Back to Geocentric coordinates. Forward to Support for high precision arithmetic. Up to Contents.

This material is now written up as

C. F. F. Karney,
On auxiliary latitudes,
Technical Report, SRI International, December 2022.
arxiv:2212.05818

An implementation of the methods described in this paper is available in the files examples/AuxLatitude.[hc]pp which provides the classes AuxAngle and AuxLatitude. These classes are not yet part of GeographicLib. The series expansions described in the paper are available in

C. F. F. Karney,
Series expansions for converting between auxiliary latitudes,
December 2022.
doi:10.5281/zenodo.7382666

Go to

Six latitudes are used by GeographicLib:

φ, the (geographic) latitude;
β, the parametric latitude;
θ, the geocentric latitude;
μ, the rectifying latitude;
χ, the conformal latitude;
ξ, the authalic latitude.

The last five of these are called auxiliary latitudes. These quantities are all defined in the Wikipedia article on latitudes. The Ellipsoid class contains methods for converting all of these and the geographic latitude.

In addition there's the isometric latitude, ψ, defined by ψ = gd⁻¹χ = sinh⁻¹ tanχ and χ = gdψ = tan⁻¹ sinhψ. This is not an angle-like variable (for example, it diverges at the poles) and so we don't treat it further here. However conversions between ψ and any of the auxiliary latitudes is easily accomplished via an intermediate conversion to χ.

The relations between φ, β, and θ are all simple elementary functions. The latitudes χ and ξ can be expressed as elementary functions of φ; however, these functions can only be inverted iteratively. The rectifying latitude μ as a function of φ (or β) involves the incomplete elliptic integral of the second kind (which is not an elementary function) and this needs to be inverted iteratively. The Ellipsoid class evaluates all the auxiliary latitudes (and the corresponding inverse relations) in terms of their basic definitions.

An alternative method of evaluating these auxiliary latitudes is in terms of trigonometric series. This offers some advantages:

these series give a uniform way of expressing any latitude in terms of any other latitude;
the evaluation may be faster, particularly if Clenshaw summation is used;
provided that the flattening is sufficiently small, the result may be more accurate (because the round-off errors are smaller).

Here we give the complete matrix of relations between all six latitudes; there are 30 (= 6 × 5) such relations. These expansions complement the work of

O. S. Adams, Latitude developments connected with geodesy and cartography, Spec. Pub. 67 (US Coast and Geodetic Survey, 1921).
P. Osborne, The Mercator Projections (2013), Chap. 5.
S. Orihuela, Funciones de Latitud (2013).

Here, the expansions are in terms of the third flattening n = (a − b)/(a + b). This choice of expansion parameter results in expansions in which half the coefficients vanish for all relations between φ, β, θ, and μ. In addition, the expansions converge for |n| < 1 or b/a ∈ (0, ∞). These expansions were obtained with the the maxima code, auxlat.mac.

Adams (1921) uses the eccentricity squared e² as the expansion parameter, but the resulting series only converge for |e²| < 1 or b/a ∈ (0, √2). In addition, it is shown in Truncation errors, that the errors when the series are truncated are much worse than for the corresponding series in n.

NOTE: The assertions about convergence above are too optimistic. Some of the series do indeed converge for |n| < 1. However others have a smaller radius of convergence. More on this later…

Series approximations for conversions

Here are the relations between φ, β, θ, and μ carried out to 4th order in n:

\[ \begin{align} \beta-\phi&=\textstyle{} -n\sin 2\phi +\frac{1}{2}n^{2}\sin 4\phi -\frac{1}{3}n^{3}\sin 6\phi +\frac{1}{4}n^{4}\sin 8\phi -\ldots\\ \phi-\beta&=\textstyle{} +n\sin 2\beta +\frac{1}{2}n^{2}\sin 4\beta +\frac{1}{3}n^{3}\sin 6\beta +\frac{1}{4}n^{4}\sin 8\beta +\ldots\\ \theta-\phi&=\textstyle{} -\bigl(2n-2n^{3}\bigr)\sin 2\phi +\bigl(2n^{2}-4n^{4}\bigr)\sin 4\phi -\frac{8}{3}n^{3}\sin 6\phi +4n^{4}\sin 8\phi -\ldots\\ \phi-\theta&=\textstyle{} +\bigl(2n-2n^{3}\bigr)\sin 2\theta +\bigl(2n^{2}-4n^{4}\bigr)\sin 4\theta +\frac{8}{3}n^{3}\sin 6\theta +4n^{4}\sin 8\theta +\ldots\\ \theta-\beta&=\textstyle{} -n\sin 2\beta +\frac{1}{2}n^{2}\sin 4\beta -\frac{1}{3}n^{3}\sin 6\beta +\frac{1}{4}n^{4}\sin 8\beta -\ldots\\ \beta-\theta&=\textstyle{} +n\sin 2\theta +\frac{1}{2}n^{2}\sin 4\theta +\frac{1}{3}n^{3}\sin 6\theta +\frac{1}{4}n^{4}\sin 8\theta +\ldots\\ \mu-\phi&=\textstyle{} -\bigl(\frac{3}{2}n-\frac{9}{16}n^{3}\bigr)\sin 2\phi +\bigl(\frac{15}{16}n^{2}-\frac{15}{32}n^{4}\bigr)\sin 4\phi -\frac{35}{48}n^{3}\sin 6\phi +\frac{315}{512}n^{4}\sin 8\phi -\ldots\\ \phi-\mu&=\textstyle{} +\bigl(\frac{3}{2}n-\frac{27}{32}n^{3}\bigr)\sin 2\mu +\bigl(\frac{21}{16}n^{2}-\frac{55}{32}n^{4}\bigr)\sin 4\mu +\frac{151}{96}n^{3}\sin 6\mu +\frac{1097}{512}n^{4}\sin 8\mu +\ldots\\ \mu-\beta&=\textstyle{} -\bigl(\frac{1}{2}n-\frac{3}{16}n^{3}\bigr)\sin 2\beta -\bigl(\frac{1}{16}n^{2}-\frac{1}{32}n^{4}\bigr)\sin 4\beta -\frac{1}{48}n^{3}\sin 6\beta -\frac{5}{512}n^{4}\sin 8\beta -\ldots\\ \beta-\mu&=\textstyle{} +\bigl(\frac{1}{2}n-\frac{9}{32}n^{3}\bigr)\sin 2\mu +\bigl(\frac{5}{16}n^{2}-\frac{37}{96}n^{4}\bigr)\sin 4\mu +\frac{29}{96}n^{3}\sin 6\mu +\frac{539}{1536}n^{4}\sin 8\mu +\ldots\\ \mu-\theta&=\textstyle{} +\bigl(\frac{1}{2}n+\frac{13}{16}n^{3}\bigr)\sin 2\theta -\bigl(\frac{1}{16}n^{2}-\frac{33}{32}n^{4}\bigr)\sin 4\theta -\frac{5}{16}n^{3}\sin 6\theta -\frac{261}{512}n^{4}\sin 8\theta -\ldots\\ \theta-\mu&=\textstyle{} -\bigl(\frac{1}{2}n+\frac{23}{32}n^{3}\bigr)\sin 2\mu +\bigl(\frac{5}{16}n^{2}-\frac{5}{96}n^{4}\bigr)\sin 4\mu +\frac{1}{32}n^{3}\sin 6\mu +\frac{283}{1536}n^{4}\sin 8\mu +\ldots\\ \end{align} \]

Here are the remaining relations (including χ and ξ) carried out to 3rd order in n:

\[ \begin{align} \chi-\phi&=\textstyle{} -\bigl(2n-\frac{2}{3}n^{2}-\frac{4}{3}n^{3}\bigr)\sin 2\phi +\bigl(\frac{5}{3}n^{2}-\frac{16}{15}n^{3}\bigr)\sin 4\phi -\frac{26}{15}n^{3}\sin 6\phi +\ldots\\ \phi-\chi&=\textstyle{} +\bigl(2n-\frac{2}{3}n^{2}-2n^{3}\bigr)\sin 2\chi +\bigl(\frac{7}{3}n^{2}-\frac{8}{5}n^{3}\bigr)\sin 4\chi +\frac{56}{15}n^{3}\sin 6\chi +\ldots\\ \chi-\beta&=\textstyle{} -\bigl(n-\frac{2}{3}n^{2}\bigr)\sin 2\beta +\bigl(\frac{1}{6}n^{2}-\frac{2}{5}n^{3}\bigr)\sin 4\beta -\frac{1}{15}n^{3}\sin 6\beta +\ldots\\ \beta-\chi&=\textstyle{} +\bigl(n-\frac{2}{3}n^{2}-\frac{1}{3}n^{3}\bigr)\sin 2\chi +\bigl(\frac{5}{6}n^{2}-\frac{14}{15}n^{3}\bigr)\sin 4\chi +\frac{16}{15}n^{3}\sin 6\chi +\ldots\\ \chi-\theta&=\textstyle{} +\bigl(\frac{2}{3}n^{2}+\frac{2}{3}n^{3}\bigr)\sin 2\theta -\bigl(\frac{1}{3}n^{2}-\frac{4}{15}n^{3}\bigr)\sin 4\theta -\frac{2}{5}n^{3}\sin 6\theta -\ldots\\ \theta-\chi&=\textstyle{} -\bigl(\frac{2}{3}n^{2}+\frac{2}{3}n^{3}\bigr)\sin 2\chi +\bigl(\frac{1}{3}n^{2}-\frac{4}{15}n^{3}\bigr)\sin 4\chi +\frac{2}{5}n^{3}\sin 6\chi +\ldots\\ \chi-\mu&=\textstyle{} -\bigl(\frac{1}{2}n-\frac{2}{3}n^{2}+\frac{37}{96}n^{3}\bigr)\sin 2\mu -\bigl(\frac{1}{48}n^{2}+\frac{1}{15}n^{3}\bigr)\sin 4\mu -\frac{17}{480}n^{3}\sin 6\mu -\ldots\\ \mu-\chi&=\textstyle{} +\bigl(\frac{1}{2}n-\frac{2}{3}n^{2}+\frac{5}{16}n^{3}\bigr)\sin 2\chi +\bigl(\frac{13}{48}n^{2}-\frac{3}{5}n^{3}\bigr)\sin 4\chi +\frac{61}{240}n^{3}\sin 6\chi +\ldots\\ \xi-\phi&=\textstyle{} -\bigl(\frac{4}{3}n+\frac{4}{45}n^{2}-\frac{88}{315}n^{3}\bigr)\sin 2\phi +\bigl(\frac{34}{45}n^{2}+\frac{8}{105}n^{3}\bigr)\sin 4\phi -\frac{1532}{2835}n^{3}\sin 6\phi +\ldots\\ \phi-\xi&=\textstyle{} +\bigl(\frac{4}{3}n+\frac{4}{45}n^{2}-\frac{16}{35}n^{3}\bigr)\sin 2\xi +\bigl(\frac{46}{45}n^{2}+\frac{152}{945}n^{3}\bigr)\sin 4\xi +\frac{3044}{2835}n^{3}\sin 6\xi +\ldots\\ \xi-\beta&=\textstyle{} -\bigl(\frac{1}{3}n+\frac{4}{45}n^{2}-\frac{32}{315}n^{3}\bigr)\sin 2\beta -\bigl(\frac{7}{90}n^{2}+\frac{4}{315}n^{3}\bigr)\sin 4\beta -\frac{83}{2835}n^{3}\sin 6\beta -\ldots\\ \beta-\xi&=\textstyle{} +\bigl(\frac{1}{3}n+\frac{4}{45}n^{2}-\frac{46}{315}n^{3}\bigr)\sin 2\xi +\bigl(\frac{17}{90}n^{2}+\frac{68}{945}n^{3}\bigr)\sin 4\xi +\frac{461}{2835}n^{3}\sin 6\xi +\ldots\\ \xi-\theta&=\textstyle{} +\bigl(\frac{2}{3}n-\frac{4}{45}n^{2}+\frac{62}{105}n^{3}\bigr)\sin 2\theta +\bigl(\frac{4}{45}n^{2}-\frac{32}{315}n^{3}\bigr)\sin 4\theta -\frac{524}{2835}n^{3}\sin 6\theta -\ldots\\ \theta-\xi&=\textstyle{} -\bigl(\frac{2}{3}n-\frac{4}{45}n^{2}+\frac{158}{315}n^{3}\bigr)\sin 2\xi +\bigl(\frac{16}{45}n^{2}-\frac{16}{945}n^{3}\bigr)\sin 4\xi -\frac{232}{2835}n^{3}\sin 6\xi +\ldots\\ \xi-\mu&=\textstyle{} +\bigl(\frac{1}{6}n-\frac{4}{45}n^{2}-\frac{817}{10080}n^{3}\bigr)\sin 2\mu +\bigl(\frac{49}{720}n^{2}-\frac{2}{35}n^{3}\bigr)\sin 4\mu +\frac{4463}{90720}n^{3}\sin 6\mu +\ldots\\ \mu-\xi&=\textstyle{} -\bigl(\frac{1}{6}n-\frac{4}{45}n^{2}-\frac{121}{1680}n^{3}\bigr)\sin 2\xi -\bigl(\frac{29}{720}n^{2}-\frac{26}{945}n^{3}\bigr)\sin 4\xi -\frac{1003}{45360}n^{3}\sin 6\xi -\ldots\\ \xi-\chi&=\textstyle{} +\bigl(\frac{2}{3}n-\frac{34}{45}n^{2}+\frac{46}{315}n^{3}\bigr)\sin 2\chi +\bigl(\frac{19}{45}n^{2}-\frac{256}{315}n^{3}\bigr)\sin 4\chi +\frac{248}{567}n^{3}\sin 6\chi +\ldots\\ \chi-\xi&=\textstyle{} -\bigl(\frac{2}{3}n-\frac{34}{45}n^{2}+\frac{88}{315}n^{3}\bigr)\sin 2\xi +\bigl(\frac{1}{45}n^{2}-\frac{184}{945}n^{3}\bigr)\sin 4\xi -\frac{106}{2835}n^{3}\sin 6\xi -\ldots\\ \end{align} \]

Series approximations in tabular form

Finally, this is a listing of all the coefficients for the expansions carried out to 8th order in n. Here's how to interpret this data: the 5th line for φ − θ is [32/5, 0, -32, 0]; this means that the coefficient of sin(10θ) is [(32/5)n⁵ − 32n⁷ + O(n⁹)].

β − φ:
[-1, 0, 0, 0, 0, 0, 0, 0] [1/2, 0, 0, 0, 0, 0, 0] [-1/3, 0, 0, 0, 0, 0] [1/4, 0, 0, 0, 0] [-1/5, 0, 0, 0] [1/6, 0, 0] [-1/7, 0] [1/8]

φ − β:
[1, 0, 0, 0, 0, 0, 0, 0] [1/2, 0, 0, 0, 0, 0, 0] [1/3, 0, 0, 0, 0, 0] [1/4, 0, 0, 0, 0] [1/5, 0, 0, 0] [1/6, 0, 0] [1/7, 0] [1/8]

θ − φ:
[-2, 0, 2, 0, -2, 0, 2, 0] [2, 0, -4, 0, 6, 0, -8] [-8/3, 0, 8, 0, -16, 0] [4, 0, -16, 0, 40] [-32/5, 0, 32, 0] [32/3, 0, -64] [-128/7, 0] [32]

φ − θ:
[2, 0, -2, 0, 2, 0, -2, 0] [2, 0, -4, 0, 6, 0, -8] [8/3, 0, -8, 0, 16, 0] [4, 0, -16, 0, 40] [32/5, 0, -32, 0] [32/3, 0, -64] [128/7, 0] [32]

θ − β:
[-1, 0, 0, 0, 0, 0, 0, 0] [1/2, 0, 0, 0, 0, 0, 0] [-1/3, 0, 0, 0, 0, 0] [1/4, 0, 0, 0, 0] [-1/5, 0, 0, 0] [1/6, 0, 0] [-1/7, 0] [1/8]

β − θ:
[1, 0, 0, 0, 0, 0, 0, 0] [1/2, 0, 0, 0, 0, 0, 0] [1/3, 0, 0, 0, 0, 0] [1/4, 0, 0, 0, 0] [1/5, 0, 0, 0] [1/6, 0, 0] [1/7, 0] [1/8]

μ − φ:
[-3/2, 0, 9/16, 0, -3/32, 0, 57/2048, 0] [15/16, 0, -15/32, 0, 135/2048, 0, -105/4096] [-35/48, 0, 105/256, 0, -105/2048, 0] [315/512, 0, -189/512, 0, 693/16384] [-693/1280, 0, 693/2048, 0] [1001/2048, 0, -1287/4096] [-6435/14336, 0] [109395/262144]

φ − μ:
[3/2, 0, -27/32, 0, 269/512, 0, -6607/24576, 0] [21/16, 0, -55/32, 0, 6759/4096, 0, -155113/122880] [151/96, 0, -417/128, 0, 87963/20480, 0] [1097/512, 0, -15543/2560, 0, 2514467/245760] [8011/2560, 0, -69119/6144, 0] [293393/61440, 0, -5962461/286720] [6459601/860160, 0] [332287993/27525120]

μ − β:
[-1/2, 0, 3/16, 0, -1/32, 0, 19/2048, 0] [-1/16, 0, 1/32, 0, -9/2048, 0, 7/4096] [-1/48, 0, 3/256, 0, -3/2048, 0] [-5/512, 0, 3/512, 0, -11/16384] [-7/1280, 0, 7/2048, 0] [-7/2048, 0, 9/4096] [-33/14336, 0] [-429/262144]

β − μ:
[1/2, 0, -9/32, 0, 205/1536, 0, -4879/73728, 0] [5/16, 0, -37/96, 0, 1335/4096, 0, -86171/368640] [29/96, 0, -75/128, 0, 2901/4096, 0] [539/1536, 0, -2391/2560, 0, 1082857/737280] [3467/7680, 0, -28223/18432, 0] [38081/61440, 0, -733437/286720] [459485/516096, 0] [109167851/82575360]

μ − θ:
[1/2, 0, 13/16, 0, -15/32, 0, 509/2048, 0] [-1/16, 0, 33/32, 0, -1673/2048, 0, 2599/4096] [-5/16, 0, 349/256, 0, -2989/2048, 0] [-261/512, 0, 963/512, 0, -43531/16384] [-921/1280, 0, 5545/2048, 0] [-6037/6144, 0, 16617/4096] [-19279/14336, 0] [-490925/262144]

θ − μ:
[-1/2, 0, -23/32, 0, 499/1536, 0, -14321/73728, 0] [5/16, 0, -5/96, 0, 6565/12288, 0, -201467/368640] [1/32, 0, -77/128, 0, 2939/4096, 0] [283/1536, 0, -4037/7680, 0, 1155049/737280] [1301/7680, 0, -19465/18432, 0] [17089/61440, 0, -442269/286720] [198115/516096, 0] [48689387/82575360]

χ − φ:
[-2, 2/3, 4/3, -82/45, 32/45, 4642/4725, -8384/4725, 1514/1323] [5/3, -16/15, -13/9, 904/315, -1522/945, -2288/1575, 142607/42525] [-26/15, 34/21, 8/5, -12686/2835, 44644/14175, 120202/51975] [1237/630, -12/5, -24832/14175, 1077964/155925, -1097407/187110] [-734/315, 109598/31185, 1040/567, -12870194/1216215] [444337/155925, -941912/184275, -126463/72765] [-2405834/675675, 3463678/467775] [256663081/56756700]

φ − χ:
[2, -2/3, -2, 116/45, 26/45, -2854/675, 16822/4725, 189416/99225] [7/3, -8/5, -227/45, 2704/315, 2323/945, -31256/1575, 141514/8505] [56/15, -136/35, -1262/105, 73814/2835, 98738/14175, -2363828/31185] [4279/630, -332/35, -399572/14175, 11763988/155925, 14416399/935550] [4174/315, -144838/6237, -2046082/31185, 258316372/1216215] [601676/22275, -115444544/2027025, -2155215124/14189175] [38341552/675675, -170079376/1216215] [1383243703/11351340]

χ − β:
[-1, 2/3, 0, -16/45, 2/5, -998/4725, -34/4725, 1384/11025] [1/6, -2/5, 19/45, -22/105, -2/27, 1268/4725, -12616/42525] [-1/15, 16/105, -22/105, 116/567, -1858/14175, 1724/51975] [17/1260, -8/105, 2123/14175, -26836/155925, 115249/935550] [-1/105, 128/4455, -424/6237, 140836/1216215] [149/311850, -31232/2027025, 210152/4729725] [-499/225225, 30208/6081075] [-68251/113513400]

β − χ:
[1, -2/3, -1/3, 38/45, -1/3, -3118/4725, 4769/4725, -25666/99225] [5/6, -14/15, -7/9, 50/21, -247/270, -14404/4725, 193931/42525] [16/15, -34/21, -5/3, 17564/2835, -36521/14175, -1709614/155925] [2069/1260, -28/9, -49877/14175, 2454416/155925, -637699/85050] [883/315, -28244/4455, -20989/2835, 48124558/1216215] [797222/155925, -2471888/184275, -16969807/1091475] [2199332/225225, -1238578/42525] [87600385/4540536]

χ − θ:
[0, 2/3, 2/3, -2/9, -14/45, 1042/4725, 18/175, -1738/11025] [-1/3, 4/15, 43/45, -4/45, -712/945, 332/945, 23159/42525] [-2/5, 2/105, 124/105, 274/2835, -1352/945, 13102/31185] [-55/126, -16/105, 21068/14175, 1528/4725, -2414843/935550] [-22/45, -9202/31185, 20704/10395, 60334/93555] [-90263/155925, -299444/675675, 40458083/14189175] [-8962/12285, -3818498/6081075] [-4259027/4365900]

θ − χ:
[0, -2/3, -2/3, 4/9, 2/9, -3658/4725, 76/225, 64424/99225] [1/3, -4/15, -23/45, 68/45, 61/135, -2728/945, 2146/1215] [2/5, -24/35, -46/35, 9446/2835, 428/945, -95948/10395] [83/126, -80/63, -34712/14175, 4472/525, 29741/85050] [52/45, -2362/891, -17432/3465, 280108/13365] [335882/155925, -548752/96525, -48965632/4729725] [51368/12285, -197456/15795] [1461335/174636]

χ − μ:
[-1/2, 2/3, -37/96, 1/360, 81/512, -96199/604800, 5406467/38707200, -7944359/67737600] [-1/48, -1/15, 437/1440, -46/105, 1118711/3870720, -51841/1209600, -24749483/348364800] [-17/480, 37/840, 209/4480, -5569/90720, -9261899/58060800, 6457463/17740800] [-4397/161280, 11/504, 830251/7257600, -466511/2494800, -324154477/7664025600] [-4583/161280, 108847/3991680, 8005831/63866880, -22894433/124540416] [-20648693/638668800, 16363163/518918400, 2204645983/12915302400] [-219941297/5535129600, 497323811/12454041600] [-191773887257/3719607091200]

μ − χ:
[1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200] [13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400] [61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600] [49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600] [34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080] [212378941/319334400, -30705481/10378368, 175214326799/58118860800] [1522256789/1383782400, -16759934899/3113510400] [1424729850961/743921418240]

ξ − φ:
[-4/3, -4/45, 88/315, 538/4725, 20824/467775, -44732/2837835, -86728/16372125, -88002076/13956067125] [34/45, 8/105, -2482/14175, -37192/467775, -12467764/212837625, -895712/147349125, -2641983469/488462349375] [-1532/2835, -898/14175, 54968/467775, 100320856/1915538625, 240616/4209975, 8457703444/488462349375] [6007/14175, 24496/467775, -5884124/70945875, -4832848/147349125, -4910552477/97692469875] [-23356/66825, -839792/19348875, 816824/13395375, 9393713176/488462349375] [570284222/1915538625, 1980656/54729675, -4532926649/97692469875] [-496894276/1915538625, -14848113968/488462349375] [224557742191/976924698750]

φ − ξ:
[4/3, 4/45, -16/35, -2582/14175, 60136/467775, 28112932/212837625, 22947844/1915538625, -1683291094/37574026875] [46/45, 152/945, -11966/14175, -21016/51975, 251310128/638512875, 1228352/3007125, -14351220203/488462349375] [3044/2835, 3802/14175, -94388/66825, -8797648/10945935, 138128272/147349125, 505559334506/488462349375] [6059/4725, 41072/93555, -1472637812/638512875, -45079184/29469825, 973080708361/488462349375] [768272/467775, 455935736/638512875, -550000184/147349125, -1385645336626/488462349375] [4210684958/1915538625, 443810768/383107725, -2939205114427/488462349375] [387227992/127702575, 101885255158/54273594375] [1392441148867/325641566250]

ξ − β:
[-1/3, -4/45, 32/315, 34/675, 2476/467775, -70496/8513505, -18484/4343625, 29232878/97692469875] [-7/90, -4/315, 74/2025, 3992/467775, 53836/212837625, -4160804/1915538625, -324943819/488462349375] [-83/2835, 2/14175, 7052/467775, -661844/1915538625, 237052/383107725, -168643106/488462349375] [-797/56700, 934/467775, 1425778/212837625, -2915326/1915538625, 113042383/97692469875] [-3673/467775, 390088/212837625, 6064888/1915538625, -558526274/488462349375] [-18623681/3831077250, 41288/29469825, 155665021/97692469875] [-6205669/1915538625, 504234982/488462349375] [-8913001661/3907698795000]

β − ξ:
[1/3, 4/45, -46/315, -1082/14175, 11824/467775, 7947332/212837625, 9708931/1915538625, -5946082372/488462349375] [17/90, 68/945, -338/2025, -16672/155925, 39946703/638512875, 164328266/1915538625, 190673521/69780335625] [461/2835, 1102/14175, -101069/467775, -255454/1563705, 236067184/1915538625, 86402898356/488462349375] [3161/18900, 1786/18711, -189032762/638512875, -98401826/383107725, 110123070361/488462349375] [88868/467775, 80274086/638512875, -802887278/1915538625, -200020620676/488462349375] [880980241/3831077250, 66263486/383107725, -296107325077/488462349375] [37151038/127702575, 4433064236/18091198125] [495248998393/1302566265000]

ξ − θ:
[2/3, -4/45, 62/105, 778/4725, -193082/467775, -4286228/42567525, 53702182/212837625, 182466964/8881133625] [4/45, -32/315, 12338/14175, 92696/467775, -61623938/70945875, -32500616/273648375, 367082779691/488462349375] [-524/2835, -1618/14175, 612536/467775, 427003576/1915538625, -663111728/383107725, -42668482796/488462349375] [-5933/14175, -8324/66825, 427770788/212837625, 421877252/1915538625, -327791986997/97692469875] [-320044/467775, -9153184/70945875, 6024982024/1915538625, 74612072536/488462349375] [-1978771378/1915538625, -46140784/383107725, 489898512247/97692469875] [-2926201612/1915538625, -42056042768/488462349375] [-2209250801969/976924698750]

θ − ξ:
[-2/3, 4/45, -158/315, -2102/14175, 109042/467775, 216932/2627625, -189115382/1915538625, -230886326/6343666875] [16/45, -16/945, 934/14175, -7256/155925, 117952358/638512875, 288456008/1915538625, -11696145869/69780335625] [-232/2835, 922/14175, -25286/66825, -7391576/54729675, 478700902/1915538625, 91546732346/488462349375] [719/4725, 268/18711, -67048172/638512875, -67330724/383107725, 218929662961/488462349375] [14354/467775, 46774256/638512875, -117954842/273648375, -129039188386/488462349375] [253129538/1915538625, 2114368/34827975, -178084928947/488462349375] [13805944/127702575, 6489189398/54273594375] [59983985827/325641566250]

ξ − μ:
[1/6, -4/45, -817/10080, 1297/18900, 7764059/239500800, -9292991/302702400, -25359310709/1743565824000, 39534358147/2858202547200] [49/720, -2/35, -29609/453600, 35474/467775, 36019108271/871782912000, -14814966289/245188944000, -13216941177599/571640509440000] [4463/90720, -2917/56700, -4306823/59875200, 3026004511/30648618000, 99871724539/1569209241600, -27782109847927/250092722880000] [331799/7257600, -102293/1871100, -368661577/4036032000, 2123926699/15324309000, 168979300892599/1600593426432000] [11744233/239500800, -875457073/13621608000, -493031379277/3923023104000, 1959350112697/9618950880000] [453002260127/7846046208000, -793693009/9807557760, -145659994071373/800296713216000] [103558761539/1426553856000, -53583096419057/500185445760000] [12272105438887727/128047474114560000]

μ − ξ:
[-1/6, 4/45, 121/1680, -1609/28350, -384229/14968800, 12674323/851350500, 7183403063/560431872000, -375027460897/125046361440000] [-29/720, 26/945, 16463/453600, -431/17325, -31621753811/1307674368000, 1117820213/122594472000, 30410873385097/2000741783040000] [-1003/45360, 449/28350, 3746047/119750400, -32844781/1751349600, -116359346641/3923023104000, 151567502183/17863765920000] [-40457/2419200, 629/53460, 10650637121/326918592000, -13060303/766215450, -317251099510901/8002967132160000] [-1800439/119750400, 205072597/20432412000, 146875240637/3923023104000, -2105440822861/125046361440000] [-59109051671/3923023104000, 228253559/24518894400, 91496147778023/2000741783040000] [-4255034947/261534873600, 126430355893/13894040160000] [-791820407649841/42682491371520000]

ξ − χ:
[2/3, -34/45, 46/315, 2458/4725, -55222/93555, 2706758/42567525, 16676974/30405375, -64724382148/97692469875] [19/45, -256/315, 3413/14175, 516944/467775, -340492279/212837625, 158999572/1915538625, 85904355287/37574026875] [248/567, -15958/14175, 206834/467775, 4430783356/1915538625, -7597644214/1915538625, 2986003168/37574026875] [16049/28350, -832976/467775, 62016436/70945875, 851209552/174139875, -375566203/39037950] [15602/18711, -651151712/212837625, 3475643362/1915538625, 5106181018156/488462349375] [2561772812/1915538625, -10656173804/1915538625, 34581190223/8881133625] [873037408/383107725, -5150169424688/488462349375] [7939103697617/1953849397500]

χ − ξ:
[-2/3, 34/45, -88/315, -2312/14175, 27128/93555, -55271278/212837625, 308365186/1915538625, -17451293242/488462349375] [1/45, -184/945, 6079/14175, -65864/155925, 106691108/638512875, 149984636/1915538625, -101520127208/488462349375] [-106/2835, 772/14175, -14246/467775, 5921152/54729675, -99534832/383107725, 10010741462/37574026875] [-167/9450, -5312/467775, 75594328/638512875, -35573728/273648375, 1615002539/75148053750] [-248/13365, 2837636/638512875, 130601488/1915538625, -3358119706/488462349375] [-34761247/1915538625, -3196/3553875, 46771947158/488462349375] [-2530364/127702575, -18696014/18091198125] [-14744861191/651283132500]

Truncation errors

There are two sources of error when using these series. The truncation error arises from retaing terms up to a certain order in n; it is the absolute difference between the value of the truncated series compared with the exact latitude (evaluated with exact arithmetic). In addition, using standard double-precision arithmetic entails accumulating round-off errors so that at the end of a complex calculation a few of the trailing bits of the result are wrong.

Here's a table of the truncation errors. The errors are given in "units in the last place (ulp)" where 1 ulp = 2⁻⁵³ radian = 6.4 × 10⁻¹⁵ degree = 2.3 × 10⁻¹¹ arcsecond which is a measure of the round-off error for double precision. Here is some rough guidance on how to interpret these errors:

if the truncation error is less than 1 ulp, then round-off errors dominate;
if the truncation error is greater than 8 ulp, then truncation errors dominate;
otherwise, round-off and truncation errors are comparable.

The truncation errors are given accurate to 2 significant figures.

Auxiliary latitude truncation errors (ulp)
expression	[f = 1/150, order = 6]		[f = 1/297, order = 5]
expression	n series	e² series	n series	e² series
β − φ	`0.0060`	`28`	`0.035`	`41`
φ − β	`0.0060`	`28`	`0.035`	`41`
θ − φ	`2.9`	`82`	`6.0`	`120`
φ − θ	`2.9`	`82`	`6.0`	`120`
θ − β	`0.0060`	`28`	`0.035`	`41`
β − θ	`0.0060`	`28`	`0.035`	`41`
μ − φ	`0.037`	`41`	`0.18`	`60`
φ − μ	`0.98`	`59`	`2.3`	`84`
μ − β	`0.00069`	`5.8`	`0.0024`	`9.6`
β − μ	`0.13`	`12`	`0.35`	`19`
μ − θ	`0.24`	`30`	`0.67`	`40`
θ − μ	`0.099`	`23`	`0.23`	`33`
χ − φ	`0.78`	`43`	`2.1`	`64`
φ − χ	`9.0`	`71`	`17`	`100`
χ − β	`0.018`	`3.7`	`0.11`	`6.4`
β − χ	`1.7`	`16`	`3.4`	`24`
χ − θ	`0.18`	`31`	`0.56`	`43`
θ − χ	`0.87`	`23`	`1.9`	`32`
χ − μ	`0.022`	`0.56`	`0.11`	`0.91`
μ − χ	`0.31`	`1.2`	`0.86`	`2.0`
ξ − φ	`0.015`	`39`	`0.086`	`57`
φ − ξ	`0.34`	`53`	`1.1`	`75`
ξ − β	`0.00042`	`6.3`	`0.0039`	`10`
β − ξ	`0.040`	`10`	`0.15`	`15`
ξ − θ	`0.28`	`28`	`0.75`	`38`
θ − ξ	`0.040`	`23`	`0.11`	`33`
ξ − μ	`0.015`	`0.79`	`0.058`	`1.5`
μ − ξ	`0.0043`	`0.54`	`0.018`	`1.1`
ξ − χ	`0.60`	`1.9`	`1.5`	`3.6`
χ − ξ	`0.023`	`0.53`	`0.079`	`0.92`

The 2nd and 3rd columns show the results for the SRMmax ellipsoid, f = 1/150, retaining 6th order terms in the series expansion. The 4th and 5th columns show the results for the International ellipsoid, f = 1/297, retaining 5th order terms in the series expansion. The 2nd and 4th columns give the errors for the series expansions in terms of n given in this section (appropriately truncated). The 3rd and 5th columns give the errors when the series are reexpanded in terms of e² = 4n/(1 + n)² and truncated retaining the e¹² and e¹⁰ terms respectively.

Some observations:

For production use, the 6th order series in n are recommended. For f = 1/150, the resulting errors are close to the round-off limit. The errors in the 6th order series scale as f⁷; so the errors with f = 1/297 are about 120 times smaller.
It's inadvisable to use the 5th order series in n; this order is barely acceptable for f = 1/297 and the errors grow as f⁶ as f is increased.
In all cases, the expansions in terms of e² are considerably less accurate than the corresponding series in n.
For every series converting between φ and any of θ, μ, χ, or ξ, the series where β is substituted for φ is more accurate. Considering that the transformation between φ and β is so simple, tanβ = (1 - f) tanφ, it sometimes makes sense to use β internally as the basic measure of latitude. (This is the case with geodesic calculations.)

Back to Geocentric coordinates. Forward to Support for high precision arithmetic. Up to Contents.