ELLIPTIC INTEGRALS AND LANDEN’S TRANSFORMATION (WHAT I SHOULD’VE KNOWN FOR GEODESY) f F (f , k ) = ò 0 dq ( F ir st K in d ) 2 2 2 2 1 - k sin q f E (f , k ) = ò 1 - k sin q d q ( S econ d K in d ) 0 Rod Deakin What’s so special about elliptic integrals They crop up in geodetic problems: • Meridian distance on an ellipsoid. • Geodesic arc length. • Transverse Mercator projection coordinates. They cannot be evaluated in closed form by elementary methods. They may be approximated by: • Series expansion and term-by-term integration. (usual method - not bad) • Landen’s transformation. (simple and very clever!) Who was Landen? John Landen (1719–1790) was an English land surveyor (20 years) and then the Earl Fitzwilliam’s land agent (26 years). Also an amateur mathematician. Published 8 articles in Philosophical Transactions of the Royal Society between 1754 and 1785 and elected a fellow in 1766. Also published 8 books and contributed problems and solutions to the Ladies Diary (a publication designed principally for the amusement and instruction of the fair sex) Landen’s transformation Landen’s contribution to mathematics was to show how an arc of an hyperbola could be expressed as the sum of two arcs of auxiliary ellipses. He did this by using an algebraic transformation of fluents (integrals). A.M. Legendre (1752–1833) developed Landen’s work and expressed his transformation in trigonometric form as: sin (2 f n+1 - f n )= k n sin f n a n d kn + 1 = 2 kn 1 + kn Evaluating F (f 0 , k 0 ) Legendre then used Landen’s transformation in the following way: f If F (f 0 , k 0 ) = 0 dq ò 2 and t h en sin (2 f n+1 - f F (f 0 , k 0 ) = = n )= 2 1 + k0 2 k n sin f 2 1 + k0 2 kn a n d kn + 1 = n 1 + kn F (f 1 , k 1 ) × 1 + k0 = 2 1 - k 0 sin q 0 2 1 + k1 × 2 1 + k1 F (f 2 , k 2 ) × 2 1 + k2 L 2 1 + kn- 1 F (f n , k n ) As n the sequence of k n converges to unity and the sequence of f n converges to a limiting value, and F (f 0 , k 0 ) » k 1k 2 k 3 L k n k0 ln t a n ( 14 p + 1 2 f The sequences converge in only a few iterations n ) Iterative scheme An example of the iterative scheme for the evaluation of F ( 13 p , 0.08 ) n 0 1 2 3 4 5 6 7 n (degrees) kn 0.080000000000000000000000000000 0.523782800878924092148773601559 0.949910187049589210997543775191 0.999670002027321953622743219568 0.999999986383174006610440236550 0.999999999999999976822755917618 1.000000000000000000000000000000 1.000000000000000000000000000000 k 1k 2 k 3 L k 7 60.000000000000000000000000000000 31.986375290275345206426642287209 24.047430982645159265462549335921 23.410227894528127n587073492551888 23.406135016316805818730472567793 23.406134847458754170823337386119 23.406134847458753883409497070244 23.406134847458753883409497070244 = 2.493447468386180336582176623321 k0 f ln t a n ( 14 p + 7 ) F ( p , 0.08 ) 1 3 1 2 f 7 = 23.406134847458753883409497070244 d egr ee s = 0.420374825518291193367606198921 » 1.04818254446186545539848353357 1 Computer code Maxima code for the evaluation of an elliptic integral of the First Kind F(phi,k) := block([product : 1/k, tol : 1.0b-36], while (1-k) > tol do block( phi : (asin(k*sin(phi))+phi)/2, k : 2*sqrt(k)/(1+k), product : product*k), return(sqrt(product)*log(tan(pion4+phi/2)))) Evaluating E (f 0 , k 0 ) Legendre used Landen’s transformation to express and elliptic integral of the Second Kind in the following way: f If 0 ò E (f 0 , k 0 ) = 2 2 1 - k 0 sin q d q 0 and sin (2 f E (f 0 , k 0 ) = 1 - f (1 + 0 )= k 0 sin f 0 a n d k1 = 2 k0 t h en 1 + k0 k 1 ) E (f 1 , k 1 ) + (1 - k 0 ) F (f 0 , k 0 ) - k 0 sin f 0 After some manipulation a formula using the Arithmetic-Geometric Mean (AGM) sequence can be obtained as: ìï 2 1 a c + 2 2 a c + 2 3 a c ü ï 1 1 2 2 3 3ï ï E (f 0 , k 0 ) » F (f 0 , k 0 ) í 4 ý ï 2 a c + L + 2n a c ï ïïî ïïþ 4 4 n n ìï 2 0 g sin f + 2 1 g sin f + 2 2 g sin f ü ï ï 0 0 1 1 2 2ï - í ý ï + 2 3 g sin f + L + 2 n - 1 g sin f n - 1 ïï ïîï 3 3 n- 1 ï þ n + 2 a n sin f n w h er e a n + 1 = 1 2 (a n + g n ), g n + 1 = a n gn , cn + 1 = 1 2 (a n - gn ) Arithmetic-Geometric Mean (AGM) For two positive real numbers a , b with a > b put a 0 = a , g 0 = b then with an+ 1 = 1 2 (a n + g n ), g n + 1 = a n gn , cn + 1 = 1 2 (a n - gn The sequences a n , g n converge to a common limit M (a , b ) and c n converges to zero. n an 0 100.000000000000000000000000000000 1 50.500000000000000000000000000000 2 30.250000000000000000000000000000 3 26.361102527122115932299070222745 4 26.216887049460483910562429431797 5 26.216688720600012389832846351910 6 26.216688720224923669479049585065 7 26.216688720224923669477707963039 gn 1.000000000000000000000000000000 10.000000000000000000000000000000 22.472205054244231864598140445491 26.072671571798851888825788640849 26.216490391739540869103263272023 26.216688719849834949125252818220 26.216688720224923669476366341013 26.216688720224923669477707963039 cn 49.500000000000000000000000000000 20.250000000000000000000000000000 3.888897472877884067700929777255 0.144215477661632021736640790948 0.000198328860471520729583079887 0.000000000375088720353796766845 0.000000000000000000001341622026 ) Computer code Maxima code for the evaluation of an elliptic integral of the Second Kind E(phi,k) := block([a : 1.0b0, g : k, n : 0, sum1 : 0.0b0, sum2 : 0.0b0, tol : 1.0b-36], F : F(phi,k), while (a-g) > tol do block( sum2 : sum2 + (2^n)*g*sin(phi), a1 : (a + g)/2, g1 : sqrt(a*g), c1 : (a - g)/2, phi : (asin(k*sin(phi))+phi)/2, sum1 : sum1 + (2^(n+1))*a1*c1, a : a1, g : g1, k : g1/a1, n : n+1), return(F*sum1-sum2+(2^n)*a*sin(phi))) Meridian distance (series formula) ìï c f ï 0 ï ï a ïï M = í 1 + n ïï ï ï ïï î c0 = 1 + 1 4 n 2 + 1 64 n 4 + üï ï ï ï + c 6 sin 6 f + c 8 sin 8 f ïï ý ï + c 10 sin 10 f + c 12 sin 12 f ï ï + c 14 sin 14 f + c 16 sin 16 f + L ïï ïþ + c 2 sin 2 f + c 4 sin 4 f 1 n 6 + 25 8 n L 256 16384 15 2 15 4 75 105 8 6 c4 = n n n n - L 16 64 2048 8192 315 4 441 6 1323 8 c8 = n n n - L 512 2048 32768 1 001 6 1573 8 c 12 = n n - L 2048 8192 109395 8 c 16 = n - L 262144 c2 = c6 = c 10 c 14 3 n + 2 35 3 n 3 16 n 48 693 3 + 3 + n 5 15 + 128 175 n 5 + n 2048 245 n 768 6144 2079 7 5 = n + n + L 1280 10240 6435 7 = n + L 14336 7 + L 7 + L Meridian distance (Elliptic integral) ìï ïï M = a í E (f , e ) ï ïîï a flat f phi e E(phi,e) mdist ü 2 e sin f cos f ïïï ý 2 2 1 - e sin f ïïþ ï = 6378137.000000000000000000000000000000 metres = 298.257222101000000000000000000000 = 0.003352810681182318935434146126 = 60.000000000000000000000000000000(degrees) = 0.081819191042815790145895739896 = 1.046168817527900319688127443142 = 6654072.819367444406819108934413675127 metres More on Landen John Landen was one of a group of non-academic men fully employed in the ordinary businesses of life, but keenly and intelligently interested in scientific matters. He was the eldest of three sons born to parents of yeoman stock. John Landen (1719–1790) He was well educated and his chosen profession of surveying demanded independently minded people able to gather information, analyse problems and provide solutions. He was an accomplished mathematician and read extensively in both French and Latin. He apparently had little tolerance for people he disagreed with and his dogmatism and pugnacity caused him to be generally shunned in polite society. But he must have had a sense of humour. His pseudonyms in the Ladies Diary were Sir Stately Stiff, C. Bumpkin and Peter Puzzlem (and others). THE END