# 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.

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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