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ELLIPTIC INTEGRALS AND LANDEN’S TRANSFORMATION (WHAT I SHOULD’VE KNOWN FOR GEODESY) Rod Deakin

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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!)

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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 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)

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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:

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Evaluating Legendre then used Landen’s transformation in the following way:

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As n the sequence of converges to unity and the sequence of converges to a limiting value, and The sequences converge in only a few iterations

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Iterative scheme An example of the iterative scheme for the evaluation of n k n n (degrees) n

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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)))) Computer code

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Evaluating Legendre used Landen’s transformation to express and elliptic integral of the Second Kind in the following way:

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After some manipulation a formula using the Arithmetic-Geometric Mean (AGM) sequence can be obtained as:

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Arithmetic-Geometric Mean (AGM) For two positive real numbers with put then with The sequences converge to a common limit and converges to zero n a n g n c n

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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))) Computer code

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Meridian distance (series formula)

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Meridian distance (Elliptic integral) a = metres flat = f = phi = (degrees) e = E(phi,e) = mdist = metres

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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)

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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.

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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

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