A computational method for the Hufnagel pseudocylindric map projectionfamilyBernhard Jennya,b, Bojan Šavriča,c and Daniel R. Strebed

aCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, OR, USA; bSchool of Science, RMIT University,Melbourne, Australia; cEsri Inc., Redlands, CA, USA; dMapthematics LLC, Seattle, WA, USA

ABSTRACTIn 1989, Herbert Hufnagel introduced a mathematical model for a family of equal-area, pseudo-cylindric projections, which includes the Mollweide, Wagner IV, and Eckert IV projections asspecializations. Hufnagel’s contribution has received very little attention, and his projections arenot available in any of the commonly used cartographic software packages. This article makesthree contributions: The Hufnagel family is introduced to the English-speaking community;algorithms for computing the forward and reverse projections are presented; and the eightnew projections proposed by Hufnagel are analyzed.

ARTICLE HISTORYReceived 2 July 2015Accepted 24 November 2015

KEYWORDSHufnagel; Mollweide; EckertIV; Wagner IV; world mapprojection; small-scaleprojection

Introduction: the Hufnagel projection family

In 1989, in the German-language journalKartographische Nachrichten, Herbert Hufnagel intro-duced a generalization of the Mollweide projection thatincludes the Eckert IV and Wagner IV projections andother useful equal-area pseudocylindric world mapprojections (Hufnagel 1989). In the same article, healso proposed eight new equal-area pseudocylindricprojections. The method by Hufnagel, who was a pro-fessor at the Fachhochschule München (now MunichUniversity of Applied Sciences), allows for the creationof a variety of equal-area pseudocylindric projectionsthat are useful for practical map making. Additionally,smooth transitions between projections can be com-puted, which can be useful for animated or user-con-trolled transformations between projections.Hufnagel’s projections are not currently available inany cartographic software package.

The only reference to the Hufnagel projection familythat we could identify in the literature is by Čapek(2001), who compared various world map projections.While Čapek found that the distortion of Hufnagel’sprojections compares favorably to other commonlyused world map projections, he did not, however,develop code for the projection equations. Instead, hebased his evaluation of the projections on a graphicalanalysis of the graticules included in Hufnagel’s (1989)

publication (personal communication with R. Čapek,April and May 2014).

Cartographers are not unanimous as to what projec-tions are best used for world maps. Besides distortioncharacteristics, the appearance and aesthetic preferenceare important factors when selecting a world mapprojection (Jenny, Patterson, and Hurni 2008). Forexample, Šavrič et al. (2015) found that map readersprefer pseudocylindric map projections (with straightparallels) to world map projections with curved paral-lels. Hufnagel’s equal-area projections are pseudocy-lindric, and cartographers can tune their appearanceto their aesthetical preference by adjusting a set ofprojection parameters.

Hufnagel’s projection family includes the Mollweide,Eckert IV, Wagner IV, and any cylindrical equal-areaprojection. By adjusting projection parameters, a vari-ety of intermediate projections can be created, andwhen projection parameters continuously change overtime, smooth transitions between these projections canbe created. There is potential for Hufnagel’s equal-areatransformation method to be applied in modified formto other projections. It could possibly be added to theadaptive composite map projection technique by Jenny(2012), which adjusts the map projection to the geo-graphical area displayed by the map. We thereforebelieve that the details of Hufnagel’s method and pro-jections merit to be documented and presented to the

CONTACT Bernhard Jenny bernhard.jenny@rmit.edu.au School of Science, GPO Box 2476V, RMIT University, Melbourne, Victoria 3001, Australia.

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English-speaking community. It is our hope that withthis contribution, Hufnagel’s projections will beincluded in cartographic software packages, and themethods involved will help further the field and pro-vide more to build upon.

The following section introduces Hufnagel’smethod; the description is loosely based on Hufnagel(1989). The next section provides equations and analgorithm using a look-up table for the conversion ofspherical coordinates to Cartesian planar coordinates.This algorithmic method is also useful for fast imple-mentations of the Mollweide, Eckert IV, and otherprojections. The last section documents the eight pseu-docylindric equal-area projections introduced byHufnagel. Appendix 1 proves that the cylindrical pro-jection is a limiting case of the Hufnagel projection.

Hufnagel’s development

Hufnagel (1989) started with the Mollweide projectionusing a polar coordinate system. An ellipse for the anti-meridian (which forms the map boundary) is con-structed by applying an affine stretching to a circle,which results in varying the distance of the antimeridianfrom the central meridian (see Figure 1). The affinetransformation, however, is not achieved by applyingdifferent scale factors to x- and y-coordinates, but thedistance r from the projection’s origin is modulated withcosine curves with double and quadruple periods, usingamplitude factors A and B. Equation (1) expresses thisgeneral development (Hufnagel 1989). The graticule isconstructed by mirroring the first quadrant along the x-and y-axes (Figure 1). The resulting map boundary issymmetric relative to the x-axis (the equator) and they-axis (the central meridian) for any A and B, and theyintersect both axes at right angles. With A ¼ 0 andB ¼ 0, a circle is obtained with radius r ¼ 1 (Figure 1).

xπ ¼ K " r " cosΨy ¼ K " r " sinΨr2 ¼ 1þ A " cos 2Ψ þ B " cos 4Ψ

(1)

In Equation (1), xπ and y are the projected coordi-nates of the map boundary, Ψ is a parametric angle(the angle from the positive direction of the x-axis to apoint on the map boundary), r is the distance from theorigin to a point on the map boundary, A and B areamplitudes defining r, and K is a scale factor.

By changing the parameters A and B, a variety ofprojections can be created. With the selection of anappropriate mathematical relationship between theparametric angle Ψ and the geographic latitude φ,equal-area projections can be developed. The equal-area property is maintained when the sum of areahighlighted in gray in Figure 1 is equal to half of thearea between the equator and latitude φ (which isπ " sinφ). From this condition one can derive the rela-tion shown in Equation (2). Equation (3) is identical toEquation (2), but Hufnagel eliminates r2 usingEquation (1) (Hufnagel 1989). In Equations (2) and(3), φ is the geographic latitude; Ψ , r, A, B, and K aredefined as above.

π " sinφ ¼ K2

2

ðΨ

0

r2dΨ þK2

2r2 " sinΨ " cosΨ

¼ K2

42ðΨ

0

r2dΨ þ r2sin2Ψ

0

@

1

A

¼ K2

42Ψ þA " sin2Ψ þ B

2sin4Ψ þ r2 " sin2Ψ

" #;

(2)

π " sinφ ¼ K2

4

"2Ψ þ

"1þ A$ B

2

#sin2Ψ

þ Aþ B2

sin4Ψ þ B2sin6Ψ

#:

(3)

When φ ¼ 90%, the corresponding Ψ in Equation (3)is the maximum parametric angle Ψmax. ForΨmax ¼ 90%, poles are represented with points. WhenΨmax is smaller than 90%, projections have pole lines.Appendix 1 proves that when Ψmax approaches 0, theHufnagel projection transforms into an equal-areacylindrical projection.

The value of the constant K in Equations (1)–(3) canbe obtained from Equation (3) by setting φ ¼ 90%,Ψ ¼ Ψmax, and using the selected amplitudes A and B(see also Equation (7)). To project the longitude λ, thex-coordinate of the antimeridian is multiplied by λ=π(Equation (4)) (Hufnagel 1989).

y

x

Figure 1. The affine stretching by modulating the radius r withcosine curves (after Hufnagel 1989).

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x ¼ λπ" xπ ¼ K " λ

π" r " cosΨ (4)

The final step in Hufnagel’s derivation is an equal-area stretching applied to the projected x- and y-coordi-nates. The stretching factor C adjusts the map to theequator-to-central meridian ratio desired by the carto-grapher (Equation (5)). The factor C is computed withEquation (6), where α is the equator-to-central meridianratio, and Ψmax is the maximum parametric angle. K iscomputed from A, B, and Ψmax with Equation (7).

x ¼ K " Cπ

" λ " r " cosΨ; y ¼ KC" r " sinΨ (5)

C2 ¼ α " sinΨmax"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ A " cos 2Ψmax þ B " cos 4Ψmax

1þ Aþ B

r(6)

Using Hufnagel’s derivation approach, theMollweide projection is created by first setting theamplitude factors A and B to 0, which results in acircular map boundary. Then, the equal-area conditionis applied to the circle using Equation (3) to compute aparametric angle Ψ for each latitude φ. The maximumparametric angle Ψmax is set to 90%, because theMollweide projection represents poles as points. Andfinally, equal-area stretching with the factor C ¼

ffiffiffi2

p

(Equation (6)) is applied to set the equator-to-centralmeridian ratio α to 2. Computational techniques aredetailed in the next section.

Tobler’s hyperelliptical projection family alsoincludes the Mollweide projection (Tobler 1973).However, Tobler and Hufnagel’s projection families areparameterized differently and have considerably differ-ent characteristics. Tobler uses a family of curves calledthe hyperellipse, while Hufnagel uses a family of curvesthat has no particular name. Unlike Tobler’s hyperel-lipses, Hufnagel’s curves have inflection points, whichresults in a greater diversity in his projection family. Asa practical matter for computation, Hufnagel’s develop-ment also does not require numerical integration.

Some aspects of Hufnagel’s derivation are similar toWagner’s area-preserving transformation methodcalled Umbeziffern, meaning renumbering (Wagner1931, 1932, 1941, 1949, 1962). With the parametricangle Ψ , new latitude values are computed, and the

map is adjusted to the preferred equator-to-centralmeridian ratio with the stretching factor C.

Algorithms for Hufnagel’s projection family

Individual members of Hufnagel’s projection family arecreated by configuring four parameters: A 2 $1; 1½ ',B 2 $1; 1½ ', Ψmax 2 0; π2

% &, and C, which is a stretching

factor that adjusts the map to the preferred equator-to-central meridian ratio. This section details algorithmicsteps for applying the parameters.

Direct equations for transforming spherical coordi-nates to planar Cartesian coordinates do not exist forHufnagel’s projection family. To compute a parametricangle Ψ for a corresponding latitude φ with Equation(3), an iteration procedure has to be used. This is awell-known characteristic of the Mollweide projection(Mollweide 1805; Snyder 1987, 1993) and the Eckert IVprojection (Eckert 1906; Snyder 1987, 1993). Using theNewton–Raphson method, the iteration process to findΨ for a given latitude φ is as follows:

(1) Selection of a seed value Ψ0 for the unknownparametric angle Ψ .

(2) Correction !Ψ with Equation (8).(3) Improved value of the parametric

angle: Ψnþ1 ¼ Ψn $!Ψ

The calculation is repeated until !Ψj j < ε (a smallthreshold value). The selection of a seed value for theparametric angle in step 1 is critical for the iterativeprocedure to converge. We found no heuristic for aseed value that works across all parameterizations ofthe projection. Therefore, we built three lookup tables,each with the same size (we used 101 elements each).The first table contains linearly increasing values of theparametric angle Ψ 2 0;Ψmax½ '. It serves as an “index”into the other two tables, one of which goes forward tothe y value computed from Ψ , and the other of whichgoes backward to the φ value computed from the sameΨ . The tables represent the relationship between Ψ , y,and φ given fixed A, B, C, and Ψmax. For a given Ψappearing in the first table, the corresponding y valueand φ value appear at the same index in their ownrespective tables. We index off of Ψ rather than off of φ

!Ψ ¼K2

4 2 " Ψn þ 1þ A$ B2

' (" sin 2Ψn þ AþB

2 " sin 4Ψn þ B2 " sin 6Ψn

' ($ π sinφ

K2

2 1þ 1þ A$ B2

' (" cos 2Ψn þ Aþ Bð Þ " cos 4Ψn þ 3B

2 " cos 6Ψn' ( (8)

K2 ¼ 4π2 " Ψmax þ 1þ A$ B

2

' (" sin 2Ψmax þ AþB

2 " sin 4Ψmax þ B2 " sin 6Ψmax

(7)

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or y in order to avoid having to iterate when calculat-ing the tables, since y and φ both can be calculated inclosed form from Ψ . Algorithm 1 presents pseudocodefor the initialization of the lookup tables (with thenumber of elements equal to TableSize).

Algorithm 1. Initialization of the lookup tables1: for i 2 0; TableSize$ 1½ ' do2: Ψ ¼ i"Ψmax

TableSize$1

3: if i ¼ 0 then4: φ ¼ 05: else if i ¼ TableSize$ 1ð Þ then6: φ ¼ 90%

7: else8: φ from Equation (3)9: end if10: r from Equation (1)11: y from Equation (5)12: if i > 0 then13: if y < yTable i $ 1½ ' or φ < latitudeTable

i$ 1½ ' then14: raise folding graticule exception15: end if16: end if17: paramAngleTable i½ ' ¼ Ψ

18: latitudeTable i½ ' ¼ φ

19: yTable i½ ' ¼ y20: end for

Algorithm 1 uses the lookup table for the y-coordi-nate to identify combinations of the parameters A, B,and Ψmax that result in y values failing to increasemonotonically with φ, in which case an exception israised (lines 13–15). In other words, δy/δφ goes negative,causing some higher latitudes to end up closer to theequator on the map than lower latitudes do. This couldbe detected more analytically but was not deemed worththe effort; given the size of the look-up tables, only a tinyuncertainty remains in detecting failure of monotonicitywhile also yielding robust seed values.

To select a robust seed value for the parametricangle Ψ for a latitude φ, we first find the two closestlatitudes in the latitude lookup table using a binarysearch. The seed value for the parametric angle Ψ isfound with linear interpolation between the corre-sponding parametric angles extracted from the lookuptable. With this approach, the seed value of Ψ is veryclose to the exact value. Algorithm 2 presents pseudo-code for the selection of a seed value Ψ0 of the para-metric angle Ψ from the lookup tables.

Algorithm 2. Finding seed value Ψ0 of the para-metric angle Ψ

Input: φ1: imin ¼ 02: imax = TableSize3: repeat4: imid ¼ imaxþimin

2

) *

5: if imid equals imin then6: break loop7: else if φj j > latitudeTable imid½ ' then8: imin ¼ imid

9: else10: imax ¼ imid

11: end if12: end loop13: φ1 ¼ latitudeTable imin½ '14: φ2 ¼ latitudeTable imin þ 1½ '15: weight ¼ φj j$φ1

φ2$φ1

16: Ψ1 ¼ paramAngleTable imin½ '17: Ψ2 ¼ paramAngleTable imin þ 1½ '18: Ψ0 ¼ weight " Ψ2 $ Ψ1ð Þ þ Ψ1

19: return φ < 0 ? $Ψ0: Ψ0

After calculating the parametric angle Ψ , theCartesian coordinates are obtained with Equation(5). Algorithm 3 presents pseudocode for the com-putation of Cartesian coordinates from sphericalcoordinates.

Algorithm 3. Projecting spherical coordinates toCartesian coordinates

Inputs: λ and φ

1: Find seed value Ψ0: (Algorithm 2)2: repeat3: !Ψnumerator of Equation (8)4: if !Ψnumeratorj j < ε

5: break loop6: end if7: !Ψdenominator of Equation (8)8: !Ψ ¼ Ψnumerator

!Ψdenominator

9: Ψnþ1 ¼ Ψn $!Ψ

10: end loop11: r with Equation (1)12: x with Equation (5)13: y with Equation (5)14: return x; y½ '

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The Mollweide, Eckert IV, and Wagner IV projec-tions have direct reverse equations. However, generalmembers of the Hufnagel projection family require aniteration procedure. The Newton–Raphson method canbe used with Equation (9) to find the correction !Ψ ofthe parametric angle for a given y-coordinate (r isdefined in Equation (1)).

For the selection of a seed value Ψ0, the parametricangle and y-coordinate look-up tables are used. Thealgorithm used to find the seed value is similar toAlgorithm 2, except the y-coordinate is used insteadof the latitude value. After the parametric angle Ψ iscomputed, the latitude can be obtained from Equation(3) and the longitude is computed from Equation (5).Algorithm 4 shows pseudocode for determining sphe-rical coordinates from Cartesian coordinates.

Algorithm 4. Projecting Cartesian coordinates tospherical coordinates

Inputs: x and y1: Find seed value Ψ0: (Algorithm 2, using y-coordi-

nate lookup table)2: repeat3: !Ψnumerator of Equation (9)4: if !Ψnumeratorj j < ε

5: break loop6: end if7: !Ψdenominator of Equation (9)8: !Ψ ¼ Ψnumerator

!Ψdenominator

9: Ψnþ1 ¼ Ψn $!Ψ

10: end loop11: r with Equation (1)12: φ with Equation (3)13: λ with Equation (5)14: return λ;φ½ '

Hufnagel’s eight projections

Hufnagel introduced eight new projections for worldmaps, which are all members of his map projectionfamily (Hufnagel 1989). The eight projections arenamed Hufnagel II, III, IV, VII, IX, X, XI, and XII.The numbering is not continuous, as Hufnagel did not

use the numerals I, V, VI, and VIII. Those respectivepositions are occupied by the Mollweide, Wagner IV(Snyder 1993; Wagner 1949, 1962), Eckert IV projec-tions, and an approximation of the Eckert VI projec-tion (Hufnagel 1989). Figure 2 illustrates theprojections proposed by Hufnagel, and Table 1 detailsthe parameters A, B, and Ψmax, and the equator-to-central meridian ratio α (which is defined as a functionof the stretching factor C; see Equation (5)).

All projections in Figure 2 have an equator-to-cen-tral meridian ratio of 2:1, except for the Hufnagel XIIprojection, which has a ratio of 2.44. The Mollweideand the Hufnagel II, III and IV projections show thepoles as points. Hufnagel (1989) noted that his II, III,and IV projections have less overall distortion than theMollweide projection. This can be confirmed by theweighted mean error in the overall scale distortionindices and the mean angular deformation indices(Canters and Decleir 1989), as well as by Čapek’sacceptance index (Čapek 2001). Results are shown inTable 2. Figure 3 shows local distortion patterns for the12 projections by means of lines of constant maximumangular deformation.

The eight projections in the second and third col-umns in Figure 2 show the pole points as lines. Thelength of the pole lines of the Eckert IV, Wagner IV,and Hufnagel VII, VIII, and X projections is half theequator length. The length of the pole line of theHufnagel IX projection is slightly shorter. Meridiansof the Wagner IV, the approximated Eckert VI, and theHufnagel VII projections meet the pole lines with anobtuse angle. The ends of the meridians of the EckertIV and the Hufnagel IX and X projections are tangentwith the pole lines, that is, they smoothly transitioninto the pole lines. The distortion indices of theseprojections are very similar (Table 2). Hufnagel com-ments that in his opinion, number IX is the mostaesthetically balanced projection (Hufnagel 1989).

The Hufnagel XI projects the poles as points, butmeridional convergence is so rapid toward the polesthat the poles can look like lines. Due to the unusualboundary of this projection, Hufnagel states that itsvisual appearance is debatable (Hufnagel 1989).

The Hufnagel XII projection has pole lines that havealmost the same length as the equator. Meridiansslightly bend where they approach the pole line, result-ing in a projection that loosely resembles cylindricalequal-area projections. The Hufnagel XII projection

!Ψ ¼r2 " sin2Ψn $ y"C

K

+ ,2

sin 2Ψn " 1$ Aþ 2Bþ 2A$ 4Bð Þ " cos 2Ψn þ 3B " cos 4Ψnð Þ(9)

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has the smallest mean angular deformation index of the12 projections (Table 2).

Conclusion

Hufnagel proposes an interesting family of map projec-tions that includes the Mollweide, Eckert IV, and WagnerIV projections as special cases. Additional projections canbe created by varying the parameters A, B, and Ψmax, andthe equator-to-central meridian ratio of the map.Hufnagel’s method is very flexible and allows for creatinga variety of equal-area pseudocylindric projections. Many

parameter combinations result in absurd distortion char-acteristics or folding graticules that are of little value forpractical map making (Figure 4). We explored validranges of the A, B, and Ψmax parameters that result innon-folded graticules, but the distribution pattern of validcombinations is very irregular and difficult to visualize.Many projections in this pseudocylindric projectionfamily have nevertheless reasonable distortion properties,and seem to be useful for a variety of equal-area world

Figure 2. The 12 projections presented by Hufnagel (1989) ordered by columns (Hufnagel did not use the numerals I, V, VI, VIII; theMollweide, Wagner IV, Eckert IV and an approximation of the Eckert VI projection occupy the corresponding positions).

Table 1. The 12 projections presented by Hufnagel (1989) withtheir parameters.Projection A B Ψmax α

Mollweide 0 0 90 2Hufnagel II 1/18 −1/18 90 2Hufnagel III 0.5 1/18 90 2Hufnagel IV 1/12 −1/12 90 2Eckert VI (approx.) −2/21 2/21 60 2Wagner IV 0 0 60 2Hufnagel VII 1/12 −1/12 60 2Eckert IV 1 0 45 2Hufnagel IX 2/3 1/3 45 2Hufnagel X −2/3 2/3 30 2Hufnagel XI 0 −1/9 90 2Hufnagel XII 0 −1/9 40 2.44

Table 2. Distortion indices for the projections presented byHufnagel (1989): the weighted mean error in the overall scaledistortion index Dab (Canters and Decleir 1989), the meanangular deformation index Dan (Canters and Decleir 1989),and the area percentage with acceptable areal and angulardistortion Q (Čapek 2001).Projection Dab Dan Q

Mollweide 0.39 32.28 70.6Hufnagel II 0.37 30.33 75.8Hufnagel III 0.37 30.27 76.7Hufnagel IV 0.36 29.52 77.8Eckert VI (approx.) 0.40 32.43 69.5Wagner IV 0.38 30.39 76.3Hufnagel VII 0.36 28.97 79.7Eckert IV 0.36 28.73 81.9Hufnagel IX 0.36 28.80 81.6Hufnagel X 0.36 28.22 83.2Hufnagel XI 0.36 28.81 78.3Hufnagel XII 0.42 25.79 79.7

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maps. We recommend using a specialized graphical userinterface coupled with a WYSIWYG map visualizationfor selecting projection parameters when creating a cus-tomized Hufnagel projection (an example for theHufnagel projection family is available at https://www.mapthematics.com/interactive/hufnagel.html).

The Hufnagel projections will be available in thenext version of the Geocart software (https://www.mapthematics.com). The computational method andalgorithms presented in this article were implementedin the open-source Java Map Projection Library(https://github.com/OSUCartography/JMapProjLib).We hope that this contribution will help Hufnagel’sprojection family find its place in other cartographic

projection libraries and software applications, andeventually on world maps.

Acknowledgements

The authors would like to thank Richard Čapek, TomasBayer (Charles University, Prague), Jan D. Bláha (JanEvangelista Purkyně University, Ústí nad Labem, Czechia),and Rolf Böhm (Ingenieurbüro für Kartographie, BadSchandau, Germany) for their help and comments. The sup-port of Esri, Inc. is greatly acknowledged. We also thankAbby Metzger and Gareth Baldrica-Franklin, Oregon StateUniversity, for editing this text and figures, and the anon-ymous reviewers for their valuable comments.

Figure 3. Lines of constant maximum angular deformation for the 12 projections, 6° increments.

Figure 4. Outlandish projections illustrating the flexibility of Hufnagel’s method.

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Disclosure statement

No potential conflict of interest was reported by the authors.

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Appendix 1. Equal-area cylindrical projection as limiting case of the Hufnagel projection

This appendix proves that Hufnagel’s projection family transforms into the equal-area cylindrical projection, when Ψmaxapproaches 0. An expression of φ for Hufnagel’s projection is:

sinφ ¼ 4ψ þ 2þ 2Aþ 2A cosð2ψÞ þ 2B cosð4ψÞ½ ' sinð2ψÞ þ B sinð4ψÞ4ψmax þ 2þ 2A$ B½ ' sinð2ψmaxÞ þ Aþ B½ ' sin ð4ψmaxÞ þ B sinð6ψmaxÞ

:

Any particular value of ψ is some fraction k of ψmax, where [0 ≤ k ≤ 1]. In finding the limit as both ψ and ψmax approach 0,then, we first change the function to one variable by substituting k ψmax everywhere for ψ:

sinφ ¼4kψmax þ 2þ 2Aþ 2A cosð2kψmaxÞ þ 2B cosð4kψmaxÞ

% &sinð2kψmaxÞ þ B sinð4kψmaxÞ

4ψmax þ 2þ 2A$ B½ ' sinð2ψmaxÞ þ Aþ B½ ' sin ð4ψmaxÞ þ B sinð6ψmaxÞ:

We cannot directly evaluate sin φ when ψmax = 0 because the fraction is indeterminate. L’Hôpital’s Rule states:limx!c

f ðxÞgðxÞ ¼ lim

x!c

f 0ðxÞg0ðxÞ

Therefore to find the value of sin φ as ψmax → 0,

sinφ¼ limψmax!0

ddψmax

4kψmax þ 2þ 2Aþ 2Acosð2kψmaxÞþ 2Bcosð4kψmaxÞ% &

sinð2kψmaxÞþB sinð4kψmaxÞ' (

ddψmax

4ψmax þ 2þ 2A$B½ ' sinð2ψmaxÞþ AþB½ ' sin ð4ψmaxÞþB sinð6ψmaxÞ' (

¼ limψmax!0

k4$ 4Aþ 8Acos2ð2kψmaxÞ$ 8B sinð2kψmaxÞ sinð4kψmaxÞþ 4cosð2kψmaxÞþ 4Acosð2kψmaxÞþ 4Bcosð2kψmaxÞcosð4kψmaxÞþ 4Bcosð4kψmaxÞ'

4$ 2B cosð2ψmaxÞþ 4cosð2ψmaxÞþ 4Acosð2kψmaxÞ4Acosð4ψmaxÞþ 4Bcosð4ψmaxÞþ 6Bcosð6ψmaxÞ

-

¼ k:

Now, to find the limit of x and y as ψmax → 0, we express x and y as series with the same ψ = k ψmax substitution, andtruncating all terms that are not constant, since they evaluate to 0:

x ¼ λffiffiaπ

p

y ¼ kffiffiπa

p :

But as we just saw, k = sin φ, so

x ¼ λ

ffiffiffiaπ

r

y ¼ sinφ

ffiffiffiπa

r

in the cylindrical case, as expected.

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