In geometry, a frustum is the portion of a solid that lies between one or two parallel planes cutting it. A right frustum is a parallel truncation of a right right cone. In computer graphics, the viewing frustum is the three-dimensional region, visible on the screen, it is formed by a clipped pyramid. In the aerospace industry, a frustum is the fairing between two stages of a multistage rocket, shaped like a truncated cone. If all the edges are forced to be identical, a frustum becomes a uniform prism. A frustum's axis is that of the original pyramid. A frustum is circular; the height of a frustum is the perpendicular distance between the planes of the two bases. Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the apex; the pyramidal frusta are a subclass of the prismatoids. Two frusta joined at their bases make a bifrustum; the volume formula of a frustum of a square pyramid was introduced by the ancient Egyptian mathematics in what is called the Moscow Mathematical Papyrus, written in the 13th dynasty: V = 1 3 h. where a and b are the base and top side lengths of the truncated pyramid, h is the height.
The Egyptians knew the correct formula for obtaining the volume of a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus. The volume of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex: V = h 1 B 1 − h 2 B 2 3 where B1 is the area of one base, B2 is the area of the other base, h1, h2 are the perpendicular heights from the apex to the planes of the two bases. Considering that B 1 h 1 2 = B 2 h 2 2 = B 1 B 2 h 1 h 2 = α,the formula for the volume can be expressed as a product of this proportionality α/3 and a difference of cubes of heights h1 and h2 only. V = h 1 α h 1 2 − h 2 α h 2 2 3 = α 3 By factoring the difference of two cubes one gets h1−h2 = h, the height of the frustum, α/3. Distributing α and substituting from its definition, the Heronian mean of areas B1 and B2 is obtained; the alternative formula is therefore V = h 3. Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary number, the square root of negative one.
In particular, the volume of a circular cone frustum is V = π h 3 where π is 3.14159265... and r1, r2 are the radii of the two bases. The volume of a pyramidal frustum whose bases are n-sided regular polygons is V = n h 12 cot π n where a1 and a2 are the sides of the two bases. For a right circular conical frustum Lateral surface area = π s = π 2 +
Phthinosuchus is an extinct genus of therapsids from the Middle Permian of Russia. Phthinosuchus is the sole member of the family Phthinosuchidae. Phthinosuchus may have been one of the most primitive therapsids, meaning that its ancestors may have branched off early from the main therapsid line. Phthinosuchus was named in 1954 by Ivan Yefremov, it is only known from the back of the skull, as the front of the skull was lost after description. Two species, P. discors and P. horissiaki, have been described. Phthinosuchus was 1.5 m long with a 20 cm skull, looked much like the Sphenacodontids, such as Dimetrodon and Sphenacodon. Its temporal fenestrae were larger than those of the Sphenacodontids, its jaw was slender, unlike other predatory therapsids, but like the other early therapsids, it was sprawling and carnivorous. Phthinosuchus is a member of the suborder Phthinosuchia, made for Phthinosuchus given that it did not fit into any other suborder. List of therapsids Evolution of mammals Dinocephalians
Fidelio, Op. 72, is Ludwig van Beethoven's only opera. The German libretto was prepared by Joseph Sonnleithner from the French of Jean-Nicolas Bouilly, with the work premiering at Vienna's Theater an der Wien on 20 November 1805; the following year, Stephan von Breuning helped shorten the work from three acts to two. After further work on the libretto by Georg Friedrich Treitschke, a final version was performed at the Kärntnertortheater on 23 May 1814. By convention, both of the first two versions are referred to as Leonore; the libretto, with some spoken dialogue, tells how Leonore, disguised as a prison guard named "Fidelio", rescues her husband Florestan from death in a political prison. Bouilly's scenario fits Beethoven's aesthetic and political outlook: a story of personal sacrifice and eventual triumph. With its underlying struggle for liberty and justice mirroring contemporary political movements in Europe, such topics are typical of Beethoven's "middle period". Notable moments in the opera include the "Prisoners' Chorus", an ode to freedom sung by a chorus of political prisoners, Florestan's vision of Leonore come as an angel to rescue him, the scene in which the rescue takes place.
The finale celebrates Leonore's bravery with alternating contributions of soloists and chorus. The work has a long and complicated history of composition: it went through three versions during Beethoven's career, some of the music was first written as part of an earlier, never-completed opera; the distant origin of Fidelio dates from 1803, when the librettist and impresario Emanuel Schikaneder worked out a contract with Beethoven to write an opera. The contract included free housing for Beethoven in the apartment complex, part of Schikaneder's large suburban theater, the Theater an der Wien. Beethoven was to set a new libretto by Schikaneder, entitled Vestas Feuer, he spent about a month composing music for it abandoned it when the libretto for Fidelio came to his attention. The time Beethoven spent on Vestas Feuer was not wasted, as two important numbers from Fidelio, Pizarro's "'Ha! Welch’ ein Augenblick!" and the duet "O namenlose Freude" for Leonora and Florestan, both originated as music for Vestas Feuer.
Beethoven remained as a resident of the Theater an der Wien for some time after he had abandoned Vestas Feuer for Fidelio, was freed from his obligations to Schikaneder when the latter was fired from his post as theater director in 1804. Fidelio itself, which Beethoven began in 1804 after giving up on Vestas Feuer, was first performed in 1805 and was extensively revised by the composer for subsequent performances in 1806 and 1814. Although Beethoven used the title Leonore, oder Der Triumph der ehelichen Liebe, the 1805 performances were billed as Fidelio at the theatre's insistence, to avoid confusion with the 1798 opera Léonore, ou L’amour conjugal by Pierre Gaveaux, the 1804 opera Leonora by Ferdinando Paer. Beethoven published the 1806 libretto and, in 1810, a vocal score under the title Leonore, the current convention is to use the name Leonore for both the 1805 and 1806 versions and Fidelio only for the final 1814 revision; the first version with a three-act German libretto adapted by Joseph Sonnleithner from the French of Jean-Nicolas Bouilly premiered at the Theater an der Wien on 20 November 1805, with additional performances the following two nights.
The success of these performances was hindered by the fact that Vienna was under French military occupation, most of the audience were French military officers. After this premiere, Beethoven was pressured by friends to revise and shorten the opera into just two acts, he did so with the help of Stephan von Breuning; the composer wrote a new overture. In this form the opera was first performed on 10 April 1806, with greater success. Further performances were prevented by a dispute between the theatre management. In 1814 Beethoven revised his opera yet again, with additional work on the libretto by Georg Friedrich Treitschke; this version was first performed at the Kärntnertortheater on 23 May 1814, again under the title Fidelio. The 17-year-old Franz Schubert was in the audience; the deaf Beethoven led the performance, "assisted" by Michael Umlauf, who performed the same task for Beethoven at the premiere of the Ninth Symphony. The role of Pizarro was taken by Johann Michael Vogl, who became known for his collaborations with Schubert.
This version of the opera was a great success, Fidelio has been part of the operatic repertory since. Although critics have noted the similarity in plot with Gluck's opera Orfeo ed Euridice—the underground rescue mission in which the protagonist must control, or conceal, his emotions in order to retrieve his or her spouse, we do not know whether or not Beethoven or any of the librettists had this in mind while constructing the opera. Beethoven can not be said to have enjoyed the difficulties posed by producing an opera. In a letter to Treitschke he said, "I assure you, dear Treitschke, that this opera will win me a martyr's crown. You have by your co-operation saved. For all this I shall be eternally grateful to you."The full score was not published until 1826, all three versions are known as Beethoven's Opus 72. The first performance outside Vienna took place in Prague on 21 November 1814, with a revival in