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Equations of the Mayr projection.

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 Lambert in Figure3b (Mayr 1964; Snyder and Voxland 1989; Richardus and Adler 1972).

Equations of the Mayr Projection

Franz Mayr arranged the multipliers x and dy in Equation (2b) as the multiplication of two variables

x = [lambda]R[square root of cos[phi]]

dy = R[square root of [phi]]d[phi] (7)

equally distributing the effect of the factor (cos [phi]) to the coordinates x and y (Mayr 1964). Rearranging Equation (7) we obtain the mapping equations of the Mayr Projection as follows (Snyder 1993; Oztan, et al. 2001):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The differentials of these quantities can be written as:

dx = R [square root of cos[phi]] d[lambda] dy = R [square root of cos[phi]] d[phi] (9)

The equal-area condition is then realized for a unit sphere (R = 1) as:

[[partial derivative]y/[partial derivative][phi]][[partial derivative]x/ [partial derivative][lambda] = cos[phi] (10)

where: [[partial derivative]y/[partial derivative][lambda]] = 0, and this indicates that the parallels are Horizontal lines (Tobler 1973).

Equation (9) denotes that if [phi] increases as d[phi] and [lambda] increases as d[lambda] at a point P([phi], [lambda]) on the sphere, the increases correspond to the increases dy and dx at x and y on the map, respectively. In this case, a grid with d[phi] = d[lambda] on the sphere converts to a square with dy = dx on the map. Therefore, the projection is called equivalent or equal-area. Waldo Tobler reintroduced this projection in 1973, but as an equal-area geometric mean of the x values of the cylindrical equal-area and sinusoidal projections (Snyder 1993; Tobler 1973).

The y-coordinate depends on the solution of the elliptic integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Mayr calculated the y-coordinates using the following equations (Bryd and Friedman 1954; Mayr 1964):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Here, the functions F([phi],k) and E([phi],k) are the incomplete elliptic integrals of the first and second kind, respectively (Weisstein 2007a, 2007b). Mayr used the tabulated values of Legendre for solving the elliptic functions E and F (Legendre 1826; Mayr 1964). His coordinates are given in Tablel, columns x1, y1. The solution of the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

will be examined below using different mathematical methods. The y-coordinates obtained using various mathematical solutions given in the upcoming sections are labeled as y2, y3, y4, and y5 and are presented in the Table 1 after the Mayr's coordinates in the same order.

Solution Using Taylor Series Expansion

Let [phi] = [theta] + [psi], the integer function [psi] = [parallel] [phi][parallel] and (0[degrees]<[theta]< 1[degrees]). Thus, d[psi] = 0 snd d[phi] = d[theta]. and d[phi] = d[theta]. The Taylor series expansion of the function f([phi])= [square root of cos [phi]] depends on [psi] for an arbitrary variable [phi] and can be written as follows (Abramowitz and Stegun 1972; Pipes and Harvill 1970; Strubecker 1967):

f([phi]) = f ([psi] + [theta]) = f([psi])+1/1! f'([psi])[theta] + 1/2! f"([psi])[[theta].sup.2] + 1/3! f'"([psi])[[theta].sup.3] + ... + 1/10! [f.sup.(10)]([psi])[[theta].sup.10] (12)

Given the variable transformation t = tan[psi], the derivatives of the function f([psi]) until the 10th order can be calculated as (Oztan, et al. 2001):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

When R = 1 and [[psi].sub.i+1] - [[psi].sub.i] = [rho] = [pi]/ 180[degrees], then the solutions for the y-coordinate are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Substituting (12) into (14), the y-coordinate can be calculated as (Oztan, et al. 2001):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

The coordinates obtained using Equation (15) are presented as the [y.sub.2] values of Table 1.

Solution Using Maclaurin Series Expansion

The function [square root of cos[phi]] is expanded into the Maclaurin series, and after each element of the series has been integrated, it takes the form of Equation (1) (Pipes and Harvill 1970; Strubecker 1967):

[y.sub.3] = [phi] - 1/12 [[phi].sup.3] - 1/480 [[phi].sup.5]-19/40320 [[phi].sup.7]-559/5806080 [[phi].sup.9] - 29143/1277337600 [[phi].sup.11] (16)

The y-coordinates calculated using this second approach are presented as [y.sub.3] in Table 1.

Solution Using Polynomial Approximation

The projection equation for y may be expressed as a polynomial (Oztan, et al. 2001):

[y.sub.4] = [a.sub.1] [phi] + [[a.sub.2] [[phi].sup.2] + [a.sub.3] [[phi].sup.3] + [a.sub.4] [[phi].sup.4] + [a.sub.5] [[phi].sup.5] + [a.sub.6] [[phi].sup.6] + [a.sub.7] [[phi].sup.7] [a.sub.8] [[phi].sup.8] [a.sub.9] [[phi].sup.9] + ... (17)

The coefficients [a.sub.i] of the polynomial can be taken as the numerical integration of the elements of the Taylor series expansion of the function [square root of cos ([phi]) thus (Abramowitz and Stegun 1972; Strubecker 1967; Weisstein 2007c):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

The auxiliary angle [psi] will be taken in radians and the derivatives will be calculated from Equation (13). The problem is the selection of the angle [psi]. Empirical studies suggest the best result is achieved with [psi] = 32[degrees] (Oztan, et al. 2001). After calculating the coefficients [a.sub.i] using Equation (18) and substituting them in Equation (17), we obtain:

[y.sub.4] = 0.99940 [phi] + 0.00272 [[phi].sup.2] - 0.09113 [[phi].sup.3] + 0.01711 [[phi].sup.4] - 0.03397 [phi].sup.5] + 0.04945 [[phi].sup.6] + -0.05945 [[phi].sup.7] + 0.05054 [[phi].sup.8] - 0.02939 [[phi].sup.9] + 0.01040 [[phi].sup.10] - 0.00176 [[phi].sup.11] (19)

The y-coordinates calculated using this third approach are presented as [y.sub.4] in Table 1.

Solution Using Hardy's Multiquadrics

Another alternative is to apply the multiquadric interpolation method to the Mayr projection equations. According to Hardy, an irregular surface may be approximated to any desired degree of exactness by the summation of regular surfaces, particularly in quadric forms. Hardy preferred to call this method multiquadric analysis (Hardy 1972; Hardy 1977). Using the multiquadric interpolation method, a single function Z = f(x,y) is defined for the data set using n reference points. This function can be written as the summation of the second-degree equations whose coefficients are solved as:

[Z.sub.i] = [n.summation over (i=1)] [c.sub.i] [square root of [([y.sub.i] - y).sup.2] + [([x.sub.i] - x).sup.2])] c = [[A.bar].sup.-1] [Z.bar]

i = 1,2, ..., n (20)

where [c.bar] is the vector of unknown coefficients [c.sub.i]. The elements of the matrix [A.bar] are calculated as follows:

[a.sub.i] = [square root of [([y.sub.i] - y).sup.2] + [([x.sub.i] - x).sup.2]] (22)

Hardy developed this method in 1968, using the derivation of equations for topography and other irregular surfaces (Hardy 1971). Apart from digital terrain models, surveying problems where an irregular surface is defined with a single function using all reference points have benefited from this method (Yanalak 2003; Ipbuker 2005). Hardy's multiquadric function Z=f(x,y) is arranged as y=f([phi]x). Thus, another new equation of the Mayr projection takes the form using the multiquadric coefficients (Table2):

[y.sub.5] = [13.summation over (i=1)] [Cy.sub.i] [square root of [([P.sub.i] - [phi].sup.2] + [([q.sub.i] - x).sup.2]] (23)

where the variables [p.sub.i] are in radians and x is calculated first using Equation (8). The y-coordinates calculated using this fourth approach are presented as [y.sub.5] in Table 1.

Thirteen reference points with a 10[degrees] latitude interval are selected from Table 1. Their tabular coordinates (x1), as given by Mayr, are used as q(i) values in Equation (23). The unknown coefficients Cy(i) are solved using Equation (21). The reference coordinates and the multiquadric coefficients are presented in Table 2.

Distortion Characteristics

Local scale distortions along meridians (h) and parallels (k) can be calculated as follows Francula 1971 ; Bugayevski and Snyder 1995).

Distortion along the meridians:

h = 1/R [square root of [([partial derivative]x/[partial derivative] [phi]).sup.2] + [[partial derivative]y/[partial derivative][phi]).sup.2]] (24)

Distortion along the parallels:

k = 1/R cos [phi] [square root of [([partial derivative]x/ [partial derivative][lambda]).sup.2] + [([partial derivative]y/[partial derivative][lambda]).sup.2]] (25)

Area distortion:

p = 1/[R.sup.2] cos[phi] ([partial derivative]x/[partial derivative][phi] [partial derivative]y/[partial derivative][lambda] - [partial derivative]x/ [partial derivative][lambda] [partial derivative]y/[partial derivative][phi]) (26)

Maximum angular distortion:

[bar.[omega]]= 2arctan([h.sup.2] + [k.sup.2]/4p - 1/2]) (27)

The partial derivatives used in Equations (24), (25), and (26) are derived from the Mayr's projection equations (see Equation (8)) as follows:

[partial derivative]y/[partial derivative][lambda] = 0 [partial derivative]x/[partial derivative][lambda] = [square root of (cos[phi])

[partial derivative]y/[partial derivative][phi] = [square root of (cos[phi]) [partial derivative]x/[partial derivative][phi] = [lambda]sin[phi]/2[square root of (cos[phi]) (28)

The maximum and minimum scale distortions a and b may be obtained by solving the following equations:

a = K + L/2

b = K - L/2 (29)

where:

K = a + b = [([h.sup.2] + [k.sup.2] +2p).sup.1/2]

L = a - b = [([h.sup.2] + [k.sup.2] - 2p).sup.1/2] (30)

All of the above distortion quantities are calculated for latitudes with 1[degrees] intervals along the 90[degrees] meridian (R=1) and are presented in Table 3.

In cartography, it is usually desirable to choose a map projection on which distortions are tolerably small. Thus, the primary aim of such a logical choice is to select a projection in which the extreme distortions are smaller than would occur in any other projection used to map the same area. Then, the real skill in selecting a suitable projection is to arrange for the important parts of the world map to lie where the distortions are least (Maling 1992). This is not a criterion for suitable projections, but one may find that an angular deformation greater than 45[degrees] or area scales in excess of 1.5 need to be tolerated in possibly a large part of the map.

In a recent study, Richard Capek (2001) ranked hundred conventional projections into a sequence list, in compliance with a global area distortion characterization Q, which is defined as the percentage ratio of the area represented in the map with permissible distortion to the area of the whole world. But Capek did not examine the Mayr projection.

[FIGURE 4 OMITTED]

The isolines of distortion for h, k, and [omega] are drawn on a quarter hemisphere of Mayr's grid and presented in Figure 4. The shaded area under the isolines fork h = 1.5, k = 1.5 and [omega]=45[degrees] shows the area below the tolerance thresholds and covers 70 percent, 89 percent, and 83 percent of the total area, respectively. The figures are symmetric along the central meridian and the equator.

With an average of 80 percent area in permissible limits of distortions, the Mayr's projection should be listed among the first fifteen favorite world map projections on top of Capek's (2001) list.

The World

in Franz Mayr's Projection

Figure 5 shows the world with shorelines, 10[degrees] graticule, and 0[degrees] central meridian, mapped using Mayr's projection. The shorelines are generalized, based on 65,000 points that were digitized from a 1993 National Geographic Society, world wall map in the Robinson projection with a scale of 1:30,840,000. The geographic coordinates of the features were obtained from the digitized Robinson plane coordinates using the inverse transformation method explained in Ipbuker (2005). Then, the Mayr projection coordinates were calculated using Equations (8) and (16). The drawing was produced in an AutoCAD environment and visualized using Adobe Photoshop.

Conclusion

The Franz Mayr projection has not been used much in cartographic applications because of the difficulties in solving the elliptic projection Equation (*) However, the Mayr projection represents the relative size and shape of the landmasses successfully to map readers. It could be classified in the "orthophanic" group of the cartographic projections, with "orthophanic" meaning "correct looking" or "right appearing," similar to the Robinson projection (Snyder 1993).

In this study, the equations of Franz Mayr have been studied in detail and four analytical solutions were presented. When the results were compared, as seen in Table 2, the [y.sub.2] values obtained from Equation (15) best fit Franz Mayr's tabular vaules. The results from the Equation (16), however, are also sufficient for most small-scale mapping applications.

[FIGURE 5 OMITTED]

REFERENCES

Abramowitz, M., and I.A. Stegun. 1972. Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing. New York, New York: Dover. p. 591.

Bugayevski, L., and J.P. Snyder. 1995. Map projections: A reference manual. Taylor and Francis, London, U.K.: Taylor and Francis. , 328 p.

Bryd, P.F., and D. Friedman. 1954. Handbook of elliptic integrals for engineers and physicists. Berlin, Germany: Springer.

Capek, R. 2001. Which is the best projection for the world map. In: Proceedings of the 20th International Cartographic Conference, Beijing, China. Vol:5, pp.3084-93.

Delmelle, E.M. 2001. Map projection properties: Considerations for small-scale GIS applications. Master of Arts, SUNY Department of Geography. p.117.

Francula, N. 1971. Die Vorteilhaftesten Abbildungen in der Atlaskartographie. Dissertation, Rheinischen Friedrich-Wilhelms Universitaet, Bonn, Deutschland. 103p.

Hardy, R. 1971. Multiquadric equations of topography and Oother irregular Ssurfaces.Journal of Geophysical Research 76(8).

Hardy, R. 1972 Geodetic applications of multiquadric analysis, AVN, Vol.79.

Hardy, R. 1977. Least squares prediction. Photogrammetric Engineering and Remote Sensing 43(4).

Hsu, M.L. 1981. The role of projections in modern map design. In:L.Guelke (ed.), Maps in modern cartography: Geographical perspectives on the new cartography. Monograph 27, Cartographica 18(2),pp. 151-86.

Ipbuker, C. 2005. 2005. A computational approach to the Robinson projection. Survey Review 38(297): 204-17.

Legendre, A.M. 1826. Traite des fonctions elliptiques et des integrales Eulerinnes, avec des tables pour en faciliter le calcul numerique. Paris, France. 2nd vol.

Maling, D.H. 1992, Coordinate systems and map projections. Oxford, U.K.: Pergamon. 476 p.

Mayr, F. 1964. Flaechentreue Plattkarten eine bisher vernachlaessigte Gruppe unechter Zylinderprojektionen,.Internationales Jahrbuch fur Kartographie,.

Oztan, O., C. Ipbuker; and N. Ulugtekin. 2001. A numerical approach to the pseudo-projections on the example of Franz Mayr projection. Journal of General Command of Mapping 125: 37-50. (in Turkish)

Pipes, A.L., and R.L. Harvill. 1970. Applied mathematics for engineers and physicists. New York, New York: McGraw Hill Book Company.

Richardus, P., and R.K. Adler. 1972. Map projections for geodesists, cartographers and geographers. North Holland Publishing Company. 174 p.

Snyder, J.R, and P.M. Voxland. 1989. An album of map projections, U.S.Geological Survey Professional Paper 1453.

Snyder, J.P. 1993. Flattening the Earth: Two thousand years of map projections. Chicago, Illinois: The University of Chicago Press.363 p.

Strubecker; K. 1967. Einfuhrung in die hohere Matematik, Band II, R. Oldenberg Verlag, Munchen, Wien.

Tobler, W. 1973. The hyperelliptical and other new pseudo cylindrical equal area map projections. Journal of Geophysical Research 78(11): 1753-59.

Weisstein, E.W. 2007a. Elliptic integral of the second kind. From MathWorld--A Wolfram Web Resource. [http://mathworld.wolff-am.com/ EllipticIntegraloftheSecondKind.html].

Weisstein, E.W. 2007b. "Elliptic integral of the first kind." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/ EllipticIntegraloftheFirstKind.html].

Weisstein. E.W. 2007c. Maclaurin Series. From MathWorld--A Wolfram Web Resource. [http:// mathworld.wolfram.com/MaclaurinSeries.html].

Yanalak, M. 2003. Effect of gridding method on DTM profile data based on scattered data. J. Computing in Civil Engng. ASCE.

Cengizhan Ipbuker, Istanbul Technical University Faculty of Civil Engineering, Geodesy&Photogrammetry Dept., Division of Cartography, 34469-Maslak Istanbul, Turkey. E-mail: <buker@itu.edu.tr>.
Table 1. Plane coordinates of the outer meridian
([lambda]=90[degrees], R=1) P(x,y)=P(90[degrees],[phi]).

 [phi] x1 y1 y2 y3 y4 y5
[degrees]

 0 1.5708 0.0000 0.000000 0.000000 0.000000 0.0000
 1 1.5707 0.0175 0.017453 0.017453 0.017443 0.017369
 2 1.5703 0.0349 0.034903 0.034903 0.034885 0.034739
 3 1.5697 0.0524 0.052348 0.052348 0.052323 0.052115
 4 1.5689 0.0698 0.069785 0.069785 0.069754 0.069498
 5 1.5677 0.0872 0.087211 0.087211 0.087175 0.086892
 6 1.5665 0.1046 0.104624 0.104624 0.104584 0.104298
 7 1.5649 0.1220 0.122021 0.122021 0.121977 0.121720
 8 1.5631 0.1394 0.139399 0.139399 0.139353 0.139159
 9 1.5611 0.1568 0.156756 0.156756 0.156707 0.156618
 10 1.5588 0.1741 0.174090 0.174090 0.174038 0.1741
 11 1.5563 0.1914 0.191396 0.191396 0.191343 0.191092
 12 1.5535 0.2087 0.208673 0.208673 0.208618 0.208110
 13 1.5505 0.2259 0.225918 0.225918 0.225862 0.225158
 14 1.5473 0.2431 0.243129 0.243129 0.243072 0.242236
 15 1.5438 0.2603 0.260301 0.260301 0.260244 0.259349
 16 1.5401 0.2774 0.277434 0.277434 0.277376 0.276497
 17 1.5361 0.2945 0.294524 0.294524 0.294466 0.293684
 18 1.5319 0.3115 0.311569 0.311569 0.311510 0.310912
 19 1.5274 0.3285 0.328565 0.328565 0.328506 0.328184
 20 1.5227 0.3455 0.345510 0.345510 0.345451 0.3455
 21 1.5177 0.3624 0.362402 0.362402 0.362342 0.361859
 22 1.5125 0.3792 0.379237 0.379237 0.379177 0.378265
 23 1.5071 0.3960 0.396013 0.396013 0.395953 0.394719
 24 1.5014 0.4127 0.412726 0.412726 0.412666 0.411227
 25 1.4954 0.4293 0.429375 0.429375 0.429315 0.427788
 26 1.4892 0.4459 0.445957 0.445957 0.445896 0.444407
 27 1.4827 0.4624 0.462467 0.462467 0.462407 0.461085
 28 1.4760 0.4789 0.478905 0.478905 0.478845 0.477825
 29 1.4690 0.4952 0.495267 0.495267 0.495206 0.494630
 30 1.4618 0.5115 0.511549 0.511549 0.511489 0.5115
 31 1.4543 0.5277 0.527750 0.527750 0.527690 0.526960
 32 1.4465 0.5438 0.543866 0.543866 0.543805 0.542482
 33 1.4385 0.5598 0.559894 0.559894 0.559834 0.558072
 34 1.4302 0.5758 0.575832 0.575832 0.575772 0.573731
 35 1.4217 0.5916 0.591676 0.591676 0.591616 0.589462
 36 1.4129 0.6074 0.607424 0.607424 0.607364 0.605268
 37 1.4038 0.6230 0.623072 0.623072 0.623012 0.621151
 38 1.3944 0.6385 0.638618 0.638618 0.638557 0.637115
 39 1.3848 0.6540 0.654058 0.654058 0.653997 0.653162
 40 1.3748 0.6693 0.669389 0.669389 0.669329 0.6693
 41 1.3646 0.6845 0.684608 0.684609 0.684548 0.683570
 42 1.3541 0.6996 0.699712 0.699713 0.699653 0.697921
 43 1.3433 0.7146 0.714699 0.714700 0.714640 0.712353
 44 1.3323 0.7295 0.729563 0.729563 0.729505 0.726869
 45 1.3209 0.7442 0.744303 0.744303 0.744246 0.741472
 46 1.3092 0.7588 0.758915 0.758915 0.758858 0.756166
 47 1.2972 0.7733 0.773395 0.773396 0.773340 0.770952
 48 1.2849 0.7877 0.787741 0.787741 0.787687 0.785835
 49 1.2723 0.8019 0.801948 0.801949 0.801896 0.800819
 50 1.2594 0.8159 0.816013 0.816014 0.815964 0.8159
 51 1.2461 0.8298 0.829933 0.829934 0.829886 0.828655
 52 1.2325 0.8436 0.843703 0.843705 0.843660 0.841495
 53 1.2186 0.8572 0.857320 0.857323 0.857282 0.854427
 54 1.2043 0.8707 0.870781 0.870785 0.870749 0.867457
 55 1.1896 0.8840 0.884081 0.884086 0.884055 0.880586
 56 1.1746 0.8971 0.897216 0.897222 0.897198 0.893820
 57 1.1592 0.9101 0.910183 0.910190 0.910175 0.907163
 58 1.1435 0.9229 0.922976 0.922986 0.922980 0.920620
 59 1.1273 0.9355 0.935591 0.935604 0.935609 0.934196
 60 1.1107 0.9479 0.948025 0.948041 0.948060 0.9479
 61 1.0937 0.9602 0.960273 0.960292 0.960327 0.959530
 62 1.0763 0.9722 0.972329 0.972353 0.972407 0.971271
 63 1.0584 0.9841 0.984188 0.984219 0.984294 0.983127
 64 1.0400 0.9957 0.995846 0.995885 0.995985 0.995103
 65 1.0212 1.0072 1.007298 1.007346 1.007475 1.0072
 66 1.0018 1.0184 1.018537 1.018597 1.018759 1.017735
 67 0.9819 1.0295 1.029558 1.029633 1.029833 1.028382
 68 0.9614 1.0403 1.040354 1.040447 1.040691 1.039152
 69 0.9403 1.0508 1.050920 1.051035 1.051328 1.050053
 70 0.9186 1.0611 1.061249 1.061389 1.061739 1.0611
 71 0.8963 1.0712 1.071332 1.071505 1.071919 1.070388
 72 0.8732 1.0810 1.081163 1.081376 1.081862 1.079803
 73 0.8493 1.0906 1.090734 1.090995 1.091562 1.089352
 74 0.8247 1.0999 1.100035 1.100355 1.101012 1.099045
 75 0.7991 1.1089 1.109057 1.109449 1.110207 1.1089
 76 0.7726 1.1177 1.117789 1.118269 1.119140 1.116727
 77 0.7450 1.1261 1.126222 1.126808 1.127803 1.124685
 78 0.7162 1.1342 1.134341 1.135057 1.136189 1.132788
 79 0.6862 1.1420 1.142133 1.143008 1.144291 1.141053
 80 0.6546 1.1495 1.149583 1.150652 1.152099 1.1495
 81 0.6213 1.1566 1.156673 1.157980 1.159605 1.155487
 82 0.5860 1.1633 1.163382 1.164981 1.166800 1.161594
 83 0.5484 1.1696 1.169686 1.171646 1.173672 1.167844
 84 0.5079 1.1755 1.175557 1.177963 1.180213 1.174265
 85 0.4637 1.1809 1.180959 1.183921 1.186409 1.1809
 86 0.4149 1.1857 1.185845 1.189508 1.192248 1.184520
 87 0.3594 1.1900 1.190154 1.194712 1.197716 1.188126
 88 0.2935 1.1937 1.193793 1.199520 1.202799 1.191703
 89 0.2075 1.1965 1.196603 1.203917 1.207481 1.195203
 90 0.0000 1.1981 1.198153 1.207889 1.211744 1.1981

Table 2. The multiquadric cefficients for the Mayr projection.

 Cy(i) p(i) q(i)

+0.76736742258 0[degrees] 1.5708
-0.01458108798 10[degrees] 1.5588
-0.02778026834 20[degrees] 1.5227
-0.03906646743 30[degrees] 1.4618
-0.04820096865 40[degrees] 1.3748
-0.05575828627 50[degrees] 1.2594
-0.04509944841 60[degrees] 1.1107
-0.03271599859 65[degrees] 1.0212
-0.03349925205 70[degrees] 0.9186
-0.03460321575 75[degrees] 0.7991
-0.03554333374 80[degrees] 0.6546
-0.03175015375 85[degrees] 0.4637
+0.19346937537 90[degrees] 0.0000

Table 3. The distortion quantities of the Mayr projection.

 [phi]
[degrees] h k p

 0 1.000000 1.000000 1.000000
 1 1.000300 1.000076 1.000000
 2 1.001198 1.000305 1.000000
 3 1.002695 1.000686 1.000000
 4 1.004788 1.001220 1.000000
 5 1.007477 1.001908 1.000000
 6 1.010757 1.002750 1.000000
 7 1.014627 1.003748 1.000000
 8 1.019083 1.004902 1.000000
 9 1.024120 1.006213 1.000000
 10 1.029736 1.007684 1.000000
 11 1.035926 1.009315 1.000000
 12 1.042684 1.011109 1.000000
 13 1.050006 1.013067 1.000000
 14 1.057887 1.015191 1.000000
 15 1.066321 1.017485 1.000000
 16 1.075304 1.019951 1.000000
 17 1.084831 1.022591 1.000000
 18 1.094896 1.025408 1.000000
 19 1.105495 1.028407 1.000000
 20 1.116623 1.031590 1.000000
 21 1.128276 1.034961 1.000000
 22 1.140450 1.038525 1.000000
 23 1.153142 1.042286 1.000000
 24 1.166349 1.046249 1.000000
 25 1.180067 1.050418 1.000000
 26 1.194296 1.054799 1.000000
 27 1.209035 1.059399 1.000000
 28 1.224282 1.064223 1.000000
 29 1.240038 1.069277 1.000000
 30 1.256305 1.074570 1.000000
 31 1.273085 1.080108 1.000000
 32 1.290380 1.085900 1.000000
 33 1.308194 1.091954 1.000000
 34 1.326534 1.098280 1.000000
 35 1.345405 1.104887 1.000000
 36 1.364816 1.111786 1.000000
 37 1.384775 1.118989 1.000000
 38 1.405293 1.126507 1.000000
 39 1.426383 1.134354 1.000000
 40 1.448057 1.142544 1.000000
 41 1.470332 1.151092 1.000000
 42 1.493226 1.160014 1.000000
 43 1.516758 1.169328 1.000000
 44 1.540951 1.179052 1.000000
 45 1.565830 1.189207 1.000000
 46 1.591421 1.199815 1.000000
 47 1.617756 1.210900 1.000000
 48 1.644868 1.222488 1.000000
 49 1.672795 1.234607 1.000000
 50 1.701578 1.247287 1.000000
 51 1.731264 1.260562 1.000000
 52 1.761903 1.274468 1.000000
 53 1.793552 1.289046 1.000000
 54 1.826274 1.304340 1.000000
 55 1.860139 1.320396 1.000000
 56 1.895224 1.337270 1.000000
 57 1.931615 1.355020 1.000000
 58 1.969411 1.373710 1.000000
 59 2.008718 1.393414 1.000000
 60 2.049659 1.414214 1.000000
 61 2.092370 1.436198 1.000000
 62 2.137006 1.459471 1.000000
 63 2.183743 1.484146 1.000000
 64 2.232779 1.510355 1.000000
 65 2.284342 1.538246 1.000000
 66 2.338694 1.567990 1.000000
 67 2.396137 1.599783 1.000000
 68 2.457020 1.633850 1.000000
 69 2.521753 1.670457 1.000000
 70 2.590814 1.709914 1.000000
 71 2.664773 1.752585 1.000000
 72 2.744308 1.798907 1.000000
 73 2.830236 1.849406 1.000000
 74 2.923551 1.904719 1.000000
 75 3.025478 1.965631 1.000000
 76 3.137542 2.033117 1.000000
 77 3.261674 2.108414 1.000000
 78 3.400361 2.193111 1.000000
 79 3.556862 2.289289 1.000000
 80 3.735557 2.399744 1.000000
 81 3.942486 2.528330 1.000000
 82 4.186262 2.680540 1.000000
 83 4.479671 2.864526 1.000000
 84 4.842693 3.093020 1.000000
 85 5.308706 3.387287 1.000000
 86 5.938795 3.786237 1.000000
 87 6.860658 4.371192 1.000000
 88 8.405295 5.352916 1.000000
 89 11.889207 7.569590 1.000000

 [phi] [omega]
[degrees] [degrees] a b

 0 0.000000 1.000000 1.000000
 1 1.570791 1.013803 0.986385
 2 3.141291 1.027796 0.972956
 3 4.711460 1.041984 0.959707
 4 6.281010 1.056371 0.946637
 5 7.849704 1.070960 0.933742
 6 9.417439 1.085755 0.921018
 7 10.983943 1.100759 0.908464
 8 12.549034 1.115977 0.896076
 9 14.112575 1.131414 0.883850
 10 15.674330 1.147072 0.871785
 11 17.234190 1.162958 0.859876
 12 18.791941 1.179075 0.848122
 13 20.347436 1.195429 0.836520
 14 21.900557 1.212025 0.825065
 15 23.451145 1.228868 0.813757
 16 24.999072 1.245964 0.802592
 17 26.544227 1.263318 0.791566
 18 28.086492 1.280938 0.780678
 19 29.625804 1.298829 0.769924
 20 31.162056 1.316999 0.759302
 21 32.695201 1.335454 0.748809
 22 34.225167 1.354203 0.738442
 23 35.751935 1.373254 0.728198
 24 37.275462 1.392615 0.718074
 25 38.795752 1.412295 0.708067
 26 40.312799 1.432305 0.698175
 27 41.826646 1.452654 0.688395
 28 43.337316 1.473353 0.678724
 29 44.844885 1.494415 0.669158
 30 46.349410 1.515851 0.659695
 31 47.850996 1.537675 0.650332
 32 49.349758 1.559901 0.641066
 33 50.845824 1.582544 0.631894
 34 52.339358 1.605620 0.622812
 35 53.830519 1.629146 0.613818
 36 55.319518 1.653141 0.604909
 37 56.806571 1.677625 0.596081
 38 58.291919 1.702619 0.587331
 39 59.775830 1.728145 0.578655
 40 61.258596 1.754229 0.570051
 41 62.740563 1.780896 0.561515
 42 64.221979 1.808176 0.553044
 43 65.703314 1.836099 0.544633
 44 67.184947 1.864698 0.536280
 45 68.667301 1.894010 0.527980
 46 70.150848 1.924073 0.519731
 47 71.636091 1.954930 0.511527
 48 73.123560 1.986628 0.503366
 49 74.613841 2.019215 0.495242
 50 76.107541 2.052748 0.487152
 51 77.605332 2.087286 0.479091
 52 79.107914 2.122894 0.471055
 53 80.616052 2.159644 0.463039
 54 82.130560 2.197616 0.455038
 55 83.652309 2.236898 0.447048
 56 85.182241 2.277584 0.439062
 57 86.721356 2.319782 0.431075
 58 88.270746 2.363612 0.423081
 59 89.831575 2.409204 0.415075
 60 91.405107 2.456708 0.407049
 61 92.992706 2.506288 0.398996
 62 94.595850 2.558132 0.390910
 63 96.216147 2.612451 0.382782
 64 97.855344 2.669484 0.374604
 65 99.515360 2.729505 0.366367
 66 101.198285 2.792826 0.358060
 67 102.906431 2.859808 0.349674
 68 104.642343 2.930870 0.341196
 69 106.408848 3.006498 0.332613
 70 108.209096 3.087265 0.323911
 71 110.046617 3.173846 0.315075
 72 111.925391 3.267048 0.306087
 73 113.849934 3.367844 0.296926
 74 115.825408 3.477414 0.287570
 75 117.857760 3.597212 0.277993
 76 119.953911 3.729051 0.268165
 77 122.121993 3.875221 0.258050
 78 124.371689 4.038673 0.247606
 79 126.714678 4.223274 0.236783
 80 129.165293 4.434219 0.225519
 81 131.741461 4.678671 0.213736
 82 134.466141 4.966845 0.201335
 83 137.369594 5.313901 0.188186
 84 140.493193 5.743530 0.174109
 85 143.896236 6.295303 0.158849
 86 147.669351 7.041641 0.142012
 87 151.964504 8.133931 0.122942
 88 157.076944 9.964568 0.100356
 89 163.766410 14.094215 0.070951
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Title Annotation:Franz Mayr's pseudo-cylindrical projection
Author:Ipbuker, Cengizhan
Publication:Cartography and Geographic Information Science
Article Type:Report
Geographic Code:1USA
Date:Jul 1, 2008
Words:5958
Previous Article:Modeling the potential swath coverage of nadir and off-nadir pointable remote sensing satellite-sensor systems.
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