sci.math #183956 (2 + 1869 more) <1>+-<1> From: conscience \-<1>--<1>--<1>--<1> [1] Rodrigues formula Date: Tue Mar 18 03:24:49 EST 1997 Lines: 4 Hi! I am looking for the Rodrigues formula .I have heard about it in an article of robotics research and I would like more informations. thanks Mireille End of article 183956 (of 184476) -- what next? [npq] sci.math #182790 (1 + 1869 more) -[1] sci.math #183962 (4 + 1864 more) (1)+-[1] From: Ariel Scolnicov \-[1]--[1]--[1]--[1] [1] Re: Rodrigues formula Date: Tue Mar 18 04:10:16 EST 1997 ... > article of robotics research and I would like more informations. Try, e.g., Klaus M\"uller, _Spherical Harmonics_, published by Springer-Verlag sometime long ago. It's a large booklet. One slight annoyance is the author's utter refusal to use *any* form of linear algebra whatsoever. On the other hand, it does make for a very readable book. End of article 183962 (of 184476) -- what next? [npq] sci.math #183967 (3 + 1864 more) (1)+-(1) From: Pertti Lounesto \-[1]--[1]--[1]--[1] [1] Re: Rodrigues formula Date: Tue Mar 18 05:00:17 EST 1997 There are two quite different Rodrigues formulas. Since you are in robotics, you are presumably interested in the second alternative. 1. Rodrigues formula for Legendre polynomials: P_n(x) = 1/(2^n n!)D^n[(x^2-1)^n]. 2. Rodrigues formula for composition of rotations in R^3: c' = (a'+b'+a'xb')/(1-a'.b') where a' = a/alpha tan(alpha/2) and a is the axis of rotation, alpha = |a| is the angle. For more information about the second alternative, I can refer to my book "Clifford Algebras and Spinors", Cambridge University Press, London Mathematical Society pages 59 and 71. -- Pertti.Lounesto@hut.fi http://www.math.hut.fi/~lounesto/counterexamples.htm End of article 183967 (of 184476) -- what next? [npq] sci.math #184216 (2 + 1864 more) (1)+-(1) From: Robin Chapman \-(1)--[1]--[1]--[1] [1] Re: Rodrigues formula Date: Thu Mar 20 03:05:14 EST 1997 Just out of interest. Are these the same Rodrigues? -- Robin John Chapman "... needless to say, Department of Mathematics I think there should be University of Exeter, EX4 4QE, UK more sex and violence rjc@maths.exeter.ac.uk on television, not less." http://www.maths.ex.ac.uk/~rjc/rjc.html J. G. Ballard (1990) End of article 184216 (of 184476) -- what next? [npq] sci.math #184226 (1 + 1864 more) (1)+-(1) From: Pertti Lounesto \-(1)--(1)--[1]--[1] [1] Re: Rodrigues formula Date: Thu Mar 20 05:36:09 EST 1997 Lines: 13 Robin Chapman writes: > Just out of interest. Are these the same Rodrigues? The Rodrigues formula for Legendre polynomials is annotated to Olinde Rodrigues (1794-1851) by my textbook (Kreyszig AEM, 7th, p. 213). The Rodrigues formula for composition of rotations in R^3 is annotated to Olinde Rodrigues by van der Waerden (History of Algebra, 1985, p. 120). Yes, it is the same Rodrigues. -- Pertti.Lounesto@hut.fi http://www.math.hut.fi/~lounesto/counterexamples.htm End of article 184226 (of 184476) -- what next? [npq] sci.math #184267 (0 + 1864 more) (1)+-(1) From: hbaker@netcom.com (Henry Baker) \-(1)--(1)--(1)--[1] [1] Re: Rodrigues formula Date: Thu Mar 20 14:24:03 EST 1997 According to Simon Altmann, "Rotations, Quaternions and Double Groups" (Clarendon Press, Oxford, 1986, ISBN 0-19-855372-2) : on him by Jeremy Gray (Arch. for History of Exact Sciences 21 (1980), 375-85), we know next to nothing. He is given a mere one-page entry in Michaud "Biographi Universelle" (1843) as an 'economist and French reformer'. So little is he known, indeed, that Cartan (1938, p.57) invented a non-existent collaborator of Rodrigues by the surname of Olinde, a mistake repeated by Temple (1960, p.68). Booth (1871) calls him Ridrigue throughout his book, and Wilson (1941,p.100) spells his name Rodriques. Nothing that Rodrigues did on the rotation group---and he did more than any man before him, or than any would do for several decades afterwards---brought him undivided credit; and for much of his work he received no credit at all. This Invisible Man of the rotation group was probably born in Bordeaux on 16 October 1794, the son of a Jewish banker, and he was named Benjamin Olinde, although he never used his first name in later life. The family is often said to have been of Spanish origin, but the spelling of the family name rather suggests Portuguese descent (as indeed asserted by the 'Enciclopedia Universal Illustrada Espasa-Calpe'). He studied mathematics at the Ecole Normale, the \'Ecole Polytechnique not being accessible to him owing to his Jewish extraction. He took his doctorate at the new University of Paris in 1816 with a thesis that contains the famous `Rodrigues formula' for Legendre polynomials, for which he is mainly known (Grattan-Guiness 1983). The little known about him is only as a paranymph of Saint-Simon, the charismatic attempted suicide. S we read (Weill 1894,p.30) that the banker Rodrigues helped him in his illness and distitution and supported him financially until his death in 1825. That Rodrigues must have been very well off we can surmise from Weill's reference to him as belonging to high banking circles, on a par with the wealthy Laffittes (1894,p.238). When Saint-Simon died, Rodrigues shared with another disciple of Saint-Simon's, Prosper Enfantin, the headship of the movement, thus becoming 'Pere Olinde' for the acolytes; but in 1832 he repudiated Enfantin's extreme views of sexual freedom and he proclaimed himself the apostle of Saint-Simonism. In August that year he was charged with taking part in illegal meetings and outraging public morality and was fined fifty francs (Booth 1871). Neither Booth nor Weill even mention that Rodrigues was a mathematician: the single reference to this (Booth, 1871, p.100) is that in 1813 he was Enfantin's tutor in mathematics at the \'Ecole Polytechnique. Indeed, all that we know about him in the year 1840 when he published his fundamental paper on the Euclidean and rotation groups, is that he was 'speculating at the Bourse' [French stock market] (Booth 1871, p.216). "Besides his extensive writings on social and political matters, Rodrigues published several pamphlets on the theory of banking and was influential in the development of French railways. He died in Paris, however, almost 1850 according to the Biographie Universelle, or 17 December 1851 according to Larousse (1866). Sebastien Charlety (1936, pp.26,294), although hardly touching upon Rodrigues in his authoritative history of Saint-Simonism gives 1851 as the year of Rodrigues's death, a date which most modern references seem to favour." End of article 184267 (of 184476) -- what next? [npq] From askey@math.wisc.edu Sat Mar 29 19:25:58 1997 Received: from conley.math.wisc.edu (conley.math.wisc.edu [144.92.166.10]) by ns1.auto.ucl.ac.be (8.7.4/13.12.95) with SMTP id TAA06689; Sat, 29 Mar 1997 19:22:56 +0100 (MET) Received: by conley.math.wisc.edu; id AA21653; 4.1/42; Sat, 29 Mar 97 12:28:17 CST From: Richard Askey Message-Id: <9703291828.AA21653@conley.math.wisc.edu> Subject: More on Rodrigues To: Magnus@anma.ucl.ac.be (Alphonse Magnus) Date: Sat, 29 Mar 1997 12:28:17 -0600 (CST) Cc: Claude.Brezinski@univ-lille1.fr, askey@math.wisc.edu, meinguet@anma.ucl.ac.be, pali+@osu.edu In-Reply-To: <9703251034.AA28879@math.math.ucl.ac.be> from "Alphonse Magnus" at Mar 25, 97 11:34:19 am X-Mailer: ELM [version 2.4 PL25] Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Status: O Dear Alphonse, Rodrigues was one of the great short-term losers in the "game" of mathematical fame. Two examples were mentioned in your forwarded message. One thing about the Rodrigues formula for Legendre polynomials was not mention, the fact that it was lost for many decades. There is a paper by Ivory and Jacobi which is not included in Jacobi's "Collected Papers", which was published in J. de Math. in about 1835, giving a derivation (or just a proof) of this formula, which they had seperately discovered. They wrote it was important and not known in France. For a number of years this formula was known as the formula of Ivory and Jacobi. Hermite found Rodrigues's paper about 1860 and the name change came via Heine's book. The history here is essentially or completely given by Heine. However, there is another very important formula which merits Rodrigues's name. This came from a little problem posed by Maximalian Stern about 1840. There are n! permuatations of the set {1,2,...,n}. Stern asked for the total number of inversions of these n! permutations. This was solved by Terquem, and then Rodrigues gave a deeper solution by finding the generating function of the number of inversions. The permutations and number of inversions of {1,2,3} are 123 0 132 1 213 1 231 2 312 2 321 3 and the generating function if 1 + 2q + 2q^2 + q^3 = (1+q)(1+q+q^2) This was the first connection between q-series and inversions. This is very important, having eventually led to SU(2,q) and all of the connections between q-special functions and representation theory. This paper of Rodrigues was lost from 1840 to 1970 when it was mentioned in a paper by Leonard Carlitz. His student Charles Church seems to have been the first to appreciate what Rodrigues did. Unfortunately, the paper of Rodrigues played no role in the modern developments, or seemingly even in the work MacMahon did on inversions and Gaussian binomial coefficients. Netto does not seem to have known of Rodrigues's paper, although he knew the inversion problem. Dick askey@math.wisc.edu