead to a smaller deformation of large shapes. So by giving up the equal-area property it is possible to obtain an image of the earth that better resembles the shape of the continents on the globe. Nevertheless the selection or development of a projection giving a portrayal of the continents as realistic as it can be is not self-evident.

First of all, the deformation of large shapes is difficult to quantify. Tissot's well-known theory of distortion is a local theory and a projection with a low angular distortion only provides a good representation of shape for an infinitely

FRANK CANTERS is a Research Assistant in the Department of Geography, Vrije Universiteit Brussel, Belgium. The author would like to thank Hugo Decleir for his comments on an earlier draft of this paper. The research was supported financially by the Belgian National Fund for Scientific Research (NFWO) through a research grant to the author, MS submitted April 1989

CARTOGRAPHICA VOL 26 No 2 SUMMER 1989 pp 53-71

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FIGURE 1. Snyder's minimum-error pseudocyclindrical equal-area projection with pole line.

small area around every point on the map. Hence, the traditional theory ot minimum-error projections (see Snyder 1985, p. 57) provides good results when applied to relatively small areas, but fails to produce an acceptable world map (Figure 1). More recently, several attempts have been made to quantify finite distortions. Tobler (1977) developed a minimum-error projection for the United States by minimizing the mean linear distortion of all distances between the points of a regular grid, covering the area of interest (Snyder 1985, p. 78). Peters (1975) used a similar approach to optimize the parameters in the transformation formu­las of existing projections for world maps. However, instead of working with a grid, he minimized the mean linear distortion of 30,000 random distances cover­ing the surface of the earth. Although Peters' idea is very interesting, the results of his analysis show that his method does not solve the problem of the deformation of large continental shapes. From his numerical results Peters concludes that giving up the equal-area property only leads to slight improvements. However a visual comparison of the maps involved in his analysis shows the contrary. Therefore it is clear that any attempt to develop adequate world maps by quantitative techniques can only be succesful if certain perceptive requirements are taken into account during the design process. It is also possible to develop new maps solely on the ground of a perceptive evaluation. In 1974, A.H. Robinson described a new map projection especially designed for general purpose world maps (Robinson 1974). Recently the projection was adopted by the National Geographic Society to cele­brate their centennial anniversary. The publication of this new 'standard' world map forms part of their current campaign for geographic literacy, in which

NEW PROJECTIONS FOR WORLD MAPS 55

0 5000 K M

FIGURE 2. Robinson's pseudocylindrical projection with pole line.

cartography plays a key role. Within a framework of given constraints defining the general appearance of the map, Robinson's projection was developed by an iterative plotting process that was repeated until the shapes of the land masses in all except the higher latitudes were as 'realistic' as they could be. Therefore, the final map is to a very large extent the result of the designer's experience. It has no mathematical formulas in the usual sense. Although the map is very pleasing to the eye (Figure 2) it may be argued whether the total reliability on the cartog- rapher's perception does not introduce too much subjectivity in the designing process. Perception is a very complicated psychophysical process, individualistic to a certain extent, and it is likely that a given number of cartographers would produce as many different maps when applying the procedure just described. Therefore it is the author's opinion that most advantage can be derived from the coupling of a quantitative evaluation technique with a process of perceptive verification. In this way subjectivity is limited to the introduction of a number of appropriate constraints defining the general appearance of the graticule. It must however be admitted that a careful definition of these constraints remains a necessary condition for the development of acceptable maps.

This paper attempts to provide an operational tool for the development of new world maps within a set of given constraints. It will be shown how, through an iterative procedure, the results of the quantitative analysis can lead to a systematic refinement of the imposed constraints in order to obtain maps that are acceptable from a perceptive point of view but at the same time result from the minimization of an objective distortion criterion.

A PRACTICAL DISTORTION CRITERION

Considering the objectives of this study it is desirable to quantify distortion by a single measure. Since areal and angular distortion are mutually exclusive and both result from a scale distortion, varying from point to point and with direction, it is opportune to opt for a mean linear distortion value. As local distortion theory is not suited to describe the deformation of large shapes it was decided to measure the distortion of finite distances. Peters (1975) defined the linear distortion D of a finite distance S as

with S = distance between two points on a globe with same nominal scale as the

s = distance between the two corresponding points on the map.

This measure has the advantage of assigning the same weight to enlargements and reductions by the same factor. The obtained value D can be converted to a corrresponding scale factor K (= distortion of a distance of unit length) as follows

which leads to the following quadratic equation

and yields two reciprocal solutions

The scale factors K , and K 2 respectively correspond with the enlargement or reduction that will occur for a given linear distortion D.

Peters compared and optimized existing projections by averaging the value of D for 30,000 distances, randomly chosen over the entire surface of the earth. Later he refined his method by only considering distances connecting points on the continental surface (Peters 1975). The results of Peters' analysis are shown in Table 1. His so-called 'Entfernungsbezogene Weltkarte', which has the lowest mean value for D, is obtained through optimization of the parameters of the well-known Hammer-Wagner projection. In the present paper the value of K , , which can be derived from D, was averged over all distances. So a mean value of K , = 1.20 signifies that on the average distances are enlarged or reduced by 20%. Alternatively the value of K 2 could have been averaged, yielding a reciprocal mean

'

value. Since Peters generates 60,000 points evenly spread over the continental area,

the resulting 30,000 distances range from o to n. This partly explains why Peters'

NEW PROJECTIONS FOR WORLD MAPS 57

TABLE I . MEAN LINEAR DISTORTION FOR A NUMBER OF WORLD MAP PROJECTIONS (AFTER PETERS, I 978)

Peters' results 119781 D K , K ,

Equal-area projections Entfernungsbezogene Weltkarte

Wagner's polyconic equal-area projection with pole line [Hammer-Wagner projection]

Mollweide's pseudocylindrical equal-area projection

Hammer's polyconic equal-area projection [Hammer-Aitoff projection]

Sanson's pseudocylindrical equal-area projection

Behrmann's cylindrical equal-area projection with standard latitude 30" Other projections Winkel's polyconic projection with equally spaced parallels and pole line [Winkel-Tripel projection]

Aitoff s polyconic projection with equally spaczd parallels

Kavraisky's pseudocylindrical projection with equally spaced parallels, elliptical meridians and pole line [Kavraisky VII]

Cylindrical equidistant projection with standard latitude 30" Mercator's cylindrical conformal projection

results are not convincing. Indeed, Figure 3 shows how the mean linear distortion value of a projection substantially decreases with increasing distance. This results from the compensation of subsequent enlargements and reductions that occur along a line and indicates how large distances reduce the practical value of the distortion measure. The phenomenon is observed for every projection but, as illustrated in Figure 3, the distance from which compensation starts to occur depends on the nature of the projection. For pseudocylindrical (Mollweide) and polyconic projections (Winkel-Tripel) compensation already occurs for distances around 30°, while for cylindrical projections (Behrmann, Mercator) it is only observed for distances exceeding 50'. In other words, for cylindrical projections linear distortion has an impact over larger distances than for pseudocylindrical and polyconic projections. These findings suggest that the introduction of a projection dependent 'cut-off distance, which narrows the distance spectrum actually used in the analysis, should lead to a more reliable measure of the over-all distortion of continental area. Since this paper deals with the development of polyconic and pseudocylindrical projections that minimize continental shape distortion, the cut-off distance was chosen at 30'.

Peters' distortion criterion has another serious drawback. In the classical theory of map projection the whole process of representing the earth on a mapcan be schematized in two stages. First, the spherical model of the earth (a globe with radius RE = 637 1 km) is reduced to a smaller sphere with radius R, the so-called 'generating globe'. During this process no distortion occurs. In the second stage the generating globe is transformed into a map. Thereby a certain amount of

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FIGURE 3. Mean linear scale factor as a function of distance for some well-known projections: 1 Winkel-Tripel, 2 Mollweide, 3 Behrmann, 4 Mercator.

distortion is introduced depending on the nature of the transformation. The nominal scale of the resulting map is defined as the ratio of the radius of the generating globe to the radius of the spherical earth. It is preserved along those lines or in those points where no distortion occurs. Consider for example the well-known cylindrical equidistant projection with one standard parallel (Plate Carrée), which is defined as

For this projection the linear scale factor along the meridians h and along the parallels k is respectively given by

Thus the nominal scale is preserved along the equator, as well as along each meridian. All other distances are stretched. However it is clear that given the generating globe with radius R scale variation can be minimized by multiplying both equations (5) and (6) with a factor kO < 1 :

NEW PROJECTIONS FOR WORLD MAPS 59

FIGURE 4. Mean linear scale factor as a function of the scale factor along the equator for the Plate Carree.

Although equations (9) and ( lo) do not define a Plate Carree s.s. (k = k, < 1 along the equator), the new graticule is merely a reduction of the original one, which implies that the shape of the continents is identical on both. However both graticules have a substantially different mean linear scale factor. This is illustrated in Figure q which represents the mean linear scale factor as a function of the linear scale factor k, along the equator. Distortion reaches its minimum for k, = 0.9137. This optimal value of k, is, of course, projection dependent. Therefore, since it is strictly the intention to use the mean linear scale factor as an objective criterion for the comparison of continental shape distortion as it occurs on different projection systems, their graticules need to be scaled till the distortion value reaches its minimum. For the minimum-error projections developed in this paper, this scale adjustment is inherent to the minimization procedure.

The algorithm finally used for the selection of the representative distances generates a set of distances with a rectangular frequency distribution. It can be verified that 5,000 distances already yield stable results which allows reduction of computation time by a factor of six.

AN OPERATIONAL PROCEDURE FOR THE DEVELOPMENT OF NEW WORLD MAPS

Once a criterion for the evaluation of map projections is established it is possible to develop new maps or to modify existing ones through optimization of the trans- formation parameters. The complexity of the transformation system will grow with the number of parameters involved, while the mean linear distortion will

decrease. However, as will be shown, an uncontrolled optimization of a trans- formation with a high level of complexity often results in a map which is not attractive from a perceptive point of view. Limiting the degree of freedom by a well-considered imposition of constraints can lead to better maps even although the mean distortion value will increase.

In general, a cartographic transformation can be described by two polyno- mial expressions defining the relationship between the x,y coordinates in the plane and the geographical coordinates +, A on the globe:

fl(+, A) = C, + C,A + C3c$ + C4A2 + C5A+ + C&' f C7A3 f C d 2 + f CgA$* + C1db3 + c , , A ~ + C , , A ~ + + C,,A~+' + C14k+3 + + C l d 5 + C17h4+ + C,8A3+2 + C,gA2+3 + C2,h+4 + C21+5 (13)

fz(+, A) = C,' + C,'A + C3'+ + C4'A2 + C,'A+ + CG'+' + C7'A3 + C8'h2+ + CglA+' + C,,'+3 + Cl11A4 + Cl2'X3+ + C,3'X2+2 + C141Xc$3 + C151+4 + C16'A5 + C17'A4+ + C18'A3+2 + C19'AY+3 + C20'A44 + C21+5 (14)

These two fifth-order polynomials have the desired flexibility to support (some- times with minor modifications) the development of a great variety of graticules corresponding with different combinations of constraints. Starting from a high level of complexity (40 parameters) the number of parameters will be systemati- cally reduced through the introduction of new constraints, leading from relatively irregular graticules to more conventional ones. Mathematically, most constraints are easy to impose. Parameters will be optimized by minimization of the mean linear distortion. The algorithm which was developed is based on a simplex method proposed by Nelder and Mead (1 965). It will become clear that the careful observation of an optimized map may lead to a refinement of the constraints, in its turn resulting in a better map from a perceptive point of view. This interaction is a necessary condition for the development of maps that are visually acceptable but at the same time reduce distortion as far as possible.

APPLICATION OF THE DESCRIBED PROCEDURE

All graticules further described, with exception of the pointed-polar projection, minimize the mean linear distortion over continental area, Antarctica not in- cluded. The latter is situated too eccentrically so that it would only worsen the representation of the other continents while being split up anyway.

Optimization without constraints When no constraints are imposed the optimization of the transformation system defined by equations (1 i) , (iz), (13) and (14) involves not less than 40 (!) para- meters (C, and C,' becoming zero if the origin of the coordinate system is chosen to coincide with the intersection of the central meridian and the equator). Figure 5

NEW PROJECTIONS FOR WORLD MAPS 6 l

FIGURE 5. Minimum-error polyconic projection obtained after a non-constrained optimization.

TABLE 2 . POLYNOMIAL COEFFICIENTS AND MEAN

LINEAR DISTORTION FOR THE NON-CONSTRAINED

MINIMUM-ERROR POLYCONIC PROJECTION

C 2 = 1.0542 C2' = —0.0009 C 3 = -0.0173 C3 = 0.9744 C4 = 0.0088 C4' = 0.2607 C5 = -0.4146 C5' = —0.0040 C6 = 0.0036 C6' = 0.0712 C, = -0.0449 C7' = 0.0005 C8 = 0.0010 C8' = —0.0708 C9 = -0.0432 C9 = -0.0002

C10 = 0.0055 C10' = -O.0694 C11 = 0.0001 C11' = —0.0081 Cl2 = 0.0144 C12' = 0.0001 C13 = -0.0007 C13' = -0.0146

C14 = -0.0499 C14' = 0.0004 Cl5 = 0.0053 C15' = -0.0207 Cl6 = 0.0007 C16 = —0.0001 C,7 = —0.0006 C17' = 0.0047 Cl8 = -0.0042 Cl8' = 0.0003 C19 = -0.0020 C19' = —0.0298 C20 = 0.0388 C20' = 0.0014 C21 = 0.0017 C21' = 0.0491

D = 0.0227 K1 = 1.0465 K2 = 0.9556

shows the obtained graticule while Table 2 gives the values of the polynomial coefficients.

Since no constraints were imposed there is no doubt that this graticule has the lowest mean linear distortion of all. Due to the uneven distribution of the con­tinental surface the equator deflects to the north, leading to a good representation of the northern hemisphere. The southern continents however are severely stretched in the E-W direction. This bias can be partly compensated by forcing the equator to coincide with the x-axis. It is also clear from Figure 5 that the minimum-

error map is almost symmetrical with respect to the central meridian. Therefore the imposition of symmetry about the central meridian will only slightly influence the mean distortion value of the map. It will, however, result in a substantial decrease in the number of parameters to be optimized.

Also disturbing are the overlaps that occur in less important parts of the map to allow for a better representation of the regions of interest. In practice it usually suffices to impose some additional constraints on the form of the pole line in order to avoid these overlaps. All overlaps which initially occurred for some of the minimizations described in this paper were avoided on the final maps by merely imposing axlah r o and ay/aA r o in the point A = IT, + = 1~12 (as well as axlah r o and ayah 4 o in the point A = IT, + = -1~12 for graticules which are not symmetrical about the equator).

The introduction of geometric constraints If the coefficients of even powers of A in (13) and of odd powers of A in (14) are made zero the transformation will become symmetrical about they-axis, which will then coincide with a straight central meridian. This symmetry constraint will reduce the number of parameters in the optimization problem from 40 to 20. Moreover, forcing the equator to coincide with the x-axis and letting it be evenly divided by the meridians, equations (13) and (14) respectively simplify to

The optimized grid is shown in Figure 6. Table 3 gives the values of the 16 parameters defining the graticule. It is clear that the representation of both hemispheres is more balanced now. Yet maps with a straight equator which are only symmetrical about the y-axis are hardly used unless it concerns interrupted forms. Mostly the absence of symmetry about the straight equator is not appreci- ated. Practically all world maps which are actually used are symmetrical about the equator and the central meridian. By imposing this additional constraint (1 5) and (1 6) respectively become

Optimization of the l o parameters involved (Table 4) yields the map shown in Figure 7. As a result of the symmetry about the equator the concentration of continental surface in the northern hemisphere highly influences the appearance of the whole graticule. The decrease of the parallel spacing with increasing latitude leads to a stretching of the equatorial areas which is disturbing from a perceptive point of view.

Towards an 'optimal' projection for world maps It is obvious from the foregoing that additional constraints have to be imposed to obtain a map which does not depart from the familiar shapes of the continents

FIGURE 6. Minimum-error polyconic projection with pole line, straight equator and symmetry about the central meridian.

TABLE 3. POLYNOMIAL COEFFICIENTS AND MEAN

LINEAR DISTORTION FOR THE MINIMUM-ERROR POLYCONIC PROJECTION WITH POLE LINE, STRAIGHT EQUATOR A N D SYMMETRY ABOUT THE

CENTRAL MERIDIAN

C, = 0.8735 C, = -0.0793 C, = -0.1630 C,, = -0.01 11

C,, = 0.0958 C,g = -0.0030 C,, = 0.0034 C,' = I ,0065

D = 0.0514 K , = 1.1083

C6' = 0.0236 C8' = 0.0483 C,,' = -0.0583 C,,' = -0.ooog c,,' = 0.0159 C,,' = 0.0004 C,,' = -0.0104 C,,' = -0.0074

K , = 0.9022

TABLE 4. POLYNOMIAL COEFFICIENTS AND MEAN

LINEAR DISTORTION FOR THE MINIMUM-ERROR

POLYCONIC PROJECTION WITH POLE LINE

c, = 0.8202 cg'-0.0398 Cg = -0.1295 C,,' = -0.0165 CLB = -0.0091 CB7' = 0.0001

C,, = 0.0358 C,,' = -0.01 18 C,. = 1.0101 C,,' = -0.0071

D = 0.0556 K, = I. I I 78 K , = 0.8946

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TABLE 6 . MEAN LINEAR DISTORTION FOR A NUMBER OF WELI .-KNOWN POLYCONIC F 'ROJECTIONS

Polyconic projections D K1 K2

Winkel's polyconic projection with equally spaced parallels and pole line [Winkel-Tripel projection] 0.0664 1.1423 0.8754

Wagner's polyconic equal-area projection with pole line [Hammer-Wagner projection] 0.0749 1.1619 0.8607

Aitoff s polyconic projection with equally spaced parallels [Aitoff projection] 0.0894 1.1963 0.8359

Hammer's polyconic equal-area projection [Hammer-Aitoff projection] 0.0948 1.2095 0.8268

van der Grinten's polyconic projection with circular meridians and parallels [van der Grinten projection] 0 . 1 4 3 2 13342 o.7495

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FIGURE 8. Minimum-error polyconic projection with equally spaced parallels and pole line.

while at the same time having a mean linear distortion which is as low as possible. The shortcomings of the map shown in Figure 7 suggest an equal spacing of the parallels along the central meridian. This means that for λ = of2( ,λ,.) has to become a linear function of the latitude. Expression (18) will then simplify to

After optimization of the 8 remaining parameters (Table 5), a regular grati­cule is obtained (Figure 8). It has a mean linear distortion value which is lower than for other well-known polyconic projections (Table 6), i.e., projections with curved meridians and parallels according to Maling's classification scheme (1973, p. 101). Thus the term 'polyconic' is not used here in a strictly geometrical sense but refers to group A of Tobler's parametric classification (Tobler 1962). Visual comparison of the optimized graticule with the very popular Winkel-Tripel projection shows that both maps represent the continental surface in a very similar way. This resemblance is also apparent from the distortion values. Nevertheless a further improvement of the continental shapes is still possible. In consequence of the high concentration of land surface in the middle latitudes, the optimized map of Figure 8 preserves the shapes of the continents best in the temperate zones. The equato­rial areas are compressed in the E-W direction, resulting in a substantial shape distortion of Africa, South-America and even Australia. Due to the central posi­tion of these continents on conventional maps the eye is very sensitive to their

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FIGURE g. Minimum-error polyconic projection with equally spaced parallels, pole line and 2/1 ratio of the axes.

deformation. By imposing a 2/1 ratio of the axes the shape of the land masses in the lower latitudes can be improved at the cost of a more considerable E-W stretching in the higher latitudes. This can easily be done by putting C2 in (17) equal to C3' in (19). The optimized map (Figure 9, Table 7) has a slightly higher mean linear distortion value than the Winkel-Tripel projection. However, the correct ratio of the axes has a pleasing effect on the over-all representation of the continental surface.

Pseudocylindrical projections The above described procedure can be applied to other than the polyconic class of projections by imposing additional constraints on the form of meridians and/or parallels. This will be demonstrated for the pseudocylindricals, a class of projec­tions which is very popular for the mapping of the entire surface of the earth. Since pseudocylindrical projections are characterized by straight horizontal para­llels and curved meridians, the y-coordinate is a function of the latitude only. By demanding symmetry about the two coordinate axes as well as equally spaced parallels which are evenly divided by the meridians, equations (11), (12), (13) and (14) simplify to

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TABLE 7 . POLYNOMIAL COEFFICIENTS AND MEAN LINEAR DISTORTION FOR THE MINIMUM-ERROR POLYCONIC PROJECTION WITH EQUALLY SPACED PARALLELS, POLE LINE AND 2 /1 RATIO OF T H E AXES

C2 = 0.9305 C3 = 0.9305 C9 = -0.1968 C8 = 0.0394 Cl8 = -0.0067 C17' = 0.0005 C20 = 0.0076 Cl9' = -0.0115

0 = 0.0678 K1 = 1.1454 K2 = 0.8731

TABLE 8 . POLYNOMIAL COEFFICIENTS AND MEAN LINEAR DISTORTION FOR THE MINIMUM-ERROR PSEUDOCYLINDRICAL PROJECTION WITH EQUALLY SPACED PARALLELS AND POLE LINE

Optimization of the four parameters involved (Table 8) yields the graticule shown in Figure 10. It is interesting to note that it is almost identical to Kavraisky's seventh projection, which has the lowest mean linear distortion of all well-known pseudocylindrical projections that represent the pole as a line (Table 9). Com­pared with Robinson's empirically determined projection the newly developed graticule has a mean linear distortion value which is slightly lower. The equal spacing of the parallels guarantees a better shape of the continents in the higher latitudes at the cost of a more substantial areal distortion.

Pointed-polar projections It is well-known that pseudocylindrical and polyconic projections with a pole line of finite length have more balanced distortion patterns then pointed-polar projec­tions belonging to the same classes. The latter always show an excessive compress­ion of the polar areas (Canters and Decleir 1989). Yet it is very easy to develop optimized projections which represent the pole as a point. It suffices to multiply the x-coordinate as well as each term which holds X in the expression for the -coordinate by cos γ • Thus a pointed-polar equivalent of the map shown in

Figure 9 is defined as

Optimization, Antarctica included, yields the map shown in Figure 11. Table 10 gives the values of the eight parameters defining the graticule. As expected, the mean linear distortion value is considerably higher than the value obtained for the

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FIGURE 10. Minimum-error pseudocylindrical projection with equally spaced parallels and pole line.

TABLE 9 . MEAN LINEAR DISTORTION FOR A NUMBER OF WELL-KNOWN PSEUDOCYLINDRICAL PROJECTIONS

Pseudocylindrical projections D K1 K2

Kavraisky's pseudocylindrical projection with equally spaced parallels, elliptical meridians and pole line [Kavraisky VIIprojection] 0.0698 1.1501 0.8695

Robinson's pseudocylindrical projection with pole line [Robinson projection] 0.0705 1.1518 0.8682

Wagner's pseudocylindrical equal-area projection with elliptical meridians and pole line [Wagner iv projection] 0.0786 1.1707 0.8542

McBryde and Thomas' pseudocylindrical equal-area projection with quartic meridians and pole line [Flat-polar quartic authalic projection] 0.0887 1.1947 0.8370

Mollweide's pseudocylindrical equal-area projection with elliptical meridians [Mollweide projection] 0.0895 1.1967 0.8356

Sanson's pseudocylindrical equal-area projection with sinusoidal meridians [Sinusoidal projection] 0.1142 1.2578 0.7950

optimal polyconic projection with pole line (Figure 9). However, in spite of the inevitable compression of the polar areas the over-all shape of the continental area is far more pleasing than on the well-known pointed-polar polyconic projections (Aitoff, Hammer-Aitoff).

Oblique projections One of the easiest ways to reduce distortion is by centering the distortion pattern

FIGURE 11. M i n i m u m - e r r o r p o l y c o n i c p r o j e c t i o n w i t h e q u a l l y s p a c e d p a r a l l e l s and 211 r a t i o of t h e a x e s .

TABLE 10. POLYNOMIAL COEFFICIENTS AND MEAN

LINEAR DISTORTION FOR T H E MINIMUM-ERROR

POLYCONIC PROJECTION WITH EQUALLY SPACED

PARALLELS A N D 211 RATIO OF THE AXES

cs = 0.9445 c3' = 0.5445 C, = 0.1316 Cn' = 0.0485 C I S = -0.0145 C,,' = 0.0063 C,, = 0.01 7 6 C,,' = -0.0085

D = 0.0851 K, = 1.1861 K , = 0.8431

TABLE 11. POLYNOMIAL COEFFlClENTS AND MEAN

LINEAR DISTORTION FOR THE MINIMUM-ERROR

POLYCONIC PROJECTION WITH EQUALLY SPACED

PARALLELS AND ORIGIN 4 5 ' ~ , 2 0 ~ ~

c,' = 0.99 19 c8' = 0.043 1

C,,' = -0.0064 c,,' = 0.0273

of the projection on another part of the earth's surface in order to place the region of interest in the least distorted area of the map (close to the centre of the distortion pattern). Such a repositioning is denoted as a change of aspect and is accomplished by a transformation of the map graticule. While it is possible to optimize the position of the centre of the distortion pattern for a conventional projection according to the earlier described distortion criterion, it is more in-

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FIGURE 12. Minimum-error polyconic projection with equally spaced parallels and origin at 45°N,20°E.

teresting to iterate the longitude and the latitude of the centre as well as the coefficients that define the projection simultaneously. Applied to the polyconic projection with equally spaced parallels and a straight central meridian as the smaller axis a graticule is obtained with centre at 6 I °N ,24°E , which means that the southern part of Africa is split up. This is caused by the high concentration of continental area in the northern hemisphere.

Optimization of other oblique graticules, conventional or not, almost invari­ably leads to the same phenomenon. Yet it is clear that this inconvenience can be avoided by a small displacement of the centre, followed by a readjustment of the coefficients of the projection. The graticule shown in Figure 12 results from optimization under the same constraints, only the centre of the projection was fixed at 45°N,20°E (Table 11). The representation of the continents of the north­ern hemisphere is much better with respect to any normal polyconic projection. The mean linear distortion is almost half of the value obtained for the optimal polyconic projection shown in Figure 9.

CONCLUSION

Since local distortion theory can not describe the deformation of areas of finite size it is less suited to evaluate distortion occurring on world maps. A practical distor­tion criterion was therefore developed which is based on the distortion of finite distances. Starting from a fifth-order polynomial description of the general trans­formation process it is shown how through the imposition of simple geometric constraints maps can be obtained that match prespecified perceptive require-

NEW PROJECTIONS FOR WORLD MAPS 7 1

ments and at the same time minimize the shape distortion of the continents. In fact, the use of traditional and very often mathematically complicated projection systems for world maps becomes more or less superfluous. For a given set of geometrical constraints (class of projection, pole = point/line, normal/oblique aspect, ...) map transformations can be obtained that give a more realistic por­trayal of the continental area than existing world map projections with a similar configuration of the graticule. Moreover, the definition of these transformations in terms of power series makes them well-suited to be integrated in an automated mapping system or a small-scale GIS since they are all described by the same general transformation formulas. If the availability of traditional projection sys­tems is explicitly demanded a least squares approximation can be applied to convert them to the polynomial format. This will speed up the transformation process for those projections defined in terms of complicated trigonometric and/or transcendental functions. Further research is in progress on the applicabil­ity of the minimum-error procedure described in this paper to the small-scale mapping of continent-sized regions.

REFERENCES

CANTERS, F.and H. DECLEIR 1989. The world in perspective, A Directory of World Map Projections, John Wiley & Sons Ltd, Chichester, Sussex. Forthcoming.

MAUNG, D.H. 1973. Coordinate systems and map projections, George Philip & Son, London, 255.

NELDER, J.A. and R. MEAD 1965. A simplex method for function minimization, The Computer Journal, 7: 308—313.

PETERS, A. 1975. Wie man unsere Weltkarten der Erdeä hnlicher machen kann, Kartographische Nachrichten, 25/5: 173—183.

1978. Über Weltkartenverzerrungen und Weltkartenmittelpunkte, Kartographische Nachrich­ten, 28/3: 106—113.

ROBINSON, A.H. 1974. A new map projection: Its development and characteristics, International Yearbook of Cartography, 14: 145-155.

SNYDER, j .p .1985 . Computer-assisted map projection research, U.S. Geological Survey Bulletin 1629, United States Government Printing Office, Washington, 157.

TOBLER, W.R. 1962. A classification of map projections, Annals of the Association of American Geographers, 52: 167-175.

RÉSUMÉ L'usage acru de cartes dans les communications de masse influence en grande partie notre conception de l'espace géographique. Ceci s'applique tout particulièrement aux cartes du monde. La préservation de la forme des continents devrait donc être considérée comme une importante propriété de telles cartes. Cet article présente une mesure quantitative de la distorsion de la forme des continents. On y propose un outil opérationnel pour le développement de cartes du monde qui soient acceptables du point de vue de la perception mais qui, en même temps, résultent de la réduction d'un critère de distorsion objectif.

KURZFASSUNG Der zunehmende Gebrauch von Karten in der Massenkommunikation beeinfluBt in groBem AusmaB unsere Vorstellung über den geographischen Raum. Das gilt besonders fur Weltkarten. Deshalb mufi die Erhaltung der Kontinentgestalt als wichtige Eigenschaft betrachtet werden. Der Artikel beschreibt ein quantitatives Maß kontinentaler Gestaltverzerrung und erörtert ein praktisches Mittel fur die Entwicklung von Weltkarten, die aus perzeptiver Sicht annehmbar sind, aber gleichzeitig nur ein Minimum an objektiver Verzerrung enthalten.