ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume III-4, 2016 XXIII ISPRS Congress, 12–19 July 2016, Prague, Czech Republic
NONZONAL EXPRESSIONS OF GAUSS- KRÜGER PROJECTION IN POLAR REGIONS
Zhongmei Li, Shaofeng Bian*, Qiang Liu, Houpu Li, Cheng Chen, Yanfeng Hu
Dept. of Navigation, Naval University of Engineering, Wuhan, 430033 China - [email protected]
Commission Ⅳ, WG Ⅳ/2
KEY WORDS: Gauss-krüger projection, transverse Mercator projection, polar surveying, reference frame, conformal colatitude, computer algebra system
ABSTRACT:
With conformal colatitude introduced, based on the mathematical relationship between exponential and logarithmic functions by complex numbers, strict equation of complex conformal colatitude is derived, and then theoretically strict nonzonal expressions of Gauss projection in polar regions are carried out. By means of the computer algebra system, correctness of these expressions is verified, and sketches of Gauss-krüger projection without bandwidth restriction in polar regions are charted. In the Arctic or Antarctic region, graticule of nonzonal Gauss projection complies with people’s reading habit and reflects real ground-object distribution. Achievements in this paper could perfect mathematical basis of Gauss projection and provide reference frame for polar surveying and photogrammetry.
1. INTRODUCTION high-precision test set based on Lee (1976) and discussed
properties of the exact mapping far from the central meridian. Polar regions have increasingly been the international focus in Obviously, researches on TM or Gauss-Krüger projection have recent decades. It is of great significance for polar navigation already obtained brilliant achievement. and scientific investigation to select the suitable projection In the development history of Gauss-krüger projection method. As one common conformal projection, the transverse theories, formulae above have different features, for example, Mercator (TM) or Gauss-Krüger projection is frequently used real power series expansions of longitude difference are for charting topographic map (e.g., Lauf 1983; Snyder 1987; l often limited in a narrow strip (e.g. l ∈− [ 3.5DD , + 3.5 ] in UTM Yang 2000). Series expansions of meridian length in Krüger projection; l ∈−[ 3DD , + 3 ] or l ∈− [ 1.5DD , + 1.5 ] in Gauss- (1912) is the basis of the most common way for calculation of Krüger projection). Expressions by complex numbers, Gauss coordinates. In the last century, scholars have carried on eliminating zoning restrictions, are difficult to be used in polar extensive researches about the projection. Lee (1976) and regions with the singularity of isometric latitude. Karney (2011) Dozier (1980) carried out formulae of UTM coordinates by improved Lee’s formulae, and provided an accuracy of 9 nm means of elliptic functions. Based on complex numbers, over the entire ellipsoid, bur not gave formulae that can entirely Bowring (1990) gave one improved solution for TM projection. express the Arctic or Antarctic region. In attempts on the With respect to Laplace-Beltrami and Korn-Lichtenstein nonzonal formulae of Gauss projection in polar regions, Bian equations, conformal coordinates of type UTM or Gauss- (2014) used a near-spherical assumption to derive complex Krüger were carried out directly in Grafarend (2006). colatitude, which would have an influence on the strictness of Additionally, Bermejo (2009) derived simple and highly his formulae. Given these, in order to perfect the mathematical accurate formulas of TM coordinates from longitude and foundations of Gauss-krüger projection specialized in polar isometric latitude, and compared truncation errors in different regions, by introducing the relationship between conformal orders by using the program Maple and Matlab. Karney (2011) colatitude and isometric latitude, an improvement measure will extended Krüger’s series to 8th order, constructed
______* Corresponding author
This contribution has been peer-reviewed. The double-blind peer-review was conducted on the basis of the full paper. doi:10.5194/isprsannals-III-4-11-2016 11 ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume III-4, 2016 XXIII ISPRS Congress, 12–19 July 2016, Prague, Czech Republic
be shown in this paper. where l indicates geodetic longitude, q means isometric latitude and is a function of geodetic latitude B 2. EXPRESSIONS OF GAUSS COORDINATES IN qB( ) =−arctanh( sinB) e arctanh( e sin B) NONPOLAR REGION e 1+− sinBeB⎛⎞ 1 sin (4) = ln ⎜⎟ 1−+ sinBeB⎝⎠ 1 sin According to Bian (2012), based on meridian arc length expansion about conformal latitude, non-iterative expressions of Gauss projection are written as in this form
⎧=φ arcsin() tanh w ⎪ ⎨z =+xiya =(αα02φφφ + sin 2 + α 4 sin 4 + (1) ⎪ sin 6φφ sin8 sin10 φ" ) ⎩ ααα6810++
where a indicates semi-major axis of the earth ellipsoid.
10 Coefficients α02,αα" 10 expanded to e are carried out by computer algebra system Mathematica. With its strong power in Figure 1. Sketch of isometric latitude q with symbolic operation, coefficients expanded to e20 or even geodetic latitude B ∈ [0DD , 90 ) e40 can be easily gotten in a similar way. And finally all the Taking arctanh(eB sin(−=−)) arctanh( eB sin ) and coefficients could be simplified as series summations of earth arctanh( sin()−=−B ) arctanh() sin B into account, qB()−=− qB( ) ellipsoid eccentricity. As our target in this paper is to improve is gotten from Eq. (4), meaning that isometric latitude q is an original formulae of Gauss projection for polar using, we place odd function of geodetic latitude B . Trend of isometric latitude great importance on the improvement measures, and do not q with geodetic latitude B ranging from 0D to 90D is discuss the expansion coefficients whether they are expanded charted in Figure 1. enough to a high precision or not. Here we take the first As shown in Fig. 1, isometric latitude q increases with eccentricity e for example, coefficients α "α expanded to 010 geodetic latitude B ranging from 0D to 90D . In e10 are listed in Eq. (2). consideration of isometric latitude q ’s odevity, q becomes
an infinitely large quantity as geodetic latitude B ⎧ 1 3 5 175 441 D α =1 −−ee24 − e 6 − e 8 − e 10 approaches to ±90 , which brings about singularity in ⎪ 0 4 64 256 16384 65536 ⎪ expressions of complex isometric latitude w in Eq. (3) as 1 1 9 901 16381 ⎪α = ee24−− e 6 − e 8 − e 10 ⎪ 2 8 96 1024 184320 5898240 well as complex conformal latitude φ in Eq. (1), and then ⎪ 13 17 311 18931 ⎪α = ee46+− e 8 − e 10 makes expressions of Gauss coordinates in Eq. (1) difficult to ⎪ 4 768 5120 737280 20643840 ⎨ (2) be used in polar zones. 61 899 14977 ⎪α = ee68++ e 10 ⎪ 6 15360 430080 27525120 ⎪ 49561 175087 ⎪α = e81+ e 0 ⎪ 8 41287680 165150720 ⎪ 34729 ⎪α = e10 ⎩ 10 82575360
Additionally, in Eq. (1) φ and w indicates complex conformal latitude and complex isometric latitude respectively, and
Figure 2. Sketch of Gauss projection in nonpolar region w =+qil (3) Moreover, as tanhw = tanh()qil+ in Eq. (1) contains
tanh(il) = i tan l , not suit for the situation where geodetic
This contribution has been peer-reviewed. The double-blind peer-review was conducted on the basis of the full paper. doi:10.5194/isprsannals-III-4-11-2016 12 ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume III-4, 2016 XXIII ISPRS Congress, 12–19 July 2016, Prague, Czech Republic
longitude l approaches to 90D , expressions of Gauss Obviously, singularity of Eq. (8) depends on the singularity coordinates in nonpolar region Eq. (1) only can be used in of the unique variable θ . To judge the singularity of θ , the zone DBll={() , :<< 90,DD B 90}. By means of computer inserting Eq. (6) into Eq. (7), equation of conformal colatitude algebra system Mathematica, sketch of Gauss projection in θ is gotten.
nonpolar region is drawn in Figure 2. e 2 ⎡ ⎛⎞⎛⎞π BeB1sin− ⎤ θπ=−2arctan⎢ tan⎜⎟⎜⎟ + ⎥ ⎣⎢ ⎝⎠⎝⎠421sin+ eB⎦⎥ 3. NONSINGULAR EXPRESSIONS OF GAUSS = π −⎡⎤=⎡⎤2arctan⎣ exp()qq⎦⎣⎦ 2arccot exp() (9) COORDINATES IN POLAR REGIONS =⎡−⎤2arctan⎣⎦ exp()q
In order to carry out the expressions of Gauss projection that Taking Eq. (4) into account, based on the relationship can be used in polar regions, Eq. (1) must be transformed between exponential and logarithmic functions exp( ln x) ≡ x , equivalently. As Eq. (1) is derived from meridian arc length Eq. (10) is gotten. expansion e ⎡ 1+− sinBeB⎛⎞ 1 sin ⎤ exp()−=q exp⎢ − ln ⎜⎟⎥ ⎢ 1−+ sinBeB⎝⎠ 1 sin ⎥ ⎣ ⎦ (10) Xa=+(α02ϕα sin 2 ϕα + 4 sin 4 ϕ + (5) e αϕαϕαϕsin 6++ sin8 sin10 +" ) 1sin1−+BeB⎛⎞ sin 6810 = ⎜⎟ 1sin1+−BeB⎝⎠ sin where X indicates meridian arc length, ϕ indicates Inserting Eq. (10) into Eq. (9), we can find when
D conformal latitude, and coefficients α010"α are the same as Eq. B = 90 , θ = 0 . Neither θ nor Eq. (8) is singular in north pole. (1). According to Xiong (1988), conformal latitude ϕ is a As nonpolar solution of Gauss projection is obtained by function of geodetic latitude B developing the relationship between meridian length X and isometric latitude q from real to complex number field, e 2 ⎡⎤⎛⎞⎛⎞π BeB1sin− π ϕ =+2arctan⎢⎥ tan⎜⎟⎜⎟ − (6) expressions used in polar regions can be achieved similarly. ⎣⎦⎢⎥⎝⎠⎝⎠421sin+ eB 2 Firstly, based on the definition of complex function, When the geodetic latitude B is on the northern hemisphere, w = qil+ replaces q in Eq. (9) to realize the extension of the conformal latitude ϕ is a positive value. Otherwise, it is a conformal colatitude θ , and then the complex conformal negative value. colatitude θ is derived.
3.1 Nonsingular Expressions of Gauss Projection in θ =−+2arctan⎣⎡ exp( ()qil)⎦⎤ (11) Complex Form
To eliminate the singularity of conformal latitude ϕ when Secondly, replacing θ in Eq. (8) by θ , and turning geodetic latitude B approaches to 90D in Eqs. (5)―(6), meridian length X in Eq. (8) into complex coordinates conformal colatitude θ is introduced, and values z = x + iy , where the real part x indicates Gauss ordinate and the imaginary part y indicates abscissa, yields to expressions θ =−πϕ2 (7) of Gauss projection. For convenient polar charting, moving zero point of the expressions from the equator to the north pole,
Afterwards, inserting Eq. (7) into Eq. (5), equivalent ordinate is reduced by 14 meridian arc length ( aα0π 2 ) and expression of meridian length X can be written with conformal abscissa remains unchanged. For clear presentation, abscissa colatitude θ . and ordinate after translation are still expressed with y and x . Omitting the derivation, nonsingular expression of Gauss Xa=−(α θα + sin 2 θα − sin 4 θα + sin 6 θ − coordinates by complex numbers in Arctic region is carried out. 02 4 6 (8) αθαθ8100sin8+−+ sin10" )a απ 2
This contribution has been peer-reviewed. The double-blind peer-review was conducted on the basis of the full paper. doi:10.5194/isprsannals-III-4-11-2016 13 ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume III-4, 2016 XXIII ISPRS Congress, 12–19 July 2016, Prague, Czech Republic
z =+xiya =( −αα02θθθ +sin 2 − α 4 sin 4 + Taking relationships sinuiv+= sin u cos iv + cos u sin iv, (12) ( ) ααα6810sin 6θθ−+ sin8 sin10 θ −") siniv = i sinh v and cosiv = cosh v into account, Eq. (11) is separated into real and imaginary parts, and Eq. (15) is gotten. D When longitude l =0 , abscissa y = 0 ， ordinate x
equals to meridian arc length integrating from the pole. ∞ ⎧ n−1 x =−auaαα + −1sin2cosh2nu nv Equality of the central meridian arc is guaranteed. Besides, ⎪ 02∑()n ( ) ( ) ⎪ n=1 (15) ⎨ ∞ transformations above are all elementary operations between n−1 ⎪y=− aαα v + a −1cos2sinh2nu nv ⎪ 02∑()n ( ) ( ) complex functions, which are monodrome and analytic ⎩ n=1 functions in the principle value, keep conformal in the whole transformation processes. It is verified that Eq. (12) satisfy By now, nonzonal expressions of Gauss projection in real Cauchy―Riemann equations, so conformality of Gauss form suit for the Arctic region have been carried out. projection is guaranteed. Actually, taking full advantage of earth’s symmetry, it is no By now, nonzonal solution of Gauss projection that can be need to derive extra expressions of Gauss coordinates for the used in Arctic region have been finished. Antarctic region. As the southern hemisphere is symmetrical to
the northern hemisphere, viewing the south pole as new north 3.2 Nonsingular Expressions of Gauss Projection in Real pole, replacing PBl( , ) on southern hemisphere by Form PBl(− , ) and inserting it into expressions used in Arctic region,
sketch of the Antarctic region in the same perspective as the As complex conformal colatitude θ is a complex variable, to Arctic region is gotten. Sketches of Gauss projection in polar separate θ into real and imaginary parts θ = uiv+ , regions are drawn in Figures 3 and 4. equations expqil++ = exp q cos li sin l and () ( )( ) As shown in Figures 3 and 4, the Arctic and Antarctic q =arctanhsinϕ are introduced. Based on the relationship () regions are completely displayed based on nonzonal expressions between complex function and arctangent function, Eq. (12) is above. Through translating the origin of Gauss coordinates from transformed equivalently. the equator to the pole, Gauss ordinates are negative when
longitude difference l < 90D , while ordinates are positive θ =−+2arctan⎡⎤ exp qil ⎣⎦( ( )) when longitude difference l > 90D . In polar circles, the 1expcossin+−iqlil()( − ) DD D =ln− i meridians l=0 ,±± 90 , 180 are shown as straight lines after 1expcossin−−iqlil()( − ) projected, and are the symmetry axes of Gauss coordinates. ⎛⎞sinh qlcos =−iiln ⎜⎟ + (13) ⎝⎠coshql−− sin cosh ql sin =−arctan()() cschqli cos arctanh sech ql sin =−arctan() tanθθ cosli arctanh () sin sinl
After transformed, complex conformal colatitude θ is devided into real and imaginary part, they equals
⎪⎧ul= arctan( tanθ cos ) ⎨ (14) ⎩⎪vl=−arctanh() sinθ sin
Obviously, when PBl( , ) approaches to the north pole, DD θ → 0 , range of l reaches to [− 180 ,180 ] , and θ has a Figure 3. Sketch of Gauss projection in Arctic region specific value and not singular at certain point PBl( , ) on the based on nonzonal expressions northern hemisphere.
This contribution has been peer-reviewed. The double-blind peer-review was conducted on the basis of the full paper. doi:10.5194/isprsannals-III-4-11-2016 14 ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume III-4, 2016 XXIII ISPRS Congress, 12–19 July 2016, Prague, Czech Republic
Program (No. 2012CB719902) and National Natural Science Foundation of China (No. 41471387, 41571441).
REFERENCES
Bermejo, M., Otero, J., 2009. Simple and highly accurate formulas for the computation of Transverse Mercator coordinates from longitude and isometric latitude. Journal of Geodesy, 83, pp. 1-12. Bian, S. F., Li, H. P., 2012. Mathematical analysis in cartography by means of computer algebra system. In: Cartography-A Tool for Spatial Analysis, InTech, Croatia, Chapter 1, pp. 1-24. Figure 4. Sketch of Gauss projection in Antarctic region Bian, S. F., Li, Z. M., Li, H. P., 2014. The non-singular formula based on nonzonal expressions of Gauss Projection in polar regions by complex numbers. Acta Geodaetica et Cartographica Sinica, 43(4), pp. 348-352. 4. CONCLUSIONS Bowring, B. R., 1990. The transverse Mercator projection―a solution by complex numbers. Survey Review, 30(237), pp. With equations of conformal colatitude and isometric latitude 325-342. introduced, based on relationship between complex exponential Dozier, J., 1980. Improved algorithm for calculation of UTM and logarithmic functions, nonzonal expressions of Gauss coordinates and geodetic coordinates. NOAA Technical Report projection in polar regions are carried out. Conclusions are NESS 81, National Oceanic and Atmospheric Administration, drawn as below. Washington. (1) With isometric latitude singular in the pole, traditional Grafarend, E., Krumm, F., 2006. Map Projections. Springer, expressions of Gauss projection can not be used in polar Berlin. regions. Through translating the origin of traditional Gauss Karney, C. F. F., 2011. Transverse Mercator with an accuracy of projection by 14meridian length from the equator to the pole, a few nanometers. Journal of Geodesy, 85(8), pp. 475-485. theoretically strict expressions that can be used in polar Krüger, L., 1912. Konforme Abbildung des Erdellipsoids in der regions are carried out. These are of great significance for Ebene. Drucj und Verlag von B.G. Teubner, Leipzig. perfecting mathematical system of Gauss projection. Lee, L. P., 1976. Conformal Projections Based on Elliptic (2) Compared with traditional Gauss projection, nonzonal Functions (No.16). BV Gutsell. formulae derived in this paper are fit for the whole polar regions Lauf, G. B., 1983. Geodesy and Map Projections. Tafe without bandwidth restriction, and could provide reference Publications, Collingwood. frame for polar surveying and photogrammetry. Snyder, J. P., 1987. Map Projections―A Working Manual. US (3) Though parallel circles and the other meridians are not Geol Surv Prof Paper 1395, Washington D C. projected to straight lines like Mercator projection, graticule of Yang, Q. H., Snyder, J. P., Tobler, W. R., 2000. Map Projection Gauss projection in the Arctic or Antarctic region could still Transformation: Principles and Applications. Taylor & Francis, comply with our reading habit, even reflect real ground-object London. distribution better and make the polar regions absolutely clear at a glance.
ACKNOWLEDGES Revised April 2016
This work is financially supported by National 973 Plan
This contribution has been peer-reviewed. The double-blind peer-review was conducted on the basis of the full paper. doi:10.5194/isprsannals-III-4-11-2016 15