# 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>.

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 |
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Author: | Ipbuker, Cengizhan |

Publication: | Cartography and Geographic Information Science |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jul 1, 2008 |

Words: | 5958 |

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