Archive:New Draft of the Week

The New Draft of the Week is a chance to highlight a recently created Citizendium article that has just started down the road of becoming a Citizendium masterpiece.
It is chosen each week by vote in a manner similar to that of its sister project, the Article of the Week.

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Table of Nominees

Nominated article	Vote Score	Supporters	Specialist supporters	Date created
The Sporting Life (album)	2	Meg Ireland; Drew R. Smith		2009-07-21
The Dead Weather	1	Drew R. Smith		2009-07-15
Screaming Lord Sutch	1	Howard C. Berkowitz		2009-07-24
The Rolling Stones	1	Daniel Mietchen 09:35, 30 July 2009 (UTC)		2009-07-28 Candidate title	add 2 for each specialist supporter, 1 for every other supporter	sign with four tildes if you are not an editor in one of the article's workgroups	sign with four tildes if you are an editor in one of the article's workgroups	add -->

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Recently created pages are listed on Special:NewPages.

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The next New Draft of the Week will be the article with the most votes at 1 AM UTC on Thursday, 6 August 2009. I did the honors this time. Milton Beychok 01:02, 30 July 2009 (UTC)

Nominated article	Supporters	Specialist supporters	Dates	Score
The Dead Weather: An American Alternative Rock band formed in Nashville, Tennessee in 2009. [e] The Dead Weather is a "a death-rattle blues quartet" formed in Nashville, Tennessee in 2009. Formed by Jack White of the White Stripes and the Raconteurs, the band is comprised of VV (Alison Mosshart) of the Kills, on guitar and vocals; Dean Fertita of Queens of the Stone Age, on guitar and organ; Little Jack (Jack Lawrence) of the Raconteurs and the Greenhornes, on bass; and Jack White on drums and vocals. The band was revealed, and performed for the first time, at the opening of Third Man Records' Nashville headquarters on March 11, 2009, immediately before releasing their debut single "Hang You from the Heavens". Band members Jack White Jack White formed the White Stripes in 1997, and they went on to have a string of critically acclaimed albums, with their third, White Blood Cells, catapulting them to international stardom. In 2005, White became a founding member of the rock band the Raconteurs. In 2009, he became a founding member of his third group, the Dead Weather. Jack White plays drums, and provides vocals for the Dead Weather. Alison Mosshart "VV" Alison Mosshart started her musical career in 1995 with the Florida punk rock outfit Discount, which disbanded in 2000. She then co-founded the Kills with British guitarist Jamie Hince in 2000. Alison Mosshart plays guitar and provides vocals for the Dead Weather. (Read more...)	Drew R. Smith		1
The Sporting Life (album): A 1994 studio album recorded by Diamanda Galás and John Paul Jones. [e] This article may be deleted soon. To oppose or discuss a nomination, please go to CZ:Proposed for deletion and follow the instructions. For the monthly nomination lists, see Category:Articles for deletion. The metadata subpage is missing. You can start it via filling in this form or by following the instructions that come up after clicking on the [show] link to the right. Do you see this on PREVIEW? then SAVE! before following a link. To find out more about the use of metadata for the management of Citizendium pages, see CZ:Metadata. You may see this box for one of two reasons. Either: A - New page has been created, or subpages template was added to an existing page B - Cluster move is in progress Click either the A or B link for further instructions. [Feel free to recommend improvements to the text or links in this template] A - For a New Cluster use the following directions Subpages format requires a metadata page. Using the following instructions will complete the process of creating this article's subpages. Click the blue "metadata template" link below to create the page. On the edit page that appears paste in the article's title across from "pagename =". You might also fill out the checklist part of the form. Ignore the rest. For background, see Using the Subpages template Don't worry--you'll get the hang of it right away. Remember to hit Save! the "metadata template". However, you can create articles without subpages. Just delete the {{subpages}} template from the top of this page and this prompt will disappear. :) Don't feel obligated to use subpages, it's more important that you write sentences, which you can always do without writing fancy code. [Feel free to recommend improvements to the text or links in this template] B - For a Cluster Move use the following directions The metadata template should be moved to the new name as the first step. Please revert this move and start by using the Move Cluster link at the top left of the talk page. The name prior to this move can be found at the following link. [Feel free to recommend improvements to the text or links in this template] The Sporting Life Image:SportingLifealbum1994.jpg Type Studio album Artist Diamanda Galás and John Paul Jones (musician) Release Date 6 September 1994 (US), 1 October 1994 (UK) Recorded 1994 at AIR Studios, London. Mastered at Metropolis Mastering, London. Genre Alternative rock, experimental rock Language English Length 55 minutes 4 seconds Label Mute Records Catalogue Mute 61672 (US), Mute STUMM 127 (UK) Producer John Paul Jones (musician) Engineer Richard Evans The Sporting Life (album)|The Sporting Life is an album by avant-garde singer Diamanda Galás and multi-instrumentalist John Paul Jones (musician)|John Paul Jones, released on 6 September 1994, on Mute Records. It is Galás' sixth studio album, produced by Jones. Overview The record was a notable shift in musical style from what was previously produced with Galás. The album's subject material incorporated various lyrical themes such as lust, revenge, violence, murder, betrayal, and death, mixed with sarcasm and dark humour. Galás' previous work centred around the theme of AIDS and subsequent deaths from it, after her brother and some of her friends succumbed to the disease. This album was a change in direction, albeit a much more commercially accessible one, with the additional input of former Led Zeppelin bass player and keyboardist John Paul Jones on songwriting, performance and production duties. Jones was working on a live project with Heart (band)|Heart at the time of this collaboration. When interviewed about his involvement Jones explained: ‘ I was immediately impressed with her voice, and the power and the emotion. Our backgrounds are very similar. We both played in our fathers' bands when we were starting out, and we're both great fans of classical music, jazz, blues, Mediterranean music and Arabic music. A mutual friend suggested that we should work together, and I think she wanted to do a Rock music ’ —rock record. Jones had previously been introduced to Galás' body of work in 1983 via his wife Maureen, who brought home and played a 1982 compact disc containing 'Wild Women with Steak Knives (The Homicidal Love Song for Solo Scream)'. Galás intimated in an interview with online magazine Convulsion that she wanted to explore other facets of life outside of the AIDS epidemic, such as 'sick relationships.' The 'sporting life' is street hustler slang for the place where affection is a commodity, where the only love that's not for sale is obsessive and destructive love. Writing and composition After an initial evening meeting in London where they both agreed to work together, the album was composed both in New York (disambiguation)|New York (Galás) and in England (Jones). Demo tapes were exchanged, with Jones providing compositional ideas on an 8 string bass, while Galás added organ and lyrics to them. This continued until they were both satisfied with the material and a rehearsal arranged. The recording process took three weeks in London, in which Thomas laid down drum tracks in the third and final week. With mixing, the entire project took two months in total. Galás described the project as a collection of homicidal love songs: ‘ I think that if you get together and decide to do an album called Homicidal Love Songs, which is what I originally wanted to call it, you've got to have a sense of humor. What makes this album possible in terms of lyrics is real life experience. Every single song, and John [Paul Jones] knows it, too, has a particular person in mind with whom I have had various provocations and entanglements. ’ On the recording, Galás utilises both Hammond B3 organ and piano. The drummer used for the album was Pete Thomas, from Elvis Costello's backing band the Attractions. Due to his commitments with The Attractions, former Heart drummer Denny Fongheiser performed in place of Thomas for subsequent live appearances. Galás has a 3½ octave vocal range which she uses to full effect on the album. Jones commented that 'With a voice like Diamanda's, a guitar would simply be a cosmetic device.' The album opens with 'Skótoseme' which is Greek for 'kill me', on which Galás meshes her frantic, operatic voice with Jones' funky bass lines. A merging of heavy rock and Ancient Greek tragedy. The song 'Do You Take This Man?' mocks wedding vows by comparing romance to imprisonment, where the only solution is submission. Galás' haunting cover of Percy Sledge's soul classic 'Dark End of the Street,' reminds lovers that death is not far away. The title track 'The Sporting Life' meanwhile, tells the tale of a group of prostitutes who stalk and kill a man for fun. Death laments melt into Hammond-driven Cajun dance music on 'You're Mine'. 'Last Man Down', is a wailing blues drenched song about isolation and despair, with lap steel guitar embellishment. Release and critical reception Image:Thesportinglifepromo.jpg|thumb|200px|right|A screenshot from the music video, depicting Diamanda Galás and John Paul Jones. Alternate cover picture outtakes with Galás and Jones in a convertible, with Jones driving, appeared in a 1994 issue of Musician (magazine)|Musician magazine. One of these images was used for the CD single cover of 'Do You Take This Man?' A video was filmed for the single and was completed on 29 July 1994, with director Jon Reiss. It featured Galás in various scenes throughout New York, with Jones playing an eight string bass guitar, and released on 22 August 1994 prior to the album launch. Jones also used 4, 5, and 6 string basses plus a lap steel guitar, while the pair toured throughout October 1994 - January 1995 to promote the album, with concerts billed as 'An Evening of Homicidal Love Songs.' This was Jones' first full tour since Led Zeppelin's 1980 Tour Over Europe 1980|Over Europe tour. The album was released to mixed reviews. Entertainment Weekly offered the most positive with a B+ rating: '... Led Zeppelin bassist Jones and Pete Thomas join the demonic diva to form a power-mad trio on Galas' most rocking album yet ...' while Musician magazine went further: '... At times, The Sporting Life sounds like the best album Led Zeppelin never made, with Galas' acrobatic, banshee-giving-birth caterwaul subbing for Jimmy Page feedback in a newfangled power trio ...' Bomb (magazine)|Bomb arts magazine described it as: '... An expanse where divisions between what is electronic and what is "natural" smear, where the instrument is only as good as its ability to scalp you ...' Track list Album information 1994 Vinyl listing: Side One: 'Skótoseme' (Diamanda Galás, John Paul Jones (musician)|John Paul Jones) – 6:27 'Do You Take This Man?' (Diamanda Galás, John Paul Jones) – 6:09 'The Dark End of the Street' (Chips Moman, Dan Penn) – 2:43 'You're Man' (Diamanda Galás) – 5:10 'Tony' (Diamanda Galás) – 5:37 Side Two: 'Devil's Rodeo' (Diamanda Galás, John Paul Jones, Pete Thomas) – 5:37 'The Sporting Life' (Diamanda Galás, John Paul Jones) – 5:45 'Baby's Insane' (Diamanda Galás) – 4:39 'Last Man Down' (Diamanda Galás, John Paul Jones) - 4:50 'Hex' (Diamanda Galás, John Paul Jones) - 8:04 Chart positions Album Chart (1994) Peak position US Billboard The 200 Albums Chart — Singles Year Single Chart Position 1994 'Do You Take This Man?' US Billboard Hot 100 Chart — Credits Personnel Musicians: Diamanda Galás – vocals, keyboards John Paul Jones – bass guitar, electric guitars, keyboards, piano, bass pedals, producer Pete Thomas - drums, percussion Production: Richard Evans – engineer, mixing Tim Young - mastering engineer Patricia Mooney - art direction Catherine McGann - photography Notes  (Read more...) Meg Ireland; Drew R. Smith 2  Screaming Lord Sutch: (10 November 1940 - 16 June 1999), English singer and politician, leader of the Official Monster Raving Loony Party. [e] The metadata subpage is missing. You can start it via filling in this form or by following the instructions that come up after clicking on the [show] link to the right. Do you see this on PREVIEW? then SAVE! before following a link. To find out more about the use of metadata for the management of Citizendium pages, see CZ:Metadata. You may see this box for one of two reasons. Either: A - New page has been created, or subpages template was added to an existing page B - Cluster move is in progress Click either the A or B link for further instructions. [Feel free to recommend improvements to the text or links in this template] A - For a New Cluster use the following directions Subpages format requires a metadata page. Using the following instructions will complete the process of creating this article's subpages. Click the blue "metadata template" link below to create the page. On the edit page that appears paste in the article's title across from "pagename =". You might also fill out the checklist part of the form. Ignore the rest. For background, see Using the Subpages template Don't worry--you'll get the hang of it right away. Remember to hit Save! the "metadata template". However, you can create articles without subpages. Just delete the {{subpages}} template from the top of this page and this prompt will disappear. :) Don't feel obligated to use subpages, it's more important that you write sentences, which you can always do without writing fancy code. [Feel free to recommend improvements to the text or links in this template] B - For a Cluster Move use the following directions The metadata template should be moved to the new name as the first step. Please revert this move and start by using the Move Cluster link at the top left of the talk page. The name prior to this move can be found at the following link. [Feel free to recommend improvements to the text or links in this template] Screaming Lord Sutch, 3rd Earl of Harrow (born David Edward Sutch, 10 November 1940 - 16 June 1999) was an English musician and politician. As founder of the Official Monster Raving Loony Party, he was the longest serving leader of a British political party. Career Sutch was born at New End Hospital, Hampstead, North West London, and grew up in the working class area of Kilburn, North London. His father, a policeman, was killed in the Blitz, while his mother worked as a cleaner. Sutch fell in love with rock 'n' roll upon hearing 'Rock Around the Clock' in 1956 on his crystal radio set. His earliest public performances were at a local biker hangout called the Cannibal Pot Coffee Bar, on Harrow Road. Inspired by one of his favourite rock and roll stars, Screaming Jay Hawkins, he changed his name to Screaming Lord Sutch, 3rd Earl of Harrow. Despite the fact that he had no connection with the peerage, the deed poll laws of England at the time permitted this. He was one of the first musicians in England in the 1960s with long hair, reputedly 18 inches long, a fashion statement of rebellion that soon caught on with other rock artists. Sutch was aware of his weaknesses as a melodic vocalist, and relied heavily on image, taking his lead from Screaming Jay Hawkins, and putting on a show that mixed rock 'n' roll, theatrics and gothic horror. Among the stage-props he used were an axe, daggers, skulls and a black coffin he emerged from dressed as Jack the Ripper. In 1961 while playing at London, United Kingdom's famed 2 I's coffee bar in Soho, he was discovered by maverick producer Joe Meek. Sutch then assembled his first backing group, the Raving Savages. Recording at Meek's home studio in Holloway Road, Islington, Sutch and the Savages often had their records banned by the BBC, a fact which only served to give them more publicity and notoriety. Although never achieving any hit records, the Savages were an accomplished live band featuring Ritchie Blackmore, Andy Wren, and Carlo Little. Their early releases include 'Til the Following Night' (their debut a-side), 'Jack the Ripper', 'Dracula's Daughter', and 'I'm a Hog for You Baby'. Sutch once changed the name of the band to Lord Caesar Sutch & the Roman Empire, with his band dressed as Roman soldiers and Sutch dressed in a toga riding around to gigs in a horse drawn chariot, although this did not last long. Sutch's relationship with Meek ended when the two fell out over money that Sutch claimed Meek owed him. The Savages went through numerous line-up changes throughout the 1960s, including the departure of guitarist Blackmore who later went on to form Deep Purple. In January 1968, Sutch left England to tour and establish himself in the United States of America, and for most of the late 1960s and early 1970s spent increasing time living in Los Angeles. During Led Zeppelin's Spring tour of the US in 1969, he met with manager Peter Grant, whom he persuaded to assist him in brokering a recording deal with Atlantic Records. Sutch also met guitarist Jimmy Page, who was scouting studios to record in during their tour, and managed to get Page and John Bonham to test the facilities at studio Mirror Sounds in May 1969, by recording a number of rock standards. He later secretly recorded a set of his own lyrics over the standards they performed. Sutch enjoyed minor success in 1970 with the album Lord Sutch and Heavy Friends, which featured Page, Bonham, Jeff Beck, Nicky Hopkins, and Noel Redding. The musicians involved were however unimpressed with Sutch, believing the recordings to be for a studio test only. In 1971, Sutch assembled a new line-up for the Carshalton Park Rock 'n' Roll Festival, and had the evening secretly taped. This later appeared in 1972 as Hands of Jack the Ripper featuring Keith Moon, Ritchie Blackmore, and Nick Simper, but it failed to chart. To promote his albums, Sutch toured the United States of America in a Union Jack painted 1955 Rolls Royce Silver Wraith. In 1972, he was also arrested and later released without charge, when he accompanied five naked women to Downing Street to protest at the shortage of rock music on the BBC. He recorded his final album Alive & Well in 1980, featuring Pat Travers, Rick Nielsen, and Klaus Voormann. Numerous compilation albums have proliferated after the 1980s. Pirate radio In May 1964, taking advantage of a loophole in the UK broadcast law, he founded Radio Sutch, one of the first pirate radio stations in Britain, often playing his own records and those of his friends, with late night readings of a bawdy nature. Early broadcasts were transmitted from The Cornucopia, a fishing trawler, and were fraught with technical problems. The vessel was used in the early morning for fishing, and after midday it would begin broadcasting its programmes. The station finally moved to Shivering Sands, a disused wartime fort, located in the Thames Estuary off Southend. Now in competition with the newly launched Radio Caroline and hampered by a weaker transmission signal to London, Sutch began losing interest in the venture and he sold the station in September of that year to his manager Reg Calvert, who re-launched it as Radio City. Political aspirations In 1963, Sutch made his first foray into politics, standing as a parliamentary candidate (as Lord Sutch) for his own National Teenage Party (NTP) in Stratford-Upon-Avon. Straford-Upon-Avon was previously held by conservative MP John Profumo, who had to resign over the Profumo Affair. The NTP's platform included reducing the voting age to 18, establishing commercial radio, abolition of the 11-plus exams, and all-day opening of pubs, all of which have since become law in Britain. Sutch received a mere 208 votes but his taste for politics had been whetted and over the next 30 years he stood in 40 elections - more than any other candidate in history, and earning him a place in the Guinness Book of Records. As founder of the Monster Raving Loony Party, which replaced the NTP, he eventually became the longest surviving leader of any UK political party, despite the fact that he lost his own deposit at every election. In total, Sutch stood for Parliament 39 times, plus 1 Euro-election in 1989, polling some 15,000 votes, forfeiting more than £10,000 in lost deposits and incurring £85,000 in campaign expenses. Sutch formerly registered the Monster Raving Loony Party in 1983, and campaigned for more than one Monopolies Commission, for the European butter mountain to be turned into a ski slope, and to breed fish in the wine lake, 'so they'd come out ready pickled!' Sutch would appear during campaigns with his trademark megaphone, top hat and leopardskin shirt. The party's official slogan was: 'Vote for insanity, you know it makes sense'. In the ITV comedy series The New Statesman, he appeared as himself in the 1987 elections, securing second place against the SDP and Labour candidates. A Heineken Pilsener commercial in the 1990s featured Sutch as Prime Minister at 10 Downing Street. In 1991 Sutch published his autobiography Life As Sutch (with Peter Chippendale) published by Harper-Collins. Sutch's strongest showing was at Rotherham in May 1994 when he polled 1,114 votes, only some 200 short of the number required for saving his deposit. His most significant result, however, was at Bootle in May 1990, when he scored 418 votes to the Social Democrats' 155, a result which helped to convince Dr David Owen that his party was finished. In 1995, Sutch was facing bankruptcy when Barclays Bank threatened to foreclose on a loan of £194,000, but William Hill, the bookmakers agreed to finance his election deposits, and the bank re-scheduled his repayments, so that in July he was able to record one of his best results (782 votes) at the Littleborough and Saddleworth by-election. In the 1997 general election, English betting agency Ladbrokes gave odds of fifteen million to one against Sutch ever being Prime Minister of England, their greatest odds ever. They only gave fourteen million to one against little green men being found on Mars. Although he didn't win the seat, Sutch in the end scored 10 times the vote of Dr Alan Sked's UK Independence Party. It would also prove to be the party's final national election campaign as Sutch could no longer afford the deposits, which had been increased for every candidate from £150 to £500. Death Despite his seemingly light-hearted antics, Sutch in reality suffered from periods of depression and committed suicide by hanging himself at home on 16 June 1999, affected by the sudden death of his mother Annie Emily Sutch in late 1997. He had been previously booked to appear at a show in Las Vegas, and tour the US. At the coroner's inquest into his death, his fiancée Yvonne Elwood stated that he had 'manic depression'. Indeed, her descriptions, both of his symptoms and their treatment by doctors with antidepressants, suggest that Sutch suffered from clinical depression. Sutch never married, but is survived by a son, Tristan Lord Gwynne Sutch, born in 1975 to the American model Thann Rendessy. Party chairman Alan 'Howlin Laud' Hope, took over as Monster Raving Loony Party leader after Sutch's death. A spokesman for Prime Minister Tony Blair stated: Screaming Lord Sutch will be much missed. For many years he made a unique contribution to British politics. Our elections will never be quite the same without him. The 2008 Joe Meek biopic Telstar featured singer Justin Hawkins as Screaming Lord Sutch. Notes  (Read more...) Howard C. Berkowitz 1 Current Winner (to be selected and implemented by an Administrator) To change, click edit and follow the instructions, or see documentation at {{Featured Article}}. The metadata subpage is missing. You can start it via filling in this form or by following the instructions that come up after clicking on the [show] link to the right. Do you see this on PREVIEW? then SAVE! before following a link. To find out more about the use of metadata for the management of Citizendium pages, see CZ:Metadata. You may see this box for one of two reasons. Either: A - New page has been created, or subpages template was added to an existing page B - Cluster move is in progress Click either the A or B link for further instructions. [Feel free to recommend improvements to the text or links in this template] A - For a New Cluster use the following directions Subpages format requires a metadata page. Using the following instructions will complete the process of creating this article's subpages. Click the blue "metadata template" link below to create the page. On the edit page that appears paste in the article's title across from "pagename =". You might also fill out the checklist part of the form. Ignore the rest. For background, see Using the Subpages template Don't worry--you'll get the hang of it right away. Remember to hit Save! the "metadata template". However, you can create articles without subpages. Just delete the {{subpages}} template from the top of this page and this prompt will disappear. :) Don't feel obligated to use subpages, it's more important that you write sentences, which you can always do without writing fancy code. [Feel free to recommend improvements to the text or links in this template] B - For a Cluster Move use the following directions The metadata template should be moved to the new name as the first step. Please revert this move and start by using the Move Cluster link at the top left of the talk page. The name prior to this move can be found at the following link. [Feel free to recommend improvements to the text or links in this template] PD Image Figure 1. Euler angles. From left to right: initial configuration, after rotation over angle α, after rotation over angle β, and after rotation over angle γ. In physics, mathematics, and engineering, Euler angles are three rotation angles, often denoted by 0 ≤ α ≤ 2π, 0 ≤ β ≤ π, and 0 ≤ γ ≤ 2π, although the notation φ, θ, ψ is also common. Any rotation of a 3-dimensional object can be performed by three consecutive rotations over the three Euler angles. Different conventions are in use: a rotation can be active (the object is rotated, the system of axes is fixed in space), or passive (the object is fixed in space, the axes are rotated). Also the choice of rotation axes may vary; an active convention common in quantum mechanical applications is the z-y′-z′ convention. Attach a system of Cartesian coordinate axes to the body that is to be rotated (the coordinate frame is fixed to the body and is rotated simultaneously with it); in the figure the body-fixed frame is shown in red and labeled by lowercase letters. First rotate around z, then around the new body-fixed y-axis, y′, and finally around z′. Another convention often used is the z-x′-z′ convention, where instead of over the new y-axis the second rotation is over the new x-axis. Also the z-y-x convention is used (and will be discussed below). The right-hand screw rule is practically always followed: the rotation axis is a directed line and a positive rotation is as a cork screw driven into the positive direction of the axis. In older literature left-handed Cartesian coordinate frames appear sometimes, but in modern literature right-handed frames are used exclusively. Euler angles are used in many different branches of physics and engineering. The present article is written from the point of view of molecular physics, where the objects to be rotated are molecules and applications are often of quantum mechanical nature. The angles are named after the 18th century mathematician Leonhard Euler who introduced in 1765 two of the three for an axially symmetric body where the third angle, γ, does not play a role. Geometric discussion In Figure 1 the space-fixed (laboratory) axes are labeled by capital X, Y, and Z and are shown in black. The body to be rotated is not shown, but a system of axes fixed to it is shown in red. One may use any convenient orthonormal frame as a body-fixed frame. Often the body-fixed axes are principal axes, that means that they are eigenvectors of the inertia tensor of the body. Also symmetry axes, when present, may be used. When the body has symmetry axes, the principal axes often coincide with these. PD Image Figure 2. Rotation of r to r′. On the left around z-axis over α (φ increases), on the right around y-axis over β (φ decreases). Both rotation axes point toward the reader. The z-y′-z′ convention will be followed. Initially, the two frames coincide, and the path to a final arbitrary orientation of the body—and its frame—is depicted on Figure 1. The first rotation is around the z-axis, which coincides with the Z-axis. The x- and y-axis move in a plane perpendicular to the z-axis over an angle α. The second rotation is in a plane through the origin perpendicular to the y′-axis. The angle is β. The present convention has the practical advantage that the z′-axis has the usual spherical polar coordinates α ≡ φ (longitude angle) and β ≡ θ (colatitude angle) with respect to the space-fixed frame. The final rotation is in a plane perpendicular to the z′-axis over an angle γ. From geometric considerations follows that any orientation of the body-fixed frame in space may be obtained. Write $\mathbf {R} (\varphi ,{\hat {n}})$ for the rotation matrix that describes a rotation around the unit vector ${\hat {n}}$ over an angle $\varphi$. Clearly the three consecutive Euler rotations correspond to rotations around the unit vectors along the body-fixed axes z, y′, and z′ over angles α β, and γ, respectively. Because a matrix acts on a column vector to its right, the order in the matrix product is as in the leftmost term in the following equation. It will be shown that the corresponding matrix product can be written in reverse order (but around fixed, unprimed, axes z, y, z), that is, ${\begin{aligned}\mathbf {R} (\gamma ,{\hat {e}}_{z'})\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\alpha ,{\hat {e}}_{z})&=\mathbf {R} (\alpha ,{\hat {e}}_{z})\mathbf {R} (\beta ,{\hat {e}}_{y})\mathbf {R} (\gamma ,{\hat {e}}_{z})\\&={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}\cos \beta &0&\sin \beta \\0&1&0\\-\sin \beta &0&\cos \beta \\\end{pmatrix}}{\begin{pmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\\\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \beta \cos \gamma -\sin \alpha \cos \gamma \;&\;-\cos \alpha \cos \beta \sin \gamma -\sin \alpha \cos \gamma \;&\;\cos \alpha \sin \beta \\\sin \alpha \cos \beta \cos \gamma +\cos \alpha \sin \gamma \;&\;-\sin \alpha \cos \beta \sin \gamma +\cos \alpha \cos \gamma \;&\;\sin \alpha \sin \beta \\-\sin \beta \cos \gamma \;&\;\sin \beta \sin \gamma \;&\;\cos \beta \end{pmatrix}}\end{aligned}}$ Note that the third column contains the Cartesian coordinates with respect to the space-fixed frame of ${\hat {e}}_{z''}$ expressed in sines and cosines of spherical polar angles. The first and second column contain by definition expressions for the Cartesian coordinates of ${\hat {e}}_{x'''}$ and ${\hat {e}}_{y''}$, respectively, but evidently these are not solely in terms of spherical polar angles, γ also enters. Before proving the first equality in the above equation (reversal of order), we derive the matrix for a rotation around the z-axis, see the left drawing in Figure 2. The rotated vector has components ${\begin{pmatrix}\cos(\alpha +\phi )\\\sin(\alpha +\phi )\\0\end{pmatrix}}={\begin{pmatrix}\cos \alpha \cos \phi -\sin \alpha \sin \phi \\\sin \alpha \cos \phi +\cos \alpha \sin \phi \\0\end{pmatrix}}={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}\cos \phi \\\sin \phi \\0\\\end{pmatrix}}\equiv \mathbf {R} (\alpha ,{\hat {e}}_{z}){\begin{pmatrix}\cos \phi \\\sin \phi \\0\end{pmatrix}}.$ We used here the relations well-known from trigonometry for the sine and cosine of a sum angle. The derivation of the matrix for a rotation around the y-axis proceeds along the same lines. Note, however, that the angle of a vector with the x-axis decreases by a rotation around the positive y-axis (see right-hand drawing in Figure 2). To prove the first equality (reversal of the order in the angles), a property of rotation matrices is used. A rotation (orthogonal 3×3) matrix A, transforming a rotation axis, gives rise to the following similarity equation, $\mathbf {R} (\varphi ,\mathbf {A} {\hat {n}})=\mathbf {A} \mathbf {R} (\varphi ,{\hat {n}})\mathbf {A} ^{\mathrm {T} },$ where the superscript T indicates the transpose of the matrix. For rotation matrices the transposed matrix is equal to the inverse of the matrix. From this similarity relation follows that $\mathbf {R} (\gamma ,{\hat {e}}_{z'})=\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\gamma ,{\hat {e}}_{z})\mathbf {R} (\beta ,{\hat {e}}_{y'})^{\mathrm {T} },$ so that $\mathbf {R} (\gamma ,{\hat {e}}_{z'})\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\alpha ,{\hat {e}}_{z})=\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\gamma ,{\hat {e}}_{z})\mathbf {R} (\beta ,{\hat {e}}_{y'})^{\mathrm {T} }\;\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\alpha ,{\hat {e}}_{z})=\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\gamma ,{\hat {e}}_{z})\mathbf {R} (\alpha ,{\hat {e}}_{z}).$ Also $\mathbf {R} (\beta ,{\hat {e}}_{y'})=\mathbf {R} (\alpha ,{\hat {e}}_{z})\mathbf {R} (\beta ,{\hat {e}}_{y})\mathbf {R} (\alpha ,{\hat {e}}_{z})^{\mathrm {T} },$ so that $\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\gamma ,{\hat {e}}_{z})\mathbf {R} (\alpha ,{\hat {e}}_{z})=\mathbf {R} (\alpha ,{\hat {e}}_{z})\mathbf {R} (\beta ,{\hat {e}}_{y})\mathbf {R} (\alpha ,{\hat {e}}_{z})^{\mathrm {T} }\;\mathbf {R} (\gamma ,{\hat {e}}_{z})\mathbf {R} (\alpha ,{\hat {e}}_{z})=\mathbf {R} (\alpha ,{\hat {e}}_{z})\mathbf {R} (\beta ,{\hat {e}}_{y})\mathbf {R} (\gamma ,{\hat {e}}_{z}),$ where it is used that rotations around the same axis commute, that is, $\mathbf {R} (\alpha ,{\hat {e}}_{z})^{\mathrm {T} }\mathbf {R} (\gamma ,{\hat {e}}_{z})\mathbf {R} (\alpha ,{\hat {e}}_{z})=\mathbf {R} (\gamma ,{\hat {e}}_{z})\mathbf {R} (\alpha ,{\hat {e}}_{z})^{\mathrm {T} }\mathbf {R} (\alpha ,{\hat {e}}_{z})=\mathbf {R} (\gamma ,{\hat {e}}_{z})$ and the required result is proved. Algebraic treatment In the proof that any rotation can be written as three consecutive rotations, an appeal was made to the geometric insight of the reader. The same result can be proved more rigorously by algebraic means. To that end the notation is somewhat shortened: $\mathbf {R} _{z}(\omega )\equiv \mathbf {R} (\omega ,{\hat {e}}_{z})\quad {\hbox{and}}\quad \mathbf {R} _{y}(\omega )\equiv \mathbf {R} (\omega ,{\hat {e}}_{y}).$ Theorem A proper rotation matrix R can be factorized thus $\mathbf {R} =\mathbf {R} _{z}(\omega _{3})\;\mathbf {R} _{y}(\omega _{2})\;\mathbf {R} _{x}(\omega _{1})$ which is referred to as the Euler z-y-x parametrization, or also as $\mathbf {R} =\mathbf {R} _{z}(\alpha )\;\mathbf {R} _{y}(\beta )\;\mathbf {R} _{z}(\gamma )\quad$ the Euler z-y-z parametrization. Proof First the Euler z-y-x-parametrization will be proved by an algorithm for the factorization of a given matrix R ≡ (r1, r2, r3). Second the z-y-z parametrization will be proved; this parametrization is—as shown above—equivalent to the z′-y′-z parametrization with angles in reverse order. :A Fortran subroutine based on the algorithm is given on the code page. To prove the z-y-x parametrization we consider the matrix product $\mathbf {R} _{z}(\omega _{3})\,\mathbf {R} _{y}(\omega _{2})={\begin{pmatrix}\cos \omega _{3}\cos \omega _{2}&-\sin \omega _{3}&\cos \omega _{3}\sin \omega _{2}\\\sin \omega _{3}\cos \omega _{2}&\cos \omega _{3}&\sin \omega _{3}\sin \omega _{2}\\-\sin \omega _{2}&0&\cos \omega _{2}\\\end{pmatrix}}\equiv (\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3}).$ The columns of the matrix product are for ease of reference designated by a1, a2, and a3. Note that the multiplication by $\mathbf {R} _{x}(\omega _{1})\equiv {\begin{pmatrix}1&0&0\\0&\cos \omega _{1}&-\sin \omega _{1}\\0&\sin \omega _{1}&\cos \omega _{1}\end{pmatrix}}$ on the right does not affect the first column, so that a1 = r1 (the first column of R). Solve $\omega _{2}\;$ and $\omega _{3}\;$ from the first column of R (which is known), $\mathbf {a} _{1}={\begin{pmatrix}\cos \omega _{3}\;\cos \omega _{2}\\\sin \omega _{3}\;\cos \omega _{2}\\-\sin \omega _{2}\\\end{pmatrix}}={\begin{pmatrix}R_{11}\\R_{21}\\R_{31}\\\end{pmatrix}}\equiv \mathbf {r} _{1}.$ This is possible. First solve $\omega _{2}\;$ for $-\pi /2\leq \omega _{2}\leq \pi /2$ from $\sin \omega _{2}=-R_{31}.\,$ Then solve $\omega _{3}\;$ for $0\leq \omega _{3}\leq 2\pi$ from the two equations: $\cos \omega _{3}={R_{11} \over \cos \omega _{2}},\qquad \sin \omega _{3}={R_{21} \over \cos \omega _{2}}.$ The angles $\omega _{2}\;$ and $\omega _{3}\;$ determine fully the vectors a2 and a3. Since a1, a2 and a3 are the columns of a proper rotation matrix they form an orthonormal right-handed system. The plane spanned by a2 and a3 is orthogonal to $\mathbf {a} _{1}\equiv \mathbf {r} _{1}$ and hence the plane contains $\mathbf {r} _{2}$ and $\mathbf {r} _{3}$. Thus the latter two vectors are a linear combination of the first two, $(\mathbf {r} _{2},\mathbf {r} _{3})=(\mathbf {a} _{2},\mathbf {a} _{3}){\begin{pmatrix}\cos \omega _{1}&-\sin \omega _{1}\\\sin \omega _{1}&\cos \omega _{1}\\\end{pmatrix}}.$ Since $\mathbf {r} _{2},\;\mathbf {a} _{2},\;\mathbf {a} _{3}$ are known orthonormal vectors, we can compute ${\begin{aligned}\mathbf {a} _{2}\cdot \mathbf {r} _{2}=&\cos \omega _{1}\\\mathbf {a} _{3}\cdot \mathbf {r} _{2}=&\sin \omega _{1}.\end{aligned}}$ These equations give $\omega _{1}\;$ with $0\leq \omega _{1}\leq 2\pi$. The angle ω1 gives the matrix $\mathbf {R} _{x}(\omega _{1})$ with ${\begin{aligned}\mathbf {R} \equiv (\mathbf {r} _{1},\mathbf {r} _{2},\mathbf {r} _{3})=(\mathbf {r} _{1},\mathbf {a} _{2},\mathbf {a} _{3})\mathbf {R} _{x}(\omega _{1})=(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3})\mathbf {R} _{x}(\omega _{1})=\mathbf {R} _{z}(\omega _{3})\,\mathbf {R} _{y}(\omega _{2})\,\mathbf {R} _{x}(\omega _{1}).\end{aligned}}$ This gives the required z-y-x factorization of the arbitrary proper orthogonal matrix R. The proof of the Euler z-y-z parametrization is obtained by a small modification of the previous proof. We start by retrieving the spherical polar coordinates $\omega _{2}\;$ and $\omega _{3}\;$ of the unit vector $\mathbf {r} _{3}=\mathbf {a} _{3}$, the third column [the rightmost multiplication by Rz(ω1) does not affect r3]. Then consider $(\mathbf {r} _{1},\;\mathbf {r} _{2})=(\mathbf {a} _{1},\;\mathbf {a} _{2}){\begin{pmatrix}\cos \omega _{1}&-\sin \omega _{1}\\\sin \omega _{1}&\cos \omega _{1}\\\end{pmatrix}}$ or, $\mathbf {a} _{1}\cdot \mathbf {r} _{1}=\cos \omega _{1}\;,\quad \mathbf {a} _{2}\cdot \mathbf {r} _{1}=\sin \omega _{1}.$ The equation for R can be written as $(\mathbf {r} _{1},\mathbf {r} _{2},\mathbf {r} _{3})=(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {r} _{3})\,\mathbf {R} _{z}(\omega _{1})=\mathbf {R} _{z}(\omega _{3})\,\mathbf {R} _{y}(\omega _{2})\,\mathbf {R} _{z}(\omega _{1})\;,$ which proves the Euler z-y-z parametrization. Clearly, this factorization is equal to the one given in the previous section, with $\omega _{3}\equiv \alpha ,\quad \omega _{2}\equiv \beta ,\quad \omega _{1}\equiv \gamma .$ Note (Read more...) 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