GeographicLib
1.43
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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 = sinh−1 tanχ; but this is not an angle-like variable and we don't consider it further here.) 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. The expansions are in terms of the third flattening n = (a − b)/(a + b). This 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, ∞). (Some authors use the eccentricity as the expansion parameter, but the resulting series only converge for b/a ∈ (0, √2). These expansions were obtained with the the maxima code, auxlat.mac.
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} \]
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]