Research ArticleOn Conformal Conic Mappings of Spherical Domains

Andrei Bourchtein and Ludmila Bourchtein

Institute of Physics and Mathematics Pelotas State University Brazil

Correspondence should be addressed to Andrei Bourchtein andbursteingmailcom

Received 16 August 2013 Accepted 22 October 2013 Published 14 January 2014

Academic Editors H Bulut B Carpentieri and G Dassios

Copyright copy 2014 A Bourchtein and L Bourchtein This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The problem of the generation of homogeneous grids for spherical domains is considered in the class of conformal conic mappingsThe equivalence between secant and tangent projections is shown and splitting the set of conformal conicmappings into equivalenceclasses is presentedTheproblemofminimization of themapping factor variation is solved in the class of conformal conicmappingsObtained results can be used in applied sciences such as geophysical fluid dynamics and cartography where the flattening of theEarth surface is required

1 Introduction

The problem of the generation of homogeneous grids forspherical domains is one of the oldest problems of car-tography and geodesy and it is also the important partof developing efficient numerical schemes for geophysicalsimulations in particular for atmosphere-ocean dynamicsModeling large-scale atmosphere-ocean processes impliesthe use of the spherical geometry for the formulation ofthe governing equations Computational grids based on thespherical coordinates are highly nonhomogeneous that causethe problems for both dynamical andphysical parts of numer-ical schemes [1ndash3] The most efficient way to circumvent thisproblem is an application of conformal mappings from asphere onto a plane because these transformations usuallymaintain a simpler form of the governing equations andalso assure local isotropy and smoothness of variation of thephysical mesh sizes on computational grids [1ndash6]

Each conformalmapping can be characterized by itsmap-ping factor 119898 representing the ratio between elementary arclengths along a projective curve (image) and correspondingspherical curve (original) If a physical problem requiresthe use of the physical space mesh size ℎ

0 then the ideal

grid is physically homogeneous with the same mesh sizeℎ0over the entire domain On homogeneous computational

grid the physical mesh size usually varies providing betteractual (physical) approximation in the regions where themapping factor 119898 has the maximum values (119898max) and

worse approximation in the regions with the minimummapping factor (119898min) As a measure of the homogeneity ofthe computational grid one can use the ratio between themaximum and minimum values of the mapping factor overthe considered domain

120572 =

119898max119898min

(1)

In particular this criterion is suitable for generation ofcomputational grids for explicit and semi-implicit schemes[1 7 8] As far as we know the use of the variation coefficient120572 for measuring the homogeneity of the computational gridswas first proposed and studied in [1] and the further analysisof the properties of this coefficient and justification of itsapplication in the atmosphere-ocean numerical models wasperformed in different works of the same authors (eg [4 78])

Thus the problem of computational grid optimizationcan be formulated as a search for themapping of the sphericaldomain that assures the minimum values of the variationcoefficient 120572 over the considered spherical domain Ω Inthis study we consider the problem of minimization of 120572

in the class of conic mappings which are standard officialcartographic projections for intermediate and large-scaleregions of the Earth surface [9ndash12] and which are frequentlyused in the modeling of atmosphere and ocean dynamics inthe middle and low latitudes [13ndash20]

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 840280 6 pageshttpdxdoiorg1011552014840280

2 The Scientific World Journal

2 Equivalence Classes of the Conic Mappings

Let us recall the expressions involved in the definition of conicconformal mappings The formulas of the secant conformalconic projections can be written as follows [9ndash11]

120595 = 119899120582

119903 = 119886

cos1205931

119899

(

tan (1205874 minus 1205932)

tan (1205874 minus 12059312)

)

119899

= 119886

cos1205932

119899

(

tan (1205874 minus 1205932)

tan (1205874 minus 12059322)

)

119899

119898 =

cos1205931

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059312)

)

119899

=

cos1205932

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059322)

)

119899

119899 =

ln (cos1205931 cos120593

2)

ln (tan (1205874 minus 12059312) tan (1205874 minus 120593

22))

(2)

where 1205931 1205932are the secant (standard) latitudes minus1205872 lt 120593

1lt

1205932lt 1205872 that is such latitudes where elementary spherical

arch length is equal to projection arc lengthThe tangent conformal conic mappings have the follow-

ing form [9ndash11]

120595 = 119899120582 119903 = 119886

cos1205930

119899

(

tan (1205874 minus 1205932)

tan (1205874 minus 12059302)

)

119899

(3)

119898 =

cos1205930

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059302)

)

119899

(4)

119899 = sin1205930 (5)

where 1205930

isin (minus1205872 1205872) is the tangent latitude (Notethat the tangent formulas can be obtained from the secantones by calculating the limit as 120593

1and 120593

2approach 120593

0)

Although conformal conicmappings have no exact geometricmeaning the terms secant and tangent are widely used bothin cartography and in atmosphere-oceanmodeling [3 5 9 1012 21]

We will call two conformal projections equivalent if theratio between their mapping factors is a constant that isthe first projection with the mapping factor 119898 is equivalentto the second with the mapping factor 119898 on domain Ω ifthere exists a constant 119896 gt 0 such that 119898 = 119896119898 for any(120582 120593) isin Ω Obviously equivalent conformal projectionshave the same space resolution because the transformationfrom one coordinate system to an equivalent one does notaffect physical resolution but only influences the choice of thesystem of units The equivalence of two projections impliesthe equality of their variation coefficients defined by formula(1)

Two conicmappings (secant or tangent) are equivalent ona chosen domain if and only if they have the same value ofthe parameter 119899 In fact the condition

119898 =

cos1205931

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059312)

)

119899

= 119896

cos1205931

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059312)

)

119899

= 119896119898

(6)

can be transformed to

(tan(

120587

4

minus

120593

2

))

119899minus119899

= 119896

cos1205931

cos1205931

(tan (1205874 minus 12059312))119899

(tan (1205874 minus 12059312))119899= const

(7)

which implies 119899 = 119899 On the other hand the condition 119899 = 119899

results in119898 = 119896119898Now we will specify the range of variation of the param-

eter 119899 To this end we first prove two auxiliary lemmas

Lemma 1 The real-value function

119891 (119909) = 119909119899+1

+ 119909119899minus1

119909 isin (0 +infin) 119899 isin (0 1) (8)

is

(1) continuous on its domain(2) strictly decreasing on the interval (0 119909min) and strictly

increasing on the interval (119909min +infin) where

119909min = radic1 minus 119899

1 + 119899

isin (0 1) (9)

is the only minimum point of 119891(119909)(3) two-sided unbounded

lim119909rarr0

+

119891 (119909) = lim119909rarr+infin

119891 (119909) = +infin (10)

Proof In fact the property (1) is evident Moreover it is clearthat 119891(119909) is an analytic function on its domain Calculatingthe first order derivative

1198911015840(119909) = (119899 + 1) 119909

119899+ (119899 minus 1) 119909

119899minus2

= 119909119899minus2

[(119899 + 1) 1199092+ (119899 minus 1)]

(11)

and observing that 119909119899minus2 gt 0 one can obtain the only criticalpoint

119909cr = radic1 minus 119899

1 + 119899

(12)

which belongs to the interval (0 1) Since the derivative (11)is negative on the interval (0 119909cr) and positive on the interval(119909cr +infin) the property (2) holds

The Scientific World Journal 3

Finally the property (3) follows from

lim119909rarr0

+

119909119899+1

= 0 lim119909rarr0

+

119909119899minus1

= +infin

lim119909rarr+infin

119909119899+1

= +infin lim119909rarr+infin

119909119899minus1

= 0

(13)

The results of Lemma 1 together with the propertiesof continuous functions (the Intermediate Value Theorem)guarantee that 119891(119909) takes the same values in exactly twodifferent points 119909

1 1199092such that 119909

1lt 119909min lt 119909

2 The only

exception is the minimum point 119909min

It leads to the following

Corollary 2 The equation

119909119899+1

+ 119909119899minus1

= 119888 119909 isin (0 +infin) 119899 isin (0 1) (14)

has two solutions if 119888 gt 119891min has the only solution 119909min if 119888 =

119891min and has no solutions if 119888 lt 119891min Here

119891min equiv 119891 (119909min) = (radic1 minus 119899

1 + 119899

)

119899+1

+ (radic1 minus 119899

1 + 119899

)

119899minus1

= (

1 + 119899

1 minus 119899

)

(1minus119899)2

sdot

2

1 + 119899

(15)

Evidently 119891min isin (1 2) because both factors in the right-handside of (15) are greater than 1 and 119891(1) = 2

One can reformulate this corollary in the following way

Corollary 3 The equation

119909119899+1

+ 119909119899minus1

= 119905119899+1

+ 119905119899minus1

0 lt 119909 lt 119905 119899 isin (0 1) (16)

has infinite set of solutions (119909 119905) where 119909 isin (0 119909min] Therespective set of 119905 values covers the interval [119909min +infin)

Based on this result we can prove the following

Lemma 4 The equation

ln (cos1205931 cos120593

2)

ln (tan (1205874 minus 12059312) tan (1205874 minus 120593

22))

= 119899

minus

120587

2

lt 1205931lt 1205932lt

120587

2

119899 isin (0 1)

(17)

has infinite set of solutions (1205931 1205932) with any 120593

2from the

interval (120593min 1205872] where

120593min =

120587

2

minus 2 arctanradic1 minus 119899

1 + 119899

0 lt 120593min lt

120587

2

(18)

Proof Equation (17) can be rewritten as follows

ln(

(tan (1205874 minus 12059312) (tan2 (1205874 minus 120593

12) + 1))

tan (1205874 minus 12059322) (tan2 (1205874 minus 120593

22) + 1)

)

times (ln(

tan (1205874 minus 12059312)

tan (1205874 minus 12059322)

))

minus1

= 119899

(19)

Introducing the new variables

1199091= tan(

120587

4

minus

1205932

2

) 1199092= tan(

120587

4

minus

1205931

2

) (20)

where 0 lt 1199091lt 1199092lt +infin one can reduce (19) to the form

ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) = 119899 ln 1199091

1199092

(21)

or

119909119899+1

1+ 119909119899minus1

1= 119909119899+1

2+ 119909119899minus1

2 0 lt 119909

1lt 1199092 119899 isin (0 1) (22)

Thus (17) is reduced to the equivalent equation (22) whichhas infinite set of solutions (119909

1 1199092) with 119909

1isin (0 119909min] due

to Corollary 3 Therefore (17) has infinite set of solutions(1205931 1205932) with 120593

2isin (120593min 1205872] 120593min = 1205872 minus 2 arctan119909min

Besides 0 lt 120593min lt 1205872 because 0 lt 119909min lt 1

Now we can derive the main result about the parameter119899

Theorem 5 The parameter 119899 defined by the formula

119899 =

ln (cos1205931 cos120593

2)

ln (tan (1205874 minus 12059312) tan (1205874 minus 120593

22))

minus

120587

2

lt 1205931lt 1205932lt

120587

2

(23)

belongs to the interval (0 1) if and only if the condition 1205931+

1205932gt 0 is satisfied

Proof Using the change of variables (20) we rewrite (23) inthe form

ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) = 119899 ln 1199091

1199092

(24)

with 0 lt 1199091lt 1199092lt +infin The parameter 119899 belongs to (0 1) if

and only if

ln 1199091

1199092

lt ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) lt 0 (25)

These inequalities are equivalent to

1199091

1199092

lt

1199091

1199092

sdot

1 + 1199092

2

1 + 1199092

1

lt 1 (26)

Since 0 lt 1199091lt 1199092 the left inequality is satisfied The right

inequality can be simplified to the equivalent form 1199091sdot1199092lt 1

that is

tan(

120587

4

minus

1205931

2

) sdot tan(

120587

4

minus

1205932

2

) lt 1 (27)

in the original variables It can be transformed to theequivalent inequality

sin1205931+ 1205932

2

gt 0 (28)

which holds if and only if 0 lt 1205931+ 1205932lt 120587

4 The Scientific World Journal

Remark 6 Although conformal mappings have no exactgeometric representation the obtained restriction 120593

1+ 1205932gt

0 is the condition of the construction of geometric secantcone with the apex above the North Pole In many references[3 5 10 21] this condition (or even more restricted condition1205931

gt 0) is implied implicitly as a natural condition forassuring the possibility of projection on a geometric conelocated above the North Pole However it is worth notingthat ldquogeometric point of viewrdquo can not be directly appliedto conformal conic mappings and consequently the resultof the proved theorem is not evident for conformal map-pings

Remark 7 The last result together with the equivalence con-dition for conformal conic mappings means that any secantconic projection is equivalent to a certain tangent projection(with the same value of 119899) Moreover each tangent conicprojection with specific value of 119899 generates its equivalenceclass of mappings and all equivalence classes are describedby tangent projections when 119899 varies on the interval (0 1)that is for 120593

0isin (0 1205872) It is interesting to note that this

equivalence which could be ldquoevidentrdquo from ldquogeometric pointof viewrdquo is not mentioned in the references Moreover thestatement that secant projections represent a given sphericaldomain better than tangent ones can be found in varioussources [3 10ndash12 21]

Remark 8 It can be shown in a similar way that the condition1205931+ 1205932lt 0 is equivalent to 119899 isin (minus1 0) and it gives rise to

conic mappings with ldquogeometric apexrdquo above the South PoleEach projection of this family has its counterpart among theconic projections with 119899 isin (0 1) Therefore it is sufficient tostudy only the latter mappings

Based on the equivalence properties of the conic map-pings we can conclude that the problem of minimizationof the variation coefficient 120572 in some spherical domain Ω isreduced to the choice of the ldquobestrdquo projection among thetangent conic mappings with 119899 isin (0 1) or equivalently withthe tangent latitude 120593

0varying in (0 1205872)

3 Optimal Choice of Conic Mappings

First we define more precisely spherical domain Ω Sincethe expression of the mapping factor119898 for conic projectionsdoes not depend on the longitude 120582 the same is true forthe variation factorsTherefore the specification of a domainΩ can be given by its north-south extension For example wecan define two extremal latitudes 120593

1and 120593

2 that is define

the latitude interval [120593 minus 120574 120593 + 120574] where the parameters120593 = (120593

2+ 1205931)2 and 120574 = (120593

2minus 1205931)2 determine the domain

location and size with respect to latitude Note that any conicprojection with 119899 isin (0 1) neither is defined at the SouthPole nor has the mapping factor defined at the North PoleTherefore the interval [120593

1 1205932] must be located inside the

open interval (minus1205872 1205872) that is minus1205872 lt 1205931lt 1205932lt 1205872

with 1205931+ 1205932gt 0 This implies that 120593 isin (0 1205872) and 120574 lt

1205872 minus 120593

Now we can prove the following minimization theorem

Theorem9 Theminimum variation of themapping factor (4)is attained at the latitude 120593opt defined by

119899 = sin120593opt

=

ln cos 1205931minus ln cos 120593

2

ln tan (1205874 minus 12059312) minus ln tan (1205874 minus 120593

22)

(29)

For this 119899 the variation coefficient is expressed as follows

120572 =

cos120593opt

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 120593opt2))

sin120593opt

=

cos120593opt

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 120593opt2))

sin120593opt

(30)

Proof First we show that for any fixed 1205930

isin (0 1205872) thepositive function

119898(120593 1205930) =

cos1205930

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059302)

)

sin1205930

120593 isin [1205931 1205932] 120593

1+ 1205932gt 0

(31)

has the absoluteminimum value equal to 1 at the point 1205930and

the absolute maximum value at least at one of the end pointsof the interval [120593

1 1205932]

To this end let us consider the auxiliary function

119891 (120593) =

2

cos120593(tan(

120587

4

minus

120593

2

))

119899

= (tan(

120587

4

minus

120593

2

))

119899minus1

sdot (tan2 (120587

4

minus

120593

2

) + 1)

120593 isin [1205931 1205932] 119899 = sin120593

0isin (0 1)

(32)

Changing the independent variable by the formula 119909 =

tan(1205874 minus 1205932) we can rewrite (32) as follows

119891 (119909) = 119909119899+1

+ 119909119899minus1

119909 isin [1199091 1199092] 119899 isin (0 1)

1199091= tan(

120587

4

minus

1205932

2

) 1199092= tan(

120587

4

minus

1205931

2

)

(33)

By Lemma 1 the function (33) attains the absolute minimumvalue at the point

119909min = radic1 minus 119899

1 + 119899

(34)

and the absolute maximum value at one or both of the endpoints of the interval [119909

1 1199092] This means that the absolute

minimum point of the function (32) is 120593min = 1205930 because

tan(

120587

4

minus

120593min2

) = radic1 minus 119899

1 + 119899

= tan(

120587

4

minus

1205930

2

) (35)

and the absolute maximum point is 1205931or 1205932 Therefore

the same result is true for the original function (31) and

The Scientific World Journal 5

10 degree radius20 degree radius40 degree radius

10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18

Central latitude (deg)

Diff

eren

ce b

etw

een

latitu

des (

deg)

Figure 1 Differences between the optimal tangent latitude andcenter latitude for different domains

substituting 1205930instead of 120593 in this function we obtain that

119898min(1205930) = 119898(1205930 1205930) = 1 Hence

120572 (1205930) = 119898max (1205930) = max 119898 (120593

1 1205930) 119898 (120593

2 1205930) (36)

Now we should minimize the function (36) with respectto the parameter 120593

0 Obviously the solution of this problem

is found from the condition119898(1205931 1205930) = 119898(120593

2 1205930) that is

cos1205930

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 12059302)

)

sin1205930

=

cos1205930

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 12059302)

)

sin1205930

(37)

Simplifying this equation and solving with respect to 119899 =

sin1205930 we obtain formula (29)

Remark 10 Note that formula (29) defines the values of theoptimal tangent latitude in the interval (120593 120593

2)The difference

between 120593opt and 120593 increases with approximation to theNorth Pole and with increase of the radius 120574 In Figure 1 thesedifferences are shown as functions of the centerpoint 120593 fordifferent values of 120574

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by the Brazilian Science Foun-dation FAPERGS

References

[1] A Bourchtein L Bourchtein and E R Oliveira ldquoGeneralapproach to conformal mappings used in atmospheric model-ingrdquo Applied Numerical Mathematics vol 47 no 3-4 pp 305ndash324 2003[2] A Staniforth ldquoRegional modeling a theoretical discussionrdquoMeteorology and Atmospheric Physics vol 63 no 1-2 pp 15ndash291997[3] D L Williamson Difference ApproximAtions for numericalweAther prediction over a sphere vol 2 of GARP No 17 WMO-ICSU 1979[4] A Bourchtein and L Bourchtein ldquoSome problems of confor-mal mappings of spherical domainsrdquo Zeitschrift fur AngewandteMathematik und Physik vol 58 no 6 pp 926ndash939 2007[5] H R Glahn Characteristics of Map Projections and Implica-tions for AWIPS-90 vol 88 of TDL Office Note No 5 NationalWeather Service NOAA 1988[6] P Knupp and S Steinberg Fundamentals of Grid GenerationCRC Press 1993[7] L Bourchtein and A Bourchtein ldquoOn grid generation fornumerical models of geophysical fluid dynamicsrdquo Journal ofComputational andAppliedMathematics vol 218 no 2 pp 317ndash328 2008[8] L Bourchtein and A Bourchtein ldquoComparison of differentspatial grids for numerical schemes of geophysical fluid dynam-icsrdquo Journal of Computational and Applied Mathematics vol227 no 1 pp 161ndash170 2009[9] L M Bugayevskiy and J P Snyder Map Projections AReference Manual CRC Press 1995[10] F Pearson II Map Projections Theory and ApplicationsCRC Press 1990[11] F Richardus and R K AdlerMap Projections for GeodesistsCartographers and Geographers North-Holland AmsterdamThe Netherlands 1972[12] J P Snyder Flattening the Earth TwoThousandYears ofMapProjections The University of Chicago Press 1997[13] V Artale S Calmanti A Carillo et al ldquoAn atmosphere-ocean regional climate model for the Mediterranean areaassessment of a present climate simulationrdquo Climate Dynamicsvol 35 no 5 pp 721ndash740 2010[14] R Benoit M Desgagne P Pellerin S Pellerin Y Chartierand S Desjardins ldquoThe Canadian MC2 a semi-Lagrangiansemi-implicit wideband atmospheric model suited for finescaleprocess studies and simulationrdquo Monthly Weather Review vol125 no 10 pp 2382ndash2415 1997[15] G A Grell J Dudhia and D R Stauffer ldquoA descriptionof the fifth generation Penn StateNCAR mesoscale model(MM5)rdquo NCAR Tech Note NCARTN-398+STR 1994[16] E-S Im E Coppola F Giorgi and X Bi ldquoValidationof a high-resolution regional climate model for the Alpineregion and effects of a subgrid-scale topography and land userepresentationrdquo Journal of Climate vol 23 no 7 pp 1854ndash18732010[17] A Kann C Wittman Y Wang and X Ma ldquoCalibrating 2-m temperature of limited-area ensemble forecasts using high-resolution analysisrdquo Monthly Weather Review vol 137 no 10pp 3373ndash3387 2009[18] X Liang -ZMXu X Yuan et al ldquoRegional climate-weatherresearch and forecasting modelrdquo Bulletin of the AmericanMeteorological Society vol 93 no 9 pp 1363ndash1387 2012

6 The Scientific World Journal

[19] R Mei G Wang and H Gu ldquoSummer land-atmospherecoupling strength over the United States results from theregional climate model RegCM4-CLM35rdquo Journal of Hydrom-eteorology vol 14 no 3 pp 946ndash962 2013[20] K Yessad ldquoBasics about ARPEGEIFS ALADINand AROME in the cycle 40 of ARPEGEIFSrdquo Meteo-FranceCNRM Technical Notes 2013[21] W J Saucier Principles of Meteorological Analysis Dover2012

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

2 Equivalence Classes of the Conic Mappings

Let us recall the expressions involved in the definition of conicconformal mappings The formulas of the secant conformalconic projections can be written as follows [9ndash11]

120595 = 119899120582

119903 = 119886

cos1205931

119899

(

tan (1205874 minus 1205932)

tan (1205874 minus 12059312)

)

119899

= 119886

cos1205932

119899

(

tan (1205874 minus 1205932)

tan (1205874 minus 12059322)

)

119899

119898 =

cos1205931

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059312)

)

119899

=

cos1205932

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059322)

)

119899

119899 =

ln (cos1205931 cos120593

2)

ln (tan (1205874 minus 12059312) tan (1205874 minus 120593

22))

(2)

where 1205931 1205932are the secant (standard) latitudes minus1205872 lt 120593

1lt

1205932lt 1205872 that is such latitudes where elementary spherical

arch length is equal to projection arc lengthThe tangent conformal conic mappings have the follow-

ing form [9ndash11]

120595 = 119899120582 119903 = 119886

cos1205930

119899

(

tan (1205874 minus 1205932)

tan (1205874 minus 12059302)

)

119899

(3)

119898 =

cos1205930

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059302)

)

119899

(4)

119899 = sin1205930 (5)

where 1205930

isin (minus1205872 1205872) is the tangent latitude (Notethat the tangent formulas can be obtained from the secantones by calculating the limit as 120593

1and 120593

2approach 120593

0)

Although conformal conicmappings have no exact geometricmeaning the terms secant and tangent are widely used bothin cartography and in atmosphere-oceanmodeling [3 5 9 1012 21]

We will call two conformal projections equivalent if theratio between their mapping factors is a constant that isthe first projection with the mapping factor 119898 is equivalentto the second with the mapping factor 119898 on domain Ω ifthere exists a constant 119896 gt 0 such that 119898 = 119896119898 for any(120582 120593) isin Ω Obviously equivalent conformal projectionshave the same space resolution because the transformationfrom one coordinate system to an equivalent one does notaffect physical resolution but only influences the choice of thesystem of units The equivalence of two projections impliesthe equality of their variation coefficients defined by formula(1)

Two conicmappings (secant or tangent) are equivalent ona chosen domain if and only if they have the same value ofthe parameter 119899 In fact the condition

119898 =

cos1205931

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059312)

)

119899

= 119896

cos1205931

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059312)

)

119899

= 119896119898

(6)

can be transformed to

(tan(

120587

4

minus

120593

2

))

119899minus119899

= 119896

cos1205931

cos1205931

(tan (1205874 minus 12059312))119899

(tan (1205874 minus 12059312))119899= const

(7)

which implies 119899 = 119899 On the other hand the condition 119899 = 119899

results in119898 = 119896119898Now we will specify the range of variation of the param-

eter 119899 To this end we first prove two auxiliary lemmas

Lemma 1 The real-value function

119891 (119909) = 119909119899+1

+ 119909119899minus1

119909 isin (0 +infin) 119899 isin (0 1) (8)

is

(1) continuous on its domain(2) strictly decreasing on the interval (0 119909min) and strictly

increasing on the interval (119909min +infin) where

119909min = radic1 minus 119899

1 + 119899

isin (0 1) (9)

is the only minimum point of 119891(119909)(3) two-sided unbounded

lim119909rarr0

+

119891 (119909) = lim119909rarr+infin

119891 (119909) = +infin (10)

Proof In fact the property (1) is evident Moreover it is clearthat 119891(119909) is an analytic function on its domain Calculatingthe first order derivative

1198911015840(119909) = (119899 + 1) 119909

119899+ (119899 minus 1) 119909

119899minus2

= 119909119899minus2

[(119899 + 1) 1199092+ (119899 minus 1)]

(11)

and observing that 119909119899minus2 gt 0 one can obtain the only criticalpoint

119909cr = radic1 minus 119899

1 + 119899

(12)

which belongs to the interval (0 1) Since the derivative (11)is negative on the interval (0 119909cr) and positive on the interval(119909cr +infin) the property (2) holds

The Scientific World Journal 3

Finally the property (3) follows from

lim119909rarr0

+

119909119899+1

= 0 lim119909rarr0

+

119909119899minus1

= +infin

lim119909rarr+infin

119909119899+1

= +infin lim119909rarr+infin

119909119899minus1

= 0

(13)

The results of Lemma 1 together with the propertiesof continuous functions (the Intermediate Value Theorem)guarantee that 119891(119909) takes the same values in exactly twodifferent points 119909

1 1199092such that 119909

1lt 119909min lt 119909

2 The only

exception is the minimum point 119909min

It leads to the following

Corollary 2 The equation

119909119899+1

+ 119909119899minus1

= 119888 119909 isin (0 +infin) 119899 isin (0 1) (14)

has two solutions if 119888 gt 119891min has the only solution 119909min if 119888 =

119891min and has no solutions if 119888 lt 119891min Here

119891min equiv 119891 (119909min) = (radic1 minus 119899

1 + 119899

)

119899+1

+ (radic1 minus 119899

1 + 119899

)

119899minus1

= (

1 + 119899

1 minus 119899

)

(1minus119899)2

sdot

2

1 + 119899

(15)

Evidently 119891min isin (1 2) because both factors in the right-handside of (15) are greater than 1 and 119891(1) = 2

One can reformulate this corollary in the following way

Corollary 3 The equation

119909119899+1

+ 119909119899minus1

= 119905119899+1

+ 119905119899minus1

0 lt 119909 lt 119905 119899 isin (0 1) (16)

has infinite set of solutions (119909 119905) where 119909 isin (0 119909min] Therespective set of 119905 values covers the interval [119909min +infin)

Based on this result we can prove the following

Lemma 4 The equation

ln (cos1205931 cos120593

2)

ln (tan (1205874 minus 12059312) tan (1205874 minus 120593

22))

= 119899

minus

120587

2

lt 1205931lt 1205932lt

120587

2

119899 isin (0 1)

(17)

has infinite set of solutions (1205931 1205932) with any 120593

2from the

interval (120593min 1205872] where

120593min =

120587

2

minus 2 arctanradic1 minus 119899

1 + 119899

0 lt 120593min lt

120587

2

(18)

Proof Equation (17) can be rewritten as follows

ln(

(tan (1205874 minus 12059312) (tan2 (1205874 minus 120593

12) + 1))

tan (1205874 minus 12059322) (tan2 (1205874 minus 120593

22) + 1)

)

times (ln(

tan (1205874 minus 12059312)

tan (1205874 minus 12059322)

))

minus1

= 119899

(19)

Introducing the new variables

1199091= tan(

120587

4

minus

1205932

2

) 1199092= tan(

120587

4

minus

1205931

2

) (20)

where 0 lt 1199091lt 1199092lt +infin one can reduce (19) to the form

ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) = 119899 ln 1199091

1199092

(21)

or

119909119899+1

1+ 119909119899minus1

1= 119909119899+1

2+ 119909119899minus1

2 0 lt 119909

1lt 1199092 119899 isin (0 1) (22)

Thus (17) is reduced to the equivalent equation (22) whichhas infinite set of solutions (119909

1 1199092) with 119909

1isin (0 119909min] due

to Corollary 3 Therefore (17) has infinite set of solutions(1205931 1205932) with 120593

2isin (120593min 1205872] 120593min = 1205872 minus 2 arctan119909min

Besides 0 lt 120593min lt 1205872 because 0 lt 119909min lt 1

Now we can derive the main result about the parameter119899

Theorem 5 The parameter 119899 defined by the formula

119899 =

ln (cos1205931 cos120593

2)

ln (tan (1205874 minus 12059312) tan (1205874 minus 120593

22))

minus

120587

2

lt 1205931lt 1205932lt

120587

2

(23)

belongs to the interval (0 1) if and only if the condition 1205931+

1205932gt 0 is satisfied

Proof Using the change of variables (20) we rewrite (23) inthe form

ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) = 119899 ln 1199091

1199092

(24)

with 0 lt 1199091lt 1199092lt +infin The parameter 119899 belongs to (0 1) if

and only if

ln 1199091

1199092

lt ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) lt 0 (25)

These inequalities are equivalent to

1199091

1199092

lt

1199091

1199092

sdot

1 + 1199092

2

1 + 1199092

1

lt 1 (26)

Since 0 lt 1199091lt 1199092 the left inequality is satisfied The right

inequality can be simplified to the equivalent form 1199091sdot1199092lt 1

that is

tan(

120587

4

minus

1205931

2

) sdot tan(

120587

4

minus

1205932

2

) lt 1 (27)

in the original variables It can be transformed to theequivalent inequality

sin1205931+ 1205932

2

gt 0 (28)

which holds if and only if 0 lt 1205931+ 1205932lt 120587

4 The Scientific World Journal

Remark 6 Although conformal mappings have no exactgeometric representation the obtained restriction 120593

1+ 1205932gt

0 is the condition of the construction of geometric secantcone with the apex above the North Pole In many references[3 5 10 21] this condition (or even more restricted condition1205931

gt 0) is implied implicitly as a natural condition forassuring the possibility of projection on a geometric conelocated above the North Pole However it is worth notingthat ldquogeometric point of viewrdquo can not be directly appliedto conformal conic mappings and consequently the resultof the proved theorem is not evident for conformal map-pings

Remark 7 The last result together with the equivalence con-dition for conformal conic mappings means that any secantconic projection is equivalent to a certain tangent projection(with the same value of 119899) Moreover each tangent conicprojection with specific value of 119899 generates its equivalenceclass of mappings and all equivalence classes are describedby tangent projections when 119899 varies on the interval (0 1)that is for 120593

0isin (0 1205872) It is interesting to note that this

equivalence which could be ldquoevidentrdquo from ldquogeometric pointof viewrdquo is not mentioned in the references Moreover thestatement that secant projections represent a given sphericaldomain better than tangent ones can be found in varioussources [3 10ndash12 21]

Remark 8 It can be shown in a similar way that the condition1205931+ 1205932lt 0 is equivalent to 119899 isin (minus1 0) and it gives rise to

conic mappings with ldquogeometric apexrdquo above the South PoleEach projection of this family has its counterpart among theconic projections with 119899 isin (0 1) Therefore it is sufficient tostudy only the latter mappings

Based on the equivalence properties of the conic map-pings we can conclude that the problem of minimizationof the variation coefficient 120572 in some spherical domain Ω isreduced to the choice of the ldquobestrdquo projection among thetangent conic mappings with 119899 isin (0 1) or equivalently withthe tangent latitude 120593

0varying in (0 1205872)

3 Optimal Choice of Conic Mappings

First we define more precisely spherical domain Ω Sincethe expression of the mapping factor119898 for conic projectionsdoes not depend on the longitude 120582 the same is true forthe variation factorsTherefore the specification of a domainΩ can be given by its north-south extension For example wecan define two extremal latitudes 120593

1and 120593

2 that is define

the latitude interval [120593 minus 120574 120593 + 120574] where the parameters120593 = (120593

2+ 1205931)2 and 120574 = (120593

2minus 1205931)2 determine the domain

location and size with respect to latitude Note that any conicprojection with 119899 isin (0 1) neither is defined at the SouthPole nor has the mapping factor defined at the North PoleTherefore the interval [120593

1 1205932] must be located inside the

open interval (minus1205872 1205872) that is minus1205872 lt 1205931lt 1205932lt 1205872

with 1205931+ 1205932gt 0 This implies that 120593 isin (0 1205872) and 120574 lt

1205872 minus 120593

Now we can prove the following minimization theorem

Theorem9 Theminimum variation of themapping factor (4)is attained at the latitude 120593opt defined by

119899 = sin120593opt

=

ln cos 1205931minus ln cos 120593

2

ln tan (1205874 minus 12059312) minus ln tan (1205874 minus 120593

22)

(29)

For this 119899 the variation coefficient is expressed as follows

120572 =

cos120593opt

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 120593opt2))

sin120593opt

=

cos120593opt

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 120593opt2))

sin120593opt

(30)

Proof First we show that for any fixed 1205930

isin (0 1205872) thepositive function

119898(120593 1205930) =

cos1205930

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059302)

)

sin1205930

120593 isin [1205931 1205932] 120593

1+ 1205932gt 0

(31)

has the absoluteminimum value equal to 1 at the point 1205930and

the absolute maximum value at least at one of the end pointsof the interval [120593

1 1205932]

To this end let us consider the auxiliary function

119891 (120593) =

2

cos120593(tan(

120587

4

minus

120593

2

))

119899

= (tan(

120587

4

minus

120593

2

))

119899minus1

sdot (tan2 (120587

4

minus

120593

2

) + 1)

120593 isin [1205931 1205932] 119899 = sin120593

0isin (0 1)

(32)

Changing the independent variable by the formula 119909 =

tan(1205874 minus 1205932) we can rewrite (32) as follows

119891 (119909) = 119909119899+1

+ 119909119899minus1

119909 isin [1199091 1199092] 119899 isin (0 1)

1199091= tan(

120587

4

minus

1205932

2

) 1199092= tan(

120587

4

minus

1205931

2

)

(33)

By Lemma 1 the function (33) attains the absolute minimumvalue at the point

119909min = radic1 minus 119899

1 + 119899

(34)

and the absolute maximum value at one or both of the endpoints of the interval [119909

1 1199092] This means that the absolute

minimum point of the function (32) is 120593min = 1205930 because

tan(

120587

4

minus

120593min2

) = radic1 minus 119899

1 + 119899

= tan(

120587

4

minus

1205930

2

) (35)

and the absolute maximum point is 1205931or 1205932 Therefore

the same result is true for the original function (31) and

The Scientific World Journal 5

10 degree radius20 degree radius40 degree radius

10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18

Central latitude (deg)

Diff

eren

ce b

etw

een

latitu

des (

deg)

Figure 1 Differences between the optimal tangent latitude andcenter latitude for different domains

substituting 1205930instead of 120593 in this function we obtain that

119898min(1205930) = 119898(1205930 1205930) = 1 Hence

120572 (1205930) = 119898max (1205930) = max 119898 (120593

1 1205930) 119898 (120593

2 1205930) (36)

Now we should minimize the function (36) with respectto the parameter 120593

0 Obviously the solution of this problem

is found from the condition119898(1205931 1205930) = 119898(120593

2 1205930) that is

cos1205930

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 12059302)

)

sin1205930

=

cos1205930

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 12059302)

)

sin1205930

(37)

Simplifying this equation and solving with respect to 119899 =

sin1205930 we obtain formula (29)

Remark 10 Note that formula (29) defines the values of theoptimal tangent latitude in the interval (120593 120593

2)The difference

between 120593opt and 120593 increases with approximation to theNorth Pole and with increase of the radius 120574 In Figure 1 thesedifferences are shown as functions of the centerpoint 120593 fordifferent values of 120574

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by the Brazilian Science Foun-dation FAPERGS

References

[1] A Bourchtein L Bourchtein and E R Oliveira ldquoGeneralapproach to conformal mappings used in atmospheric model-ingrdquo Applied Numerical Mathematics vol 47 no 3-4 pp 305ndash324 2003[2] A Staniforth ldquoRegional modeling a theoretical discussionrdquoMeteorology and Atmospheric Physics vol 63 no 1-2 pp 15ndash291997[3] D L Williamson Difference ApproximAtions for numericalweAther prediction over a sphere vol 2 of GARP No 17 WMO-ICSU 1979[4] A Bourchtein and L Bourchtein ldquoSome problems of confor-mal mappings of spherical domainsrdquo Zeitschrift fur AngewandteMathematik und Physik vol 58 no 6 pp 926ndash939 2007[5] H R Glahn Characteristics of Map Projections and Implica-tions for AWIPS-90 vol 88 of TDL Office Note No 5 NationalWeather Service NOAA 1988[6] P Knupp and S Steinberg Fundamentals of Grid GenerationCRC Press 1993[7] L Bourchtein and A Bourchtein ldquoOn grid generation fornumerical models of geophysical fluid dynamicsrdquo Journal ofComputational andAppliedMathematics vol 218 no 2 pp 317ndash328 2008[8] L Bourchtein and A Bourchtein ldquoComparison of differentspatial grids for numerical schemes of geophysical fluid dynam-icsrdquo Journal of Computational and Applied Mathematics vol227 no 1 pp 161ndash170 2009[9] L M Bugayevskiy and J P Snyder Map Projections AReference Manual CRC Press 1995[10] F Pearson II Map Projections Theory and ApplicationsCRC Press 1990[11] F Richardus and R K AdlerMap Projections for GeodesistsCartographers and Geographers North-Holland AmsterdamThe Netherlands 1972[12] J P Snyder Flattening the Earth TwoThousandYears ofMapProjections The University of Chicago Press 1997[13] V Artale S Calmanti A Carillo et al ldquoAn atmosphere-ocean regional climate model for the Mediterranean areaassessment of a present climate simulationrdquo Climate Dynamicsvol 35 no 5 pp 721ndash740 2010[14] R Benoit M Desgagne P Pellerin S Pellerin Y Chartierand S Desjardins ldquoThe Canadian MC2 a semi-Lagrangiansemi-implicit wideband atmospheric model suited for finescaleprocess studies and simulationrdquo Monthly Weather Review vol125 no 10 pp 2382ndash2415 1997[15] G A Grell J Dudhia and D R Stauffer ldquoA descriptionof the fifth generation Penn StateNCAR mesoscale model(MM5)rdquo NCAR Tech Note NCARTN-398+STR 1994[16] E-S Im E Coppola F Giorgi and X Bi ldquoValidationof a high-resolution regional climate model for the Alpineregion and effects of a subgrid-scale topography and land userepresentationrdquo Journal of Climate vol 23 no 7 pp 1854ndash18732010[17] A Kann C Wittman Y Wang and X Ma ldquoCalibrating 2-m temperature of limited-area ensemble forecasts using high-resolution analysisrdquo Monthly Weather Review vol 137 no 10pp 3373ndash3387 2009[18] X Liang -ZMXu X Yuan et al ldquoRegional climate-weatherresearch and forecasting modelrdquo Bulletin of the AmericanMeteorological Society vol 93 no 9 pp 1363ndash1387 2012

6 The Scientific World Journal

[19] R Mei G Wang and H Gu ldquoSummer land-atmospherecoupling strength over the United States results from theregional climate model RegCM4-CLM35rdquo Journal of Hydrom-eteorology vol 14 no 3 pp 946ndash962 2013[20] K Yessad ldquoBasics about ARPEGEIFS ALADINand AROME in the cycle 40 of ARPEGEIFSrdquo Meteo-FranceCNRM Technical Notes 2013[21] W J Saucier Principles of Meteorological Analysis Dover2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

Finally the property (3) follows from

lim119909rarr0

+

119909119899+1

= 0 lim119909rarr0

+

119909119899minus1

= +infin

lim119909rarr+infin

119909119899+1

= +infin lim119909rarr+infin

119909119899minus1

= 0

(13)

The results of Lemma 1 together with the propertiesof continuous functions (the Intermediate Value Theorem)guarantee that 119891(119909) takes the same values in exactly twodifferent points 119909

1 1199092such that 119909

1lt 119909min lt 119909

2 The only

exception is the minimum point 119909min

It leads to the following

Corollary 2 The equation

119909119899+1

+ 119909119899minus1

= 119888 119909 isin (0 +infin) 119899 isin (0 1) (14)

has two solutions if 119888 gt 119891min has the only solution 119909min if 119888 =

119891min and has no solutions if 119888 lt 119891min Here

119891min equiv 119891 (119909min) = (radic1 minus 119899

1 + 119899

)

119899+1

+ (radic1 minus 119899

1 + 119899

)

119899minus1

= (

1 + 119899

1 minus 119899

)

(1minus119899)2

sdot

2

1 + 119899

(15)

Evidently 119891min isin (1 2) because both factors in the right-handside of (15) are greater than 1 and 119891(1) = 2

One can reformulate this corollary in the following way

Corollary 3 The equation

119909119899+1

+ 119909119899minus1

= 119905119899+1

+ 119905119899minus1

0 lt 119909 lt 119905 119899 isin (0 1) (16)

has infinite set of solutions (119909 119905) where 119909 isin (0 119909min] Therespective set of 119905 values covers the interval [119909min +infin)

Based on this result we can prove the following

Lemma 4 The equation

ln (cos1205931 cos120593

2)

ln (tan (1205874 minus 12059312) tan (1205874 minus 120593

22))

= 119899

minus

120587

2

lt 1205931lt 1205932lt

120587

2

119899 isin (0 1)

(17)

has infinite set of solutions (1205931 1205932) with any 120593

2from the

interval (120593min 1205872] where

120593min =

120587

2

minus 2 arctanradic1 minus 119899

1 + 119899

0 lt 120593min lt

120587

2

(18)

Proof Equation (17) can be rewritten as follows

ln(

(tan (1205874 minus 12059312) (tan2 (1205874 minus 120593

12) + 1))

tan (1205874 minus 12059322) (tan2 (1205874 minus 120593

22) + 1)

)

times (ln(

tan (1205874 minus 12059312)

tan (1205874 minus 12059322)

))

minus1

= 119899

(19)

Introducing the new variables

1199091= tan(

120587

4

minus

1205932

2

) 1199092= tan(

120587

4

minus

1205931

2

) (20)

where 0 lt 1199091lt 1199092lt +infin one can reduce (19) to the form

ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) = 119899 ln 1199091

1199092

(21)

or

119909119899+1

1+ 119909119899minus1

1= 119909119899+1

2+ 119909119899minus1

2 0 lt 119909

1lt 1199092 119899 isin (0 1) (22)

Thus (17) is reduced to the equivalent equation (22) whichhas infinite set of solutions (119909

1 1199092) with 119909

1isin (0 119909min] due

to Corollary 3 Therefore (17) has infinite set of solutions(1205931 1205932) with 120593

2isin (120593min 1205872] 120593min = 1205872 minus 2 arctan119909min

Besides 0 lt 120593min lt 1205872 because 0 lt 119909min lt 1

Now we can derive the main result about the parameter119899

Theorem 5 The parameter 119899 defined by the formula

119899 =

ln (cos1205931 cos120593

2)

ln (tan (1205874 minus 12059312) tan (1205874 minus 120593

22))

minus

120587

2

lt 1205931lt 1205932lt

120587

2

(23)

belongs to the interval (0 1) if and only if the condition 1205931+

1205932gt 0 is satisfied

Proof Using the change of variables (20) we rewrite (23) inthe form

ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) = 119899 ln 1199091

1199092

(24)

with 0 lt 1199091lt 1199092lt +infin The parameter 119899 belongs to (0 1) if

and only if

ln 1199091

1199092

lt ln(

(1199091 (1199092

1+ 1))

(1199092 (1199092

2+ 1))

) lt 0 (25)

These inequalities are equivalent to

1199091

1199092

lt

1199091

1199092

sdot

1 + 1199092

2

1 + 1199092

1

lt 1 (26)

Since 0 lt 1199091lt 1199092 the left inequality is satisfied The right

inequality can be simplified to the equivalent form 1199091sdot1199092lt 1

that is

tan(

120587

4

minus

1205931

2

) sdot tan(

120587

4

minus

1205932

2

) lt 1 (27)

in the original variables It can be transformed to theequivalent inequality

sin1205931+ 1205932

2

gt 0 (28)

which holds if and only if 0 lt 1205931+ 1205932lt 120587

4 The Scientific World Journal

Remark 6 Although conformal mappings have no exactgeometric representation the obtained restriction 120593

1+ 1205932gt

0 is the condition of the construction of geometric secantcone with the apex above the North Pole In many references[3 5 10 21] this condition (or even more restricted condition1205931

gt 0) is implied implicitly as a natural condition forassuring the possibility of projection on a geometric conelocated above the North Pole However it is worth notingthat ldquogeometric point of viewrdquo can not be directly appliedto conformal conic mappings and consequently the resultof the proved theorem is not evident for conformal map-pings

Remark 7 The last result together with the equivalence con-dition for conformal conic mappings means that any secantconic projection is equivalent to a certain tangent projection(with the same value of 119899) Moreover each tangent conicprojection with specific value of 119899 generates its equivalenceclass of mappings and all equivalence classes are describedby tangent projections when 119899 varies on the interval (0 1)that is for 120593

0isin (0 1205872) It is interesting to note that this

equivalence which could be ldquoevidentrdquo from ldquogeometric pointof viewrdquo is not mentioned in the references Moreover thestatement that secant projections represent a given sphericaldomain better than tangent ones can be found in varioussources [3 10ndash12 21]

Remark 8 It can be shown in a similar way that the condition1205931+ 1205932lt 0 is equivalent to 119899 isin (minus1 0) and it gives rise to

conic mappings with ldquogeometric apexrdquo above the South PoleEach projection of this family has its counterpart among theconic projections with 119899 isin (0 1) Therefore it is sufficient tostudy only the latter mappings

Based on the equivalence properties of the conic map-pings we can conclude that the problem of minimizationof the variation coefficient 120572 in some spherical domain Ω isreduced to the choice of the ldquobestrdquo projection among thetangent conic mappings with 119899 isin (0 1) or equivalently withthe tangent latitude 120593

0varying in (0 1205872)

3 Optimal Choice of Conic Mappings

First we define more precisely spherical domain Ω Sincethe expression of the mapping factor119898 for conic projectionsdoes not depend on the longitude 120582 the same is true forthe variation factorsTherefore the specification of a domainΩ can be given by its north-south extension For example wecan define two extremal latitudes 120593

1and 120593

2 that is define

the latitude interval [120593 minus 120574 120593 + 120574] where the parameters120593 = (120593

2+ 1205931)2 and 120574 = (120593

2minus 1205931)2 determine the domain

location and size with respect to latitude Note that any conicprojection with 119899 isin (0 1) neither is defined at the SouthPole nor has the mapping factor defined at the North PoleTherefore the interval [120593

1 1205932] must be located inside the

open interval (minus1205872 1205872) that is minus1205872 lt 1205931lt 1205932lt 1205872

with 1205931+ 1205932gt 0 This implies that 120593 isin (0 1205872) and 120574 lt

1205872 minus 120593

Now we can prove the following minimization theorem

Theorem9 Theminimum variation of themapping factor (4)is attained at the latitude 120593opt defined by

119899 = sin120593opt

=

ln cos 1205931minus ln cos 120593

2

ln tan (1205874 minus 12059312) minus ln tan (1205874 minus 120593

22)

(29)

For this 119899 the variation coefficient is expressed as follows

120572 =

cos120593opt

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 120593opt2))

sin120593opt

=

cos120593opt

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 120593opt2))

sin120593opt

(30)

Proof First we show that for any fixed 1205930

isin (0 1205872) thepositive function

119898(120593 1205930) =

cos1205930

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059302)

)

sin1205930

120593 isin [1205931 1205932] 120593

1+ 1205932gt 0

(31)

has the absoluteminimum value equal to 1 at the point 1205930and

the absolute maximum value at least at one of the end pointsof the interval [120593

1 1205932]

To this end let us consider the auxiliary function

119891 (120593) =

2

cos120593(tan(

120587

4

minus

120593

2

))

119899

= (tan(

120587

4

minus

120593

2

))

119899minus1

sdot (tan2 (120587

4

minus

120593

2

) + 1)

120593 isin [1205931 1205932] 119899 = sin120593

0isin (0 1)

(32)

Changing the independent variable by the formula 119909 =

tan(1205874 minus 1205932) we can rewrite (32) as follows

119891 (119909) = 119909119899+1

+ 119909119899minus1

119909 isin [1199091 1199092] 119899 isin (0 1)

1199091= tan(

120587

4

minus

1205932

2

) 1199092= tan(

120587

4

minus

1205931

2

)

(33)

By Lemma 1 the function (33) attains the absolute minimumvalue at the point

119909min = radic1 minus 119899

1 + 119899

(34)

and the absolute maximum value at one or both of the endpoints of the interval [119909

1 1199092] This means that the absolute

minimum point of the function (32) is 120593min = 1205930 because

tan(

120587

4

minus

120593min2

) = radic1 minus 119899

1 + 119899

= tan(

120587

4

minus

1205930

2

) (35)

and the absolute maximum point is 1205931or 1205932 Therefore

the same result is true for the original function (31) and

The Scientific World Journal 5

10 degree radius20 degree radius40 degree radius

10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18

Central latitude (deg)

Diff

eren

ce b

etw

een

latitu

des (

deg)

Figure 1 Differences between the optimal tangent latitude andcenter latitude for different domains

substituting 1205930instead of 120593 in this function we obtain that

119898min(1205930) = 119898(1205930 1205930) = 1 Hence

120572 (1205930) = 119898max (1205930) = max 119898 (120593

1 1205930) 119898 (120593

2 1205930) (36)

Now we should minimize the function (36) with respectto the parameter 120593

0 Obviously the solution of this problem

is found from the condition119898(1205931 1205930) = 119898(120593

2 1205930) that is

cos1205930

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 12059302)

)

sin1205930

=

cos1205930

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 12059302)

)

sin1205930

(37)

Simplifying this equation and solving with respect to 119899 =

sin1205930 we obtain formula (29)

Remark 10 Note that formula (29) defines the values of theoptimal tangent latitude in the interval (120593 120593

2)The difference

between 120593opt and 120593 increases with approximation to theNorth Pole and with increase of the radius 120574 In Figure 1 thesedifferences are shown as functions of the centerpoint 120593 fordifferent values of 120574

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by the Brazilian Science Foun-dation FAPERGS

References

[1] A Bourchtein L Bourchtein and E R Oliveira ldquoGeneralapproach to conformal mappings used in atmospheric model-ingrdquo Applied Numerical Mathematics vol 47 no 3-4 pp 305ndash324 2003[2] A Staniforth ldquoRegional modeling a theoretical discussionrdquoMeteorology and Atmospheric Physics vol 63 no 1-2 pp 15ndash291997[3] D L Williamson Difference ApproximAtions for numericalweAther prediction over a sphere vol 2 of GARP No 17 WMO-ICSU 1979[4] A Bourchtein and L Bourchtein ldquoSome problems of confor-mal mappings of spherical domainsrdquo Zeitschrift fur AngewandteMathematik und Physik vol 58 no 6 pp 926ndash939 2007[5] H R Glahn Characteristics of Map Projections and Implica-tions for AWIPS-90 vol 88 of TDL Office Note No 5 NationalWeather Service NOAA 1988[6] P Knupp and S Steinberg Fundamentals of Grid GenerationCRC Press 1993[7] L Bourchtein and A Bourchtein ldquoOn grid generation fornumerical models of geophysical fluid dynamicsrdquo Journal ofComputational andAppliedMathematics vol 218 no 2 pp 317ndash328 2008[8] L Bourchtein and A Bourchtein ldquoComparison of differentspatial grids for numerical schemes of geophysical fluid dynam-icsrdquo Journal of Computational and Applied Mathematics vol227 no 1 pp 161ndash170 2009[9] L M Bugayevskiy and J P Snyder Map Projections AReference Manual CRC Press 1995[10] F Pearson II Map Projections Theory and ApplicationsCRC Press 1990[11] F Richardus and R K AdlerMap Projections for GeodesistsCartographers and Geographers North-Holland AmsterdamThe Netherlands 1972[12] J P Snyder Flattening the Earth TwoThousandYears ofMapProjections The University of Chicago Press 1997[13] V Artale S Calmanti A Carillo et al ldquoAn atmosphere-ocean regional climate model for the Mediterranean areaassessment of a present climate simulationrdquo Climate Dynamicsvol 35 no 5 pp 721ndash740 2010[14] R Benoit M Desgagne P Pellerin S Pellerin Y Chartierand S Desjardins ldquoThe Canadian MC2 a semi-Lagrangiansemi-implicit wideband atmospheric model suited for finescaleprocess studies and simulationrdquo Monthly Weather Review vol125 no 10 pp 2382ndash2415 1997[15] G A Grell J Dudhia and D R Stauffer ldquoA descriptionof the fifth generation Penn StateNCAR mesoscale model(MM5)rdquo NCAR Tech Note NCARTN-398+STR 1994[16] E-S Im E Coppola F Giorgi and X Bi ldquoValidationof a high-resolution regional climate model for the Alpineregion and effects of a subgrid-scale topography and land userepresentationrdquo Journal of Climate vol 23 no 7 pp 1854ndash18732010[17] A Kann C Wittman Y Wang and X Ma ldquoCalibrating 2-m temperature of limited-area ensemble forecasts using high-resolution analysisrdquo Monthly Weather Review vol 137 no 10pp 3373ndash3387 2009[18] X Liang -ZMXu X Yuan et al ldquoRegional climate-weatherresearch and forecasting modelrdquo Bulletin of the AmericanMeteorological Society vol 93 no 9 pp 1363ndash1387 2012

6 The Scientific World Journal

[19] R Mei G Wang and H Gu ldquoSummer land-atmospherecoupling strength over the United States results from theregional climate model RegCM4-CLM35rdquo Journal of Hydrom-eteorology vol 14 no 3 pp 946ndash962 2013[20] K Yessad ldquoBasics about ARPEGEIFS ALADINand AROME in the cycle 40 of ARPEGEIFSrdquo Meteo-FranceCNRM Technical Notes 2013[21] W J Saucier Principles of Meteorological Analysis Dover2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

Remark 6 Although conformal mappings have no exactgeometric representation the obtained restriction 120593

1+ 1205932gt

0 is the condition of the construction of geometric secantcone with the apex above the North Pole In many references[3 5 10 21] this condition (or even more restricted condition1205931

gt 0) is implied implicitly as a natural condition forassuring the possibility of projection on a geometric conelocated above the North Pole However it is worth notingthat ldquogeometric point of viewrdquo can not be directly appliedto conformal conic mappings and consequently the resultof the proved theorem is not evident for conformal map-pings

Remark 7 The last result together with the equivalence con-dition for conformal conic mappings means that any secantconic projection is equivalent to a certain tangent projection(with the same value of 119899) Moreover each tangent conicprojection with specific value of 119899 generates its equivalenceclass of mappings and all equivalence classes are describedby tangent projections when 119899 varies on the interval (0 1)that is for 120593

0isin (0 1205872) It is interesting to note that this

equivalence which could be ldquoevidentrdquo from ldquogeometric pointof viewrdquo is not mentioned in the references Moreover thestatement that secant projections represent a given sphericaldomain better than tangent ones can be found in varioussources [3 10ndash12 21]

Remark 8 It can be shown in a similar way that the condition1205931+ 1205932lt 0 is equivalent to 119899 isin (minus1 0) and it gives rise to

conic mappings with ldquogeometric apexrdquo above the South PoleEach projection of this family has its counterpart among theconic projections with 119899 isin (0 1) Therefore it is sufficient tostudy only the latter mappings

Based on the equivalence properties of the conic map-pings we can conclude that the problem of minimizationof the variation coefficient 120572 in some spherical domain Ω isreduced to the choice of the ldquobestrdquo projection among thetangent conic mappings with 119899 isin (0 1) or equivalently withthe tangent latitude 120593

0varying in (0 1205872)

3 Optimal Choice of Conic Mappings

First we define more precisely spherical domain Ω Sincethe expression of the mapping factor119898 for conic projectionsdoes not depend on the longitude 120582 the same is true forthe variation factorsTherefore the specification of a domainΩ can be given by its north-south extension For example wecan define two extremal latitudes 120593

1and 120593

2 that is define

the latitude interval [120593 minus 120574 120593 + 120574] where the parameters120593 = (120593

2+ 1205931)2 and 120574 = (120593

2minus 1205931)2 determine the domain

location and size with respect to latitude Note that any conicprojection with 119899 isin (0 1) neither is defined at the SouthPole nor has the mapping factor defined at the North PoleTherefore the interval [120593

1 1205932] must be located inside the

open interval (minus1205872 1205872) that is minus1205872 lt 1205931lt 1205932lt 1205872

with 1205931+ 1205932gt 0 This implies that 120593 isin (0 1205872) and 120574 lt

1205872 minus 120593

Now we can prove the following minimization theorem

Theorem9 Theminimum variation of themapping factor (4)is attained at the latitude 120593opt defined by

119899 = sin120593opt

=

ln cos 1205931minus ln cos 120593

2

ln tan (1205874 minus 12059312) minus ln tan (1205874 minus 120593

22)

(29)

For this 119899 the variation coefficient is expressed as follows

120572 =

cos120593opt

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 120593opt2))

sin120593opt

=

cos120593opt

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 120593opt2))

sin120593opt

(30)

Proof First we show that for any fixed 1205930

isin (0 1205872) thepositive function

119898(120593 1205930) =

cos1205930

cos120593(

tan (1205874 minus 1205932)

tan (1205874 minus 12059302)

)

sin1205930

120593 isin [1205931 1205932] 120593

1+ 1205932gt 0

(31)

has the absoluteminimum value equal to 1 at the point 1205930and

the absolute maximum value at least at one of the end pointsof the interval [120593

1 1205932]

To this end let us consider the auxiliary function

119891 (120593) =

2

cos120593(tan(

120587

4

minus

120593

2

))

119899

= (tan(

120587

4

minus

120593

2

))

119899minus1

sdot (tan2 (120587

4

minus

120593

2

) + 1)

120593 isin [1205931 1205932] 119899 = sin120593

0isin (0 1)

(32)

Changing the independent variable by the formula 119909 =

tan(1205874 minus 1205932) we can rewrite (32) as follows

119891 (119909) = 119909119899+1

+ 119909119899minus1

119909 isin [1199091 1199092] 119899 isin (0 1)

1199091= tan(

120587

4

minus

1205932

2

) 1199092= tan(

120587

4

minus

1205931

2

)

(33)

By Lemma 1 the function (33) attains the absolute minimumvalue at the point

119909min = radic1 minus 119899

1 + 119899

(34)

and the absolute maximum value at one or both of the endpoints of the interval [119909

1 1199092] This means that the absolute

minimum point of the function (32) is 120593min = 1205930 because

tan(

120587

4

minus

120593min2

) = radic1 minus 119899

1 + 119899

= tan(

120587

4

minus

1205930

2

) (35)

and the absolute maximum point is 1205931or 1205932 Therefore

the same result is true for the original function (31) and

The Scientific World Journal 5

10 degree radius20 degree radius40 degree radius

10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18

Central latitude (deg)

Diff

eren

ce b

etw

een

latitu

des (

deg)

Figure 1 Differences between the optimal tangent latitude andcenter latitude for different domains

substituting 1205930instead of 120593 in this function we obtain that

119898min(1205930) = 119898(1205930 1205930) = 1 Hence

120572 (1205930) = 119898max (1205930) = max 119898 (120593

1 1205930) 119898 (120593

2 1205930) (36)

Now we should minimize the function (36) with respectto the parameter 120593

0 Obviously the solution of this problem

is found from the condition119898(1205931 1205930) = 119898(120593

2 1205930) that is

cos1205930

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 12059302)

)

sin1205930

=

cos1205930

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 12059302)

)

sin1205930

(37)

Simplifying this equation and solving with respect to 119899 =

sin1205930 we obtain formula (29)

Remark 10 Note that formula (29) defines the values of theoptimal tangent latitude in the interval (120593 120593

2)The difference

between 120593opt and 120593 increases with approximation to theNorth Pole and with increase of the radius 120574 In Figure 1 thesedifferences are shown as functions of the centerpoint 120593 fordifferent values of 120574

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by the Brazilian Science Foun-dation FAPERGS

References

[1] A Bourchtein L Bourchtein and E R Oliveira ldquoGeneralapproach to conformal mappings used in atmospheric model-ingrdquo Applied Numerical Mathematics vol 47 no 3-4 pp 305ndash324 2003[2] A Staniforth ldquoRegional modeling a theoretical discussionrdquoMeteorology and Atmospheric Physics vol 63 no 1-2 pp 15ndash291997[3] D L Williamson Difference ApproximAtions for numericalweAther prediction over a sphere vol 2 of GARP No 17 WMO-ICSU 1979[4] A Bourchtein and L Bourchtein ldquoSome problems of confor-mal mappings of spherical domainsrdquo Zeitschrift fur AngewandteMathematik und Physik vol 58 no 6 pp 926ndash939 2007[5] H R Glahn Characteristics of Map Projections and Implica-tions for AWIPS-90 vol 88 of TDL Office Note No 5 NationalWeather Service NOAA 1988[6] P Knupp and S Steinberg Fundamentals of Grid GenerationCRC Press 1993[7] L Bourchtein and A Bourchtein ldquoOn grid generation fornumerical models of geophysical fluid dynamicsrdquo Journal ofComputational andAppliedMathematics vol 218 no 2 pp 317ndash328 2008[8] L Bourchtein and A Bourchtein ldquoComparison of differentspatial grids for numerical schemes of geophysical fluid dynam-icsrdquo Journal of Computational and Applied Mathematics vol227 no 1 pp 161ndash170 2009[9] L M Bugayevskiy and J P Snyder Map Projections AReference Manual CRC Press 1995[10] F Pearson II Map Projections Theory and ApplicationsCRC Press 1990[11] F Richardus and R K AdlerMap Projections for GeodesistsCartographers and Geographers North-Holland AmsterdamThe Netherlands 1972[12] J P Snyder Flattening the Earth TwoThousandYears ofMapProjections The University of Chicago Press 1997[13] V Artale S Calmanti A Carillo et al ldquoAn atmosphere-ocean regional climate model for the Mediterranean areaassessment of a present climate simulationrdquo Climate Dynamicsvol 35 no 5 pp 721ndash740 2010[14] R Benoit M Desgagne P Pellerin S Pellerin Y Chartierand S Desjardins ldquoThe Canadian MC2 a semi-Lagrangiansemi-implicit wideband atmospheric model suited for finescaleprocess studies and simulationrdquo Monthly Weather Review vol125 no 10 pp 2382ndash2415 1997[15] G A Grell J Dudhia and D R Stauffer ldquoA descriptionof the fifth generation Penn StateNCAR mesoscale model(MM5)rdquo NCAR Tech Note NCARTN-398+STR 1994[16] E-S Im E Coppola F Giorgi and X Bi ldquoValidationof a high-resolution regional climate model for the Alpineregion and effects of a subgrid-scale topography and land userepresentationrdquo Journal of Climate vol 23 no 7 pp 1854ndash18732010[17] A Kann C Wittman Y Wang and X Ma ldquoCalibrating 2-m temperature of limited-area ensemble forecasts using high-resolution analysisrdquo Monthly Weather Review vol 137 no 10pp 3373ndash3387 2009[18] X Liang -ZMXu X Yuan et al ldquoRegional climate-weatherresearch and forecasting modelrdquo Bulletin of the AmericanMeteorological Society vol 93 no 9 pp 1363ndash1387 2012

6 The Scientific World Journal

[19] R Mei G Wang and H Gu ldquoSummer land-atmospherecoupling strength over the United States results from theregional climate model RegCM4-CLM35rdquo Journal of Hydrom-eteorology vol 14 no 3 pp 946ndash962 2013[20] K Yessad ldquoBasics about ARPEGEIFS ALADINand AROME in the cycle 40 of ARPEGEIFSrdquo Meteo-FranceCNRM Technical Notes 2013[21] W J Saucier Principles of Meteorological Analysis Dover2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

10 degree radius20 degree radius40 degree radius

10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18

Central latitude (deg)

Diff

eren

ce b

etw

een

latitu

des (

deg)

Figure 1 Differences between the optimal tangent latitude andcenter latitude for different domains

substituting 1205930instead of 120593 in this function we obtain that

119898min(1205930) = 119898(1205930 1205930) = 1 Hence

120572 (1205930) = 119898max (1205930) = max 119898 (120593

1 1205930) 119898 (120593

2 1205930) (36)

Now we should minimize the function (36) with respectto the parameter 120593

0 Obviously the solution of this problem

is found from the condition119898(1205931 1205930) = 119898(120593

2 1205930) that is

cos1205930

cos1205932

(

tan (1205874 minus 12059322)

tan (1205874 minus 12059302)

)

sin1205930

=

cos1205930

cos1205931

(

tan (1205874 minus 12059312)

tan (1205874 minus 12059302)

)

sin1205930

(37)

Simplifying this equation and solving with respect to 119899 =

sin1205930 we obtain formula (29)

Remark 10 Note that formula (29) defines the values of theoptimal tangent latitude in the interval (120593 120593

2)The difference

between 120593opt and 120593 increases with approximation to theNorth Pole and with increase of the radius 120574 In Figure 1 thesedifferences are shown as functions of the centerpoint 120593 fordifferent values of 120574

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by the Brazilian Science Foun-dation FAPERGS

References

[1] A Bourchtein L Bourchtein and E R Oliveira ldquoGeneralapproach to conformal mappings used in atmospheric model-ingrdquo Applied Numerical Mathematics vol 47 no 3-4 pp 305ndash324 2003[2] A Staniforth ldquoRegional modeling a theoretical discussionrdquoMeteorology and Atmospheric Physics vol 63 no 1-2 pp 15ndash291997[3] D L Williamson Difference ApproximAtions for numericalweAther prediction over a sphere vol 2 of GARP No 17 WMO-ICSU 1979[4] A Bourchtein and L Bourchtein ldquoSome problems of confor-mal mappings of spherical domainsrdquo Zeitschrift fur AngewandteMathematik und Physik vol 58 no 6 pp 926ndash939 2007[5] H R Glahn Characteristics of Map Projections and Implica-tions for AWIPS-90 vol 88 of TDL Office Note No 5 NationalWeather Service NOAA 1988[6] P Knupp and S Steinberg Fundamentals of Grid GenerationCRC Press 1993[7] L Bourchtein and A Bourchtein ldquoOn grid generation fornumerical models of geophysical fluid dynamicsrdquo Journal ofComputational andAppliedMathematics vol 218 no 2 pp 317ndash328 2008[8] L Bourchtein and A Bourchtein ldquoComparison of differentspatial grids for numerical schemes of geophysical fluid dynam-icsrdquo Journal of Computational and Applied Mathematics vol227 no 1 pp 161ndash170 2009[9] L M Bugayevskiy and J P Snyder Map Projections AReference Manual CRC Press 1995[10] F Pearson II Map Projections Theory and ApplicationsCRC Press 1990[11] F Richardus and R K AdlerMap Projections for GeodesistsCartographers and Geographers North-Holland AmsterdamThe Netherlands 1972[12] J P Snyder Flattening the Earth TwoThousandYears ofMapProjections The University of Chicago Press 1997[13] V Artale S Calmanti A Carillo et al ldquoAn atmosphere-ocean regional climate model for the Mediterranean areaassessment of a present climate simulationrdquo Climate Dynamicsvol 35 no 5 pp 721ndash740 2010[14] R Benoit M Desgagne P Pellerin S Pellerin Y Chartierand S Desjardins ldquoThe Canadian MC2 a semi-Lagrangiansemi-implicit wideband atmospheric model suited for finescaleprocess studies and simulationrdquo Monthly Weather Review vol125 no 10 pp 2382ndash2415 1997[15] G A Grell J Dudhia and D R Stauffer ldquoA descriptionof the fifth generation Penn StateNCAR mesoscale model(MM5)rdquo NCAR Tech Note NCARTN-398+STR 1994[16] E-S Im E Coppola F Giorgi and X Bi ldquoValidationof a high-resolution regional climate model for the Alpineregion and effects of a subgrid-scale topography and land userepresentationrdquo Journal of Climate vol 23 no 7 pp 1854ndash18732010[17] A Kann C Wittman Y Wang and X Ma ldquoCalibrating 2-m temperature of limited-area ensemble forecasts using high-resolution analysisrdquo Monthly Weather Review vol 137 no 10pp 3373ndash3387 2009[18] X Liang -ZMXu X Yuan et al ldquoRegional climate-weatherresearch and forecasting modelrdquo Bulletin of the AmericanMeteorological Society vol 93 no 9 pp 1363ndash1387 2012

6 The Scientific World Journal

[19] R Mei G Wang and H Gu ldquoSummer land-atmospherecoupling strength over the United States results from theregional climate model RegCM4-CLM35rdquo Journal of Hydrom-eteorology vol 14 no 3 pp 946ndash962 2013[20] K Yessad ldquoBasics about ARPEGEIFS ALADINand AROME in the cycle 40 of ARPEGEIFSrdquo Meteo-FranceCNRM Technical Notes 2013[21] W J Saucier Principles of Meteorological Analysis Dover2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

[19] R Mei G Wang and H Gu ldquoSummer land-atmospherecoupling strength over the United States results from theregional climate model RegCM4-CLM35rdquo Journal of Hydrom-eteorology vol 14 no 3 pp 946ndash962 2013[20] K Yessad ldquoBasics about ARPEGEIFS ALADINand AROME in the cycle 40 of ARPEGEIFSrdquo Meteo-FranceCNRM Technical Notes 2013[21] W J Saucier Principles of Meteorological Analysis Dover2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of