GeographicLib  1.49
Auxiliary latitudes
Back to Geocentric coordinates. Forward to Support for high precision arithmetic. Up to Contents.

Go to

Six latitudes are used by GeographicLib:

The last five of these are called auxiliary latitudes. These quantities are all defined in the Wikipedia article on latitudes.

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:

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

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 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 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.

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:

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]
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:

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