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
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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
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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