Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 T Bern-
The KavraiskiyThe VII projection exaggerates the These two projections most closely fulfilled the fulfilled closely most projections two These NaturalThe Earth projection of is anamalgam Creating the Natural Earth projection required was slightlyincreased to give it more height. with outwardwide graticule bulging and sides width ratio closeto0.5,resulting in aslightlytoo with additional enhancements (Figure 1). Secondly, usingthe Kavraiskiy VIIasa template, ersnain f plr ein. h Robinson The regions. polar of representation requirement for representing small-scale physical the parallels were slightly increased in length. extension vertical its projection, Robinson the three major adjustments:Firstly, starting from too much shapedistortionnearthemapedges. h Kvasi VI n Rbno projections, Robinson and VII Kavraiskiy the undesirable characteristic (Jenny et al. 2008). data on world maps, but each hadat least one ieo highlatitude areas, resulting in oversized size of projection, on the other hand, hasaheight-to- Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 X =R Y =R∙sd n hrl,telnt fthe pole lines was thirdly,And thelength of Vol. 38,No. 4 364 Table 1 provides the parameter values (Jennyet any projection of the graticule shapeof The 2008). Allthree projections exaggerate the size Equation approach 1andthe associated graphical Earth projection,two parameter sets are usedfor Natural spherical Earth projection, transforming where: when he developed his eponymousprojection otedsg fsmall-scale mapprojections to thedesignof to other well known projections (Jenny et al. trial-and-error experimentation and visual the Natural Earth projection in thisway required the equator. Equation 1 defines the original the defines 1 Equation equator. the parameters. For the Natural tabular sets of but itsdistortion characteristics are comparable a compromise projection, the Natural Earth and straight parallels (Snyder 1993:189). As an iterative process(Jenny resultet al.2008).The continentsin theappearance of of assessment at polelinesarounded appearance. Designing al. 2008;2010): parallels from and (2)therelative distanceof fhighlatitude areas A (Figure 1). Appendix of thisprocedure, the Natural Earth projection, of is a true pseudocylindrical projection, i.e., a coordinates into Cartesian by defined is Projector Flex with designed theNatural Earthprojection. characteristics of decreased by asmallamounttogive the corners pcfig()terltv egho the parallels, (1) specifying the relative length of Copyright (c) Cartography and Geographic Information Society. All rights reserved.providesfurther details about the distortion norequal area,projection neither is conformal projection with regularly distributed meridians X andYare projected coordinates; i h aiso thegenerating globe; R istheradiusof λ isthelongitudewith s lᵩ d =05 stehih-owdhrtoo the k =0.52istheheight-to-widthratio of rhrH oisnpooe h tutr fArthur H. Robinson proposed the structure of = 0.8707 is an internal scalefactor; =0.8707isaninternal ᵩ i h eaielnt ftheparallelat latitude istherelative lengthof i h eaiedsac ftheparallelat latitude istherelative distanceof φ, with∈[- φ from theequator, withφ∈[- with n h lp fand theslopeof the equator; projection. ∙slᵩλ d ᵩ
= ±1for thepolelines, and ᵩ ∙kπ π/2, /2],
lᵩ is63.883°at thepoles; λ ∈[- d lᵩ ∈[0,1], ᵩ ∈[-1,1], X/Y coordinates, and lᵩ π, π];and =1for theequator π/2, /2]and d ᵩ =0for (Eq. 1) TableParameters1. Relaprojection: Earth Natural for the Earth projection ingeospatial software. Seeking aus o te aua prmtr ta define that parameters tabular the for values Robinson (Robinson 1974). In making the Natural Earth Natural the making In 1974). (Robinson equator forevery5degrees(afterJennyetal.2008). tive lengths of parallels and relative distance from the distance relative and of parallels lengths tive and prahfrtedsg ftheir pseudocylindrical approach for the design of align with the five-degree grid, values for values grid, five-degree the with align nomial expressions. the Natural the widespread implementation of application usesapiece-wise cubic spline approximates Equation 1 withtwo simplepoly- that are parameters—factors likely to impede of d intermediate spherical coordinates thatintermediate do not greater efficiency, the remainder of this paper of remainder the efficiency, greater intricate to program and requires a large number israpidtoevaluate,interpolation relatively itis the spline witheach piece of interpolation, uv oeigfv ere.Wieti yeo of type this While degrees. five covering curve discusses a discusses compact analytical expression that projection, Jennyet al.(2010)provide numerical projections. Both defined their projection by projection their defined Both projections. need to be interpolated. The Flex Projector Flex The ᵩ need to beinterpolated. [degrees] Latitude 10 70 30 40 15 75 20 35 45 90 50 60 25 80 55 65 85 d 0 5 Analytical Expressions forthe in Equation 1 for every five degrees. degrees. For five every for 1 Equation in ᵩ andPattersonused anidentical Robinson Projection Relative lengthof parallels 0.9570 0.7160 0.5630 0.6270 0.9894 0.7874 0.6754 0.9953 0.9703 0.9409 0.8763 0.7525 0.9006 0.8196 0.8492 0.9222 0.9811 0.988 1 parallels fromequator Relative distanceof 0.9394 0.4958 0.6769 0.7903 0.8435 0.7346 0.6176 0.8936 0.5571 0.9761 0.310 0.124 0.248 0.434 0.186 0.062 0.372 1 0 lᵩ and lᵩ - Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 365 Cartography and Geographic Information Science Information andGeographic 365 Cartography Patterson fine-tuning it in Flex Projector. the Flex In in it fine-tuning Patterson Polynomial approximationrecommended, is (after Jennyetal.2008). lx rjco) ad ihrsn 18) reports (1989) Richardson and Projector); Flex Robinson projection. Since the two projections two the Since projection. Robinson Robinson projection), and their relatively difficult püe (04 20) rsns mto based method a presents 2005) (2004; Ipbüker approaches both had has projection Robinson rtebur 19) ad vne (08 use (2008) Evenden and (1994), Bretterbauer (1) interpolation and(2) approximation.(1) interpolation The which isapplied to the Natural Earth projection compared to the Kavraiskiy VII and Robinson and VII Kavraiskiy the to compared projections that Robinson applied the Aitken interpolation interpolation Aitken the applied Robinson that are closely related, this sectionreviews existing original projection are an acceptable alternative, dutn te perne f te rjce five- projected the of appearance the adjusting f fnig n nltcl xrsin forthe expression analytical an finding of and the inverse projection. have Others used applied. on multiquadric for the forward interpolation implementation. For thesereasonsthey are not the large methodsis numberinterpolating of in thenext section. interpolating methods for finding continuous finding for methods interpolating ahmtcl oes f Rbno’ projection. Robinson’s of models mathematical expressionsthat do notexactly replicate the explored furtherhere. example, Snyder (1990) appliesthe central- lᵩandd expressionsof degree degree graticule in an iterative process—Robinson cally modeling graphically defined projections: defined graphically modeling cally cubic spline interpolation (whichspline interpolation cubic also usedin is (1991), Ratner Stirling; by formula difference iue 1: projection Earth The polynomialNatural Figure sketching with pen and paper, the graticule and cee iavnaeo thementioned scheme. Adisadvantage of past, various have authors tackled the problem aaees eurd mr ta 4 fr the for 40 than (more required parameters Copyright (c) Cartography andexactlypasses through the reference points. Geographic Information Society. All rights reserved. Approximating curves with parametric Two general approaches exist for mathemati Interpolating methods use a function that methodsuseafunction Interpolating Kavraiskiy VII ᵩ inEquation 1.For Natural Earth - X =R Y =R∙(A This approach uses a total of eight parameters eight parameters approach This a total usesof Equation 1 are with integrated Decleir, aswell asBeineke, useasmaller number Decleir. oisn rjcin Euto 2. o the For 2). (Equation projection Robinson (1991; 1995). For where: with areal number exponent (Beineke 1991). with even uptotheorder,sixth degrees andfor and simpletocompute. five.order three the contains Each to expression number exponent slow.is Atest with the Java ht enk’ epnnil prxmto is approximation exponential Beineke’s that approximateRobinson’sHowever,to projection. a polynomial, sucha polynomial, the one byas and Canters fparameters, andare considerably simpler of d fdeviations to theapproximated values are if four,and for the Ycoordinates oddpowersup coefficients, and the constants the and coefficients, coordinates they useeven powers upto the order more than ten times slower to evaluate than evaluating anexponential function withareal solution contains onlysixparameters, and is fast small. Canters andDecleir (1989) presentsmall. Canters two polynomial equationspolynomial for approximating the programming language, for example, shows ᵩ heproposesan exponential approximation X andYare projected coordinates; A A A A A A φ andλare thelatitude andlongitude; i h aiso thegenerating globe; R istheradiusof A sim The approximatingThe curves by and Canters 5 4 3 2 1 0 =-0.0129. =-0.0104;and =-0.0013; =-0.1450; =0.9642; =0.8507; ∙ λ(A ilar approach isproposed by Beineke 1 ∙φ+A 0 +A 3 2 ∙φ he suggests a polynomial lᵩ he suggests ∙φ 3 2 +A +A Robinson 5 4 ∙φ ∙φ 5 4 lᵩ andd ) ) , s k, o andπof . Their ᵩ. Their (Eq. 2) X Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 X/Y coordinates to spherical coordinates. Indeed, Vol. 38,No. 4 366 JavaScript programming or other interpreted oyoileutosaebs ntrso Polynomialof equationsare bestinterms Polynomialapproximations, however, sometimes Earth projection. Polynomials of varying degrees varyingdegrees Earth projection. Polynomials of In a trial-and-error process, a polynomial were used to evaluate variants developed with rounded corners with adequate smoothness.rounded The corners number of polynomial terms and the number polynomial terms number of the method using computed coefficients their terms was determined for theoriginalNatural determined was terms to program than the interpolating methods.to program thantheinterpolating too high, which must be avoided for to a graticule the Newton-Raphsonroot findingalgorithm. this iterative trial-and-error procedure: the First, were selected and terms and different number of prxmto ihamnmmnme fapproximation withaminimumnumber of persot.Aohrptnildabc fappear smooth. Anotherpotential drawbackof an analytical inverse does not generally exist for fleast squares with constraints. Two criteria of the equations.also relevant Itis for accelerating the cubic spline interpolation in Flex Projector inFlex the cubic spline interpolation the approximated projection. Differences should using fly the on maps project that applications multiplications required to evaluate the of higher-order terms might be necessary to might benecessary higher-order terms higher-order polynomial equations. To solve for bounding meridians meet bounding thehorizontal pole be minimalthroughout theentire projection. languages that are comparatively slow. The lines. It wasfound tools and that the graphical inverse from equations that projected transform sipratfrsmlfigtepormigo is important forthe simplifyingprogramming of minimize deviationsfrom the original curve. methods are necessary, suchthe asbisectionor equationcriterion need to beminimized. This computation speedandcodesimplicity, but computations, for example, for web mapping do not provide sufficient control for defining for control sufficient provide not do differences between the original projection and ufrfo nuain fthe maximum is degree fromsuffer undulations if spherical coordinates, numerical approximation oyoileutosi h ifclyo finding of difficulty the is equations polynomial second criterion aims at the minimizing absolute smoothness of the rounded corners where the rounded the corners of smoothness projection, special focuswas given to the Copyright (c) Cartography and Geographic Information Society. All rights reserved. A Polynomial Approximation forthe hn einn te rgnl aua Earth Natural original the designing When Natural Earth Projection X =R A Y =R∙(B The polynomials are of higher degrees than those higher degrees polynomials areThe of Due to these characteristics, Equation 3 contains + A + B Equation expression 1withapolynomial (with Earth projection is given in Equations 3 and 4. and 3 Equations in given is projection Earth at rjcin h e oyoilfr fof Earth projection.form newpolynomial The Natural Earth projectionmodel the tosmoothly w and Equation 4 odd only powerconsists of terms the parallels are equallydividedbymeridians. required multiplications and the number of π, andd k, s,the constant factors f ceeaigcmuain,tenme f accelerating computations, the number of of of φthat are multiplied by λ, only even powers of Naturalaccount: (1) The Earth projection is accelerating calculations. the horizontal poleline. by Canters and Decleir (1989) for the Robinsonthe for (1989) Decleir and Canters by projection’simprovingthe distortion for applied result They sensibilities. authors’ the satisfy to changes The tothesmoothness to thecorners. the projection therefore deliberately deviates of the corners were entirely esthetic and done the corners of following considerations were taken into in asubjective improvement that cannot be from the original projection byadding curvature evaluated with objective criteria. Norwere they characteristics. curved corners connecting the meridian lines to curved corners itnusigcaatrsi fthe Natural characteristicdistinguishing of a polynomial approximation development of straight but not equally spaced parallels; and (3) symmetrical about the x andy-axis;(2) it has projection. are Higherdegrees required for the provided theto possibility further improve this 1 here: = s X andYare theprojected coordinates; A φ andλare the latitude and longitude in i h aiso thegenerating globe, and R istheradiusof The polynomialexpressionThe for the Natural Equation 3 replaces both lᵩandthe factor o dvlpn Eutos ad , the 4, and 3 Equations developing For φ (Canters, 2002,p. 133ff.). For the purpose 1 to A Table 2. radians; ). The polynomials in Equation 4 include 4 Equation in polynomials The ). ∙ λ(A 5 5 ∙ φ ∙ ∙ φ ∙ 5 1 andB ∙φ+B 11 12 ) (Eq. 4) (Eq. ) ) (Eq. 3) 1 +A 1 to 2 2 ∙φ ∙φ B 5 2 3 are coefficients given in given coefficients are +A +B 3 3 ∙φ ∙φ 4 7 +B + ᵩ for reducing
A 4 4 ∙ ∙φ
φ 10 9 + + s in Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 4 has no term with degree 5. withdegree 4 hasnoterm odtrieteivreo thepolynomial To the inverse determine of The . Coefficients for the polynomial expression of the of expression polynomial the for Coefficients Table2. Equation 4, an additional constraint was added was constraint additional an 4, Equation from line pole the of distance the 4, Equation Cartesian coordinates intospherical coordinates. aua Erh rjcin Eutos ad 4 and 3 Equations projection, Earth Natural ihtemto fleast squares, two additional with themethodof was forced to equal the value of the internal the internal was forced to equal the value of reader to applythistechnique toother similar near thepoles. Bprovides Appendix details on to the method of least squares, fixing the slope the fixing squares, least of method the to the equator was forced tothe original value by the equator remains the same. In the length of h plcto f h ehdo least squares, the method of the application of between the meridians and the pole lines. For additional constraints, which should allow the Naturalpolynomial a new Earth projection that of the corners involved the corners slightly reducing the of at the polynomial to7degrees the poles. The of has no terms with degree 6 and 8, and withEquationdegree has noterms egho the pole line before computing the length of graticule. In Equation 3, the first coefficient first the 3, Equation In graticule. to the exactgraticule same size as the original increase the smoothness of the rounded corners therounded corners increase of the smoothness introducing asecondconstraint. including the technique for the integrating constraints were added to bringthe polynomial deliberately deviates from the original projection Natural Earthprojection. scale factors second measure for improving the smoothness polynomial terms has been reduced. Equation 3 polynomial terms projections. is result The 2). (Figure coefficients polynomial 367 Cartography and Geographic Information Science Information andGeographic 367 Cartography Copyright (c) Cartography and Geographic Information Society. All rights reserved. Two additional measures were required to hn siaig h plnma coefficients polynomial the estimating When inverse of a map projection transforms amapprojection transforms inverse of Coefficients for A A A A A 3 4 2 5 1 Equation 3 Natural Earth Projection Inverting thePolynomial (Equation 1). This is to ensureis (Equation that 1).This -0.001529 -0.131979 -0.013791 0.870700 0.003971 Coefficients for B B B B B 3 4 2 5 1 Equation 4 -0.044475 -0.005916 1.007226 0.028874 0.015085 A 1
φ in Equation 4, and then solving Equation 3 for The system is solved by first finding the latitude the finding first by solved is system The 05. h Nwo-aho agrtm s a is algorithm Newton-Raphson The 2005). valuedused for commonly function, andis the (the Cartesian coordinates Newton-Raphson root finding algorithm was algorithm finding root Newton-Raphson where: x the unknown longitude twohas two polynomials known variables ueia ovn fnonlinear equations. The numerical of solving successively better finding for method numerical h plnma Euto 4 os o eit but exist, not does 4 Equation polynomial the ag ubro methodsare available for a large number of prxmtost h ot rzrso areal- approximationsto theroots orzerosof and requires onlyone initial guess. Equation 5 because it converges rapidly, is easyto compute, algorithm. smoothness at the end of the pole line, which is shortened which of poleline, endthe the smoothness at projection are overlaid in (C). overlaid in projection are in changes Arrows indicate unknowns(the sphericalcoordinates φandλ). in (B). must be inverted. The system defined by these by defined system The inverted. be must chosen for inverting the Natural Earth projection, Figure 2: Figure Earth polynomial (B)Natural The original(A)and hw h eea omo the Newton-Raphson shows the of general form polynomial system solving (Elkadi and Mourrain polynomial system solving(Elkadi and Mourrain n+ F
naayia xrsino teivreo the inverse of analytical An expressionof 1 (x x =x derivative; n n andx ) andF n –F’ n+ (x 1 ’( n are the previous andthe next ) x -1 n B ∙F ) are agiven functionandits (x A n ) F(x λ. X andY)two n ) = 0 = ) A B C (Eq. 5) Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 ∙ R Y ∙ pligteivrepoeto fthe polynomial the inverseApplying projection of Vol. 38,No. 4 368 F The iterativeThe approximationrepeated is until a The function F The +A φ Equation 4toEquation 6: vergence to the solution isquadratic for Equa- (3) The finallatitude:(3) The (1) T (2) With theNewton-Raphsonapproximation (4) The finallongitude: (4) The aua at rjcincnit fthe following Natural Earth projectionof consists Newton-Raphson algorithm, as it is in the range the in is it as algorithm, Newton-Raphson with 15 degrees resolutionwith 15degrees covering the whole aregularlywhen transforming spaced graticule . The quo- φ.The which isoutside the valid range of tion 6, since the tion 6,sincederivative F’ the longitude all φ applied to compute the latitude average, lessthan four iterations are needed tient fof fthe latitude φ,anddoesnot have any local of local minimum ormaximum inthe valid range netn Euto 3 Te Newton-Raphson The 3. Equation inverting method convergeswith Equation quickly 6.On extremum inthisrange(Equation 7). sufficiently accurate solution is reached. Con- is reached. solution accurate sufficiently Copyright (c) Cartography and Geographicsteps: Information Society. All rights reserved. (x )teseso theiterative process. of nand(n+1)thesteps h Nwo-aho mto i only is method Newton-Raphson The n . The closest local extremumφ. The isat φ ε isasufficientlysmallpositive quantity, floating pointarithmetic; the functionfrom Equation 6,F yial ls otemxmmpeiino typically closetothemaximum precision of ) = h trto tp f|φ the iteration stopsif method animproved latitude derivative, m andn=0,1,2,…,m.Atstep ouino thegiven function;and solution of 0 n+1 -1 he initialguessfor theunknown latitude: ∙ R Y ∙ =
∈ ∈ 2 + B = B ∙φ [- [-s ∙ k ∙ π, s ∙ k ∙ π] Y ∙R 1 ∙φ+B φ -1 π/2, 5 2 n can be used as an initial guess for the ∙φ +A –F -1 ; a b cmue i se 4 by 4 step in computed be can λ 11 π/2], andF ’(φ 3 –Y∙R ∙φ (x 2 ∙φ n n ) ) is formed by) is formed converting -1 4 ∙F 3 + φ = +B
λ = A -1 (φ 4 = 0 = φ ∙ n 3 ∈ ), where
m+1 ∙φ m+1 φ X ∙R (x [- 10 n ; and –φ ) has therefore) has no π/2, + A (x 7 +B n (Eq.6) -1 ) is positive) is for φ iscalculated: m ∙(A 5 in step (2); φ instep | <ε,where ∙φ ’(φ F π/2] (Eq. 7) 4 (φ ∙φ φ = ±1.59, n 1 ) its 12 n + ) is ) 9 -1 + 99 f h oyoilapoiaino the approximation thepolynomial of 1989) of NaturalThe Earth projection expressed by the The Newton-Raphson method for inverting the inverting for method Newton-Raphson The o ohmtos neulnme fiterations For bothmethods, anequal number of (Appendix A).For(Appendix these reasons, the authors Newton-Raphson method is faster, as it involvesfaster,it is as method Newton-Raphson Natural Earth projection are identical to those hc mrvstevsa perneo the which improves the visual appearance of eomn sn h oyoileuto frecommend using the polynomial equation of arbitrary map projections arbitrary withoutexplicit the Newton-Raphson approach presented here. presented approach Newton-Raphson the the Natural Earth projection, this method based and Bildirici (2002). They utilize the two forward otoeo other pseudocylindrical projections to those of the areal distortion index,well as themean as as onlyseven multiplications are required for additional curvature to meridians where they the Natural Earthprojection. to allowtechnique toapplythis others tosimilar and maximum angular distortion are similar indexand Decleir, (Canters angular deformation the original Natural Earth projection. The of adjustment with constraints foradjustment withconstraints obtaining the by Projector. Flex Details on the least squares on Jacobian matrices results inthe same values as fthe original projection. areal The distortion of inverse expressionswas described by Ipbüker graticule.enhancement wasdeveloped This in is required (with an identical iterations required. scaledistortionindex, The adding by projection original Patterson’s from fewer calculations, andisalgorithmically simpler. meet the horizontal pole line. curved The expressions to calculate the geographical ahplnma ffactorized appropriately.each polynomialif coordinates collaboration with Tom Patterson, the author are smoother than in the original design, corners designed with the graphical approachdesigned with the graphical offered sphere (with polynomials are easy to code and fast to compute deviates slightly 4 and 3 Equations polynomial projection converges quickly, withonlyafew projection, which is one specific projection specific one is which projection, polynomial expressionsfor the Natural Earth polynomial formulas arepolynomial formulas provided B) (Appendix An alternativeAn general method for inverting hsatcepeet h eeomn farticle presentsThis the development of φ andλusingJacobian matrices. For ε =10 Conclusion -11 ). ε). However, the Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 A 369 Cartography and Geographic Information Science Information andGeographic 369 Cartography Evenden, G. I., 2008. libproj4: A comprehensive This research This funded bywas the Erasmusstudent Elkadi, M.andB.2005.Symbolic- Mourrain. Zürich. The authors thankthe anonymousZürich. authors The Ipbüker, C., and I. Ö. Bildirici. 2002. A general Ipbüker,computational C.2005.A approachto Ipbüker, C. 2004. Numerical evaluation of the of evaluation Numerical 2004. Ipbüker,C. ml-cl mp rjcin design projection Canters, F. map 2002.Small-scale Canters, F., andH. Decleir. world in 1989. The Bretterbauer, K. 1994. Ein Berechnungsverfahren Beineke, D.Zur 1995.KritikundDiskussion: Robinson- zur Untersuchung 1991. D. Beineke, References reviewers for theirvaluable comments. remains tobe explored how the polynomial any Projector. projection designedwithFlex approximation methodcanbegeneralized for exchange out at program,carried andwas ETH Copyright (c) Cartography and Geographic Information Society. All rights reserved. It parameters. tabular by defined projections cknowledgements 12, 2011) Abbildungsvorschrift. Abbildung undVorschlag einer analytischen Nachrichten 44(6):227–9. 45(4): 151–3. 41(3): 85–94. rceig o h Tid nentoa Symposium International Third the of Proceedings Equations: Foundations, Algorithms and Applications, and Foundations,Algorithms Equations: ahmtcl Cmuainl Applications Computational & Mathematical Mathematics Information Science Information 204–17. London: Taylor &Francis. Robinsonprojection. Robinson-Abbildung. Chichester: John Wiley andSons. 68. (preliminarydraft). Online:http://home.comcast. the Robinson projection. numeric methodsforpolynomial solving loih o h nes rnfraino algorithm for the inverse of transformation ovn Polynomial and I.Z.Emiris(eds.), Solving library of cartographic projection cartographic functions of library esetv – drcoy f wrd a projections map world of directory A – perspective ü de Robinson-Projektion. die für map projections using Jacobian matrices. equations andapplications. In:A.Dickenstein, net/~gevenden56/proj/manual.pdf oue 4 f Agrtm ad optto in Computation and Algorithms of 14 volume . Berlin, Springer, Germany: 125– 31(2): 79–88. Cartography and Geographic and Geographic Cartography Kartographische Kartographische Nachrichten Kartographische Kartographische Nachrichten Survey 38(297): Review Survey (accessed April (accessedApril Kartographische Kartographische , 175– . . Jenny, B., andT. Patterson. Projector.2007. Flex Jenny, B., T. Patterson, 2008.Flex and L.Hurni. Jenny, B., T. Patterson, 2010. andL.Hurni. Richardson, R.T. 1989. Area deformation on the the of implementation An 1991. A. Ratner,D. oisn A 17. nw a projection: map new A 1974. A. Robinson, Mikhail, E.M.,andF. 1976. Ackerman. Snyder, J. P. 1993.Flattening the Earth. Two thousand projection: Robinson The 1990. J.P. Snyder, 16: 294–6. er o a Projections Map of Years Observations and least squares . New York:least and Observations Harper & Row, Publishers. International Journal of Geographical Information Information Geographical Journal of International nentoa Yabo o Cartography of Yearbook International Systems Science 24(11):1687–702. Geographic Information Systems Information Geographic Online: rpia eino world map projections. Graphical design of Godesberg, Kirschbaum,145–55. Germany: G.M. Kirschbaum andK.-H.Meine (eds), Projector – interactive software for designing nvriyo ChicagoPress. University of 59: 12–27. Robinson projection.Robinson cubic on based projection map Robinson March 11,2011) Cartographic Perspectives world mapprojections. Cartographic 82. a computation algorithm. its development and characteristics. In: splines. 18:104–8. Cartography and Geographic Information Information and Geographic Cartography http://www.flexprojector.com The American Cartographer The , Chicago, Illinois: 17: 301–5. Cartography and Cartography . Bonn-Bad ; (accessed
Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 As a compromise projection, the Natural Earth Vol. 38,No. 4 370 values fall somewhere between those of the Table 3 were usedinthedesignprocedure. (2008) present isocols of areal distortion for 30 degrees. With increasing distance from the gerated. Figure 3omits indicatrices along pole Earth projection. 3. Tissot’sfor polynomialNatural the indicatrices Figure longitude (Table 3). All isocolsofareal distor of axes The size. infinite of are they since lines, Kavraiskiy VII andRobinsonprojections, which and increases withlatitude and doesnotchangewith but itsdistortion characteristics arecomparable tions atposition tion arethereforeparallel to theequator. Areal the NaturalEarthprojection. Areal distortion tions of parallels and meridians, except at the the indicatrices do notcoincide with the direc to otherwellknownprojections. Its distortion distortion iscomputed with σ = a cating that the size of highlatitude areas is exag- equator and the central meridian. Jenny et al. equator theareaofindicatrices increases,indi- Copyright (c) Cartography and Geographic Information Society. All rights reserved. projection is neither conformal nor equal area, Figure 3shows Tissot’s indicatrices for every b Distortion Characteristicsofthe j thescalefactors alongtheprincipal direc- . Areal distortionincreaseswithlatitude. Latitude φ[̊] Natural EarthProjection 30 60 85 0 j onthesphere. Appendix A: Appendix Area Scale 3.28 1.31 0.98 0.88 j ·b j witha j - -
Table of degrees distortion for30 angularevery 4. Maximum where: (Table 4). Jennyetal. (2008) provideisocolsof a a and increases towards the edges of the graticule imum angulardistortionω maximum angular distortion. The valuesofmax- increasing latitudeandlongitude. All valuesareindegrees. the equator. Equation8computes directions atposition φ j
30 60 Angular distortionismoderate neartheequator 85 ω λ and b 0 j = 2 arcsin 115.37 25.0 j 3.0 8.3
0 are thescalefactorsalongprinciple 115.44 26.2 30 5.4 8.3 a j j +b -b 115.67 29.5 60 9.3 8.3 j jonthesphere.
j
116.05 13.6 34.1 90 8.3 j areconstantalong 116.56 120 17.9 39.6 8.3 ω j : 117.20 150 45.4 22.1 8.3 (Eq.8) 117.96 180 26.3 51.3 8.3 Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 371 Cartography and Geographic Information Science Information andGeographic 371 Cartography 11. The vector11. The φ = φ = This The first constraint(1) first for thelength The 90 degrees is expressed is 90 degrees inasimilarwayas the constraint(1)inEquation 12.For the slope 3— eiaieo Equation 9isusedtoexpress it.Bothconditionsare describedinEquation 13. (3)—a derivative of ag fpossible latitude values withφ∈ range of eut nalna ytmta a esle sn h elkonmto fleast squares inEquation results inalinearthat system canbesolvedthe usingwell knownmethod of a at rjcintrecntanswr moe:()telnt ftheparallel ral Earthprojection three wereconstraints (1) the length imposed:of nomial, and [φ number of parameters (n =37,u parameters = 5). The matrix firstnumber of row of tional constraints. The presented approach indirectis a observationsmodified LSAwith functionally of the polynomial powers of latitude are known coefficients for the LSA. Equation 9 can be in expressed be can 9 Equation LSA. the for coefficients knownare latitude of powers polynomial the ino the model, and the differences between the original andthe approximated graticule. vector The tion of h lp fthe polynomial in Equation 4 is fixed to 7 degrees at the pole line.the slope of All three constraints are allow tofindsimilarpolynomial others expressions for otherprojections. ewe ahpi f oto ons h ymtia ragmn fthe control pointsaround the control points. of symmetrical arrangement The between each pair of be 1; (2) the relative distance between the equator and the parallel unknown polynomial coefficients. In this particular case, no parameters are included in matrix l includes lᵩ and given inEquation 9,which isderived from Equations 1,3,and4: n h hl ag fpossible latitude values between - ing the whole range of denoting the number of rows in the coefficients matrix coefficients rowsthe in of number the denoting (Equation 10),withn matrix form model, noinitialguessisrequired for theparameters. expressed withparameters, which ispossible becausetheexpressions inEquation 9are linear. equator guaranteesacontinuously differentiable function. equations, polynomial expressions with five terms were chosen. The functional model of the LSA is LSA the of model functional The chosen. were terms five with expressions polynomial equations, Copyright (c) Cartography and Geographic Information Society. All rights reserved. 1976).It is hopedthat the details provided (MikhailandAckerman parameters dependent here will A A φ =0 B The constraints (2) and(3)are constraints The applied to the relative distanced The original Natural Earth projection is defined by 37 control points distributed over the complete the distributed over points control 37 by defined is projection Earth Natural original The The coefficients of the two polynomial expressions in Equation 9 are unknown parameters, and unknown parameters, 9 are in Equation expressions twopolynomial the of coefficients The To initiate the constraints, anadditional linear matrix equation is added to this model. For the Natu For computing the polynomial coefficients of the relative length relative the of coefficients polynomial the Forcomputing l + 1 n×u 1 + π/2 dᵩ=1 π/2 B ∙φ appendixpresents the approximation method using least squares adjustment (LSA) with addi- v= ∙ x A j + 2 Least Squares Adjustment for CurveFittingwith Additional Constraints u×1 ∙φ A∙x v B = 2 i 2 ∙φ + lᵩ =1 l 1 1 v represents the minimized residuals after adjustment, for evaluating the standard devia- n×1 +3∙ , j A d 3 φ + (Eq.10) s, k ᵩ, multiplied s, with the constant factors 3 1 ∙φ 3 , B φ B T 3 i 1 4 2 ∙φ ∙ vminx=( 7 + ∙π , B φ A j A 2 1 7 1 /4 +7∙ ∙π/2+ + 9 1 4 , =s ∙φ φ B 1 11 4 i
10 ∙φ ] for the second polynomial.Vector (Eq.12) + B j B 9 3 A + ∙π lᵩ canbederived from Equation 9: 2 5 ∙π ∙φ B 6 /64 +9∙ 5 3 ∙φ i /8 + 12 [- = Appendix B: j π/2, 11 A s ∙l = T B ∙A) 3 B ∙π s ∙kdᵩ ᵩ π/2] and a control point every 5degrees. For both 4 i -1 ∙π (i,j= 1,… 37) ∙ A 7 /128 + 8 /256 +11∙ T ∙ lv=A∙x–(Eq.11) j ∙π π/2 and+ and is [1, φ A is B 4 π as inEquation 9. Since this isa linear ∙π h ie itneo the parallel at ᵩ. The fixed distance of B 9 d 5 lᵩ 37control pointsare used cover /512 + ∙π x contains the parameters, i.e. the ᵩ at must 90 degrees be 1; and(3) /2 with a distance of 5 degrees 5degrees π/2 withadistanceof 1 2 10 , /1024 =tan(7°) φ (Eq.9) 1 4 , B φ 5 ∙π 1 10 , 11 φ lᵩ at must 0degrees /2048 = 1 12 ] for the first poly- (Eq.13) — constraint A, anduthe s ∙kπ A, which - - Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 10 and 14 are solved together. The first row in Equation 17 is a normal LSA of indirect observations. indirect of LSA normal a is 17 Equation in row together.first solved areThe 14 and 10 As EquationAs 9 islinear, noiterations are needed to solve the functional model,and noinitial guesses Vol. 38,No. 4 372 h eut r aaeesnticuigtecntans ntescn o fEquation 17, the correc- resultsThe are not including the constraints. parameters Inthe second row of
with respect invector to all parameters ), as the model would otherwise become under-determined. The first con- first The under-determined. become otherwise would model the as unknowns(p < u), number of tions are required for the unknown parameters. All constraints canbe expressed withfunctionallydependent the two matrixes for thedistanceconstraints. linear constraintscanonlypartiallybefulfilled. cluded in the LSA (Equation 11). This step is presented is the step Equationincluded LSA(Equation 17,where 11).This the twoin systems from thecurve defined by the37control points. could alsobeused.Inthiscase, matrix ofiinso teplnma prxmto.Adfnly h etro residuals of finally, approximation. And vector polynomial the the of coefficients straint for therepresentedlengths is in the matrix additional constraints, which must be less thanthe same asinEquations 10and11.pisthe number of polynomial inx fulfills all additional constraints expressed in Equation 14, and minimizes the deviations parameters. three The constraints for the Natural Earth projection are linear, but non-linear constraints Copyright (c) Cartography and Geographic Information Society. All rights reserved. C = [1 The additional constraints can be written in matrix form (Equation 14), where the vector the where 14), (Equation form matrix in written be can constraints additional The C x = g = C = C ∙x=gMN Equation 14 expressestherelationship between thefunctionally dependent parameters that mustbe l +v=A p×u dx for are the parameters calculated. On the third row, the vector ∙ x 0 π/2 tan(7°) s∙kπ 13∙π +dx u×1 = ∙ xN=A g p ×1 0 0], v=A∙x–l (Eq.14) 3 /8 π 2 /4 7∙π T ∙Ax 7 /128 π 6 /64 9∙π -1 ∙C wudcnanprildrvtvso the constraints equations C would containpartial derivatives of x, calculated from parameter values invector T dx=N 9 8 /512 π g =[ /256 11∙π 0 =N s -1 C andvector ](Eq.15) ∙A -1 ∙C T 11 ∙l(Eq.17) T /2048 (Eq.16) 10 ∙M /1024 -1 ∙(g-Cx g byEquation 15.Equation 16shows x is computed containing the 0 ) v is computed. The iscomputed. The x 0 . However, non- is the x is