General solutions of the PDE can then be obtained from envelopes of such conoids.
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There are a lot of conoids with singular points, which are investigated in algebraic geometry.
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Like other ruled surfaces conoids are of high interest with architects, because they can be built using beams or bars. Right conoids can be manufactured easily: one threads bars onto an axis such that they can be rotated around this axis, only. Afterwards one deflects the bars by a directrix and generates a conoid (s. parabolic conoid).
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26 The curved ribs, extending from the same point on the wall, are spaced equidistant from each other, forming conoid shapes. The resulting conoids, however, require great compressive forces to keep shape. Spandrels usually provide pressure along the upper edge of the conoids.Trowles (2008); p.
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The development of decoration in fan vaults is notable in later forms, such as pendant vaulting. In pendant vaulting the form and ornamentation of the vault evolve as “pendants as elongated voussoirs are dropped from a constructive pointed arch, concealed above the vaulting, and form abutments to support the pendant conoids.”Banister 1905, p. 290.
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196 In the Henry VII Chapel, these spandrels are replaced with hanging pendants. The pendants still provide the compression necessary to support the conoids and add complexity to the aesthetics of the room. The pendants serve an additional structural purpose. The pendants are cut from single stones and inserted as wedge stones in the transverse arches.
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Archimedes begins On Spirals with a message to Dositheus of Pelusium mentioning the death of Conon as a loss to mathematics. He then goes on to summarize the results of On the Sphere and Cylinder (Περὶ σφαίρας καὶ κυλίνδρου) and On Conoids and Spheroids (Περὶ κωνοειδέων καὶ σφαιροειδέων). He continues to state his results of On Spirals.
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Second, the fans spring not from the walls of the Chapel, but from pendants placed about 2m from the walls.”Heyman 2000, p. 368. Here, concealed transverse arches intersect the conoids and provide support for the hanging pendants. While fan vaulting is purported to be confined to England, versions of pendant vaulting came to be characteristic of the Flamboyant period in France.
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16 Archimedes (died c. 212 BCE) is known to have studied conics, having determined the area bounded by a parabola and a chord in Quadrature of the Parabola. His main interest was in terms of measuring areas and volumes of figures related to the conics and part of this work survives in his book on the solids of revolution of conics, On Conoids and Spheroids.
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Both Colpodella and Perkinsus species have open sided truncated conoids (sometimes called pseudoconoids), rhoptries that occupy the length of the cell and smaller micronemes. Both the rhoptries and micronemes arise at the anterior portion of the cell. A three-layered pellicle lies beneath the plasma membrane and is otherwise composed of the alveolar membranes and widely separated microtubules that arise subapically. Some species have extrusive organelles (trichocysts).
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In some cases, the pendants are a large form of boss.Encyclopedia Britannica 2018. In his book on fan vaults, Walter Leedy defines the fan vault stating: “Fan vaults have the following specific interrelated visual and structural characteristics: (1) vaulting conoids of regular geometric form, (2) vertical ribs, each of consistent curvature and placement, (3) a distinct central spandrel panel, (4) ribs perpendicular to the vaulting surface, and (5) applied surface patterning.”Leedy 1980, p. 1.
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Of particular note are the pendant vaults at the Divinity School at Oxford, built in 1480 and designed by William Orchard, and at Henry VII Chapel at Westminster Abbey, built between 1503-1509 and plausibly designed by Robert and William Vertue.Heyman 2000, p. 367-369. The application of vaulting at the Divinity School at Oxford visually separates the arch from the conoids. According to Jacques Heyman, the vaulting at the divinity school intended to “astonish and delight” and possibly makes reference to “Villard’s lodge-book of c.
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On Conoids and Spheroids () is a surviving work by the Greek mathematician and engineer Archimedes ( 287 BC – 212 BC). Comprising 32 propositions, the work explores properties of and theorems related to the solids generated by revolution of conic sections about their axes, including paraboloids, hyperboloids, and spheroids. The principal result of the work is comparing the volume of any segment cut off by a plane with the volume of a cone with equal base and axis. The work is addressed to Dositheus of Pelusium.
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On the Sphere and Cylinder, Measurement of a Circle, On Conoids and Spheroids, On Spirals, On the Equilibrium of Planes, The Quadrature of the Parabola, and The Sand Reckoner. The manuscript consists of 82 folio leaves, is held in the collection of the Biblioteca Riccardiana and is a copy of the translation of the Archimedean corpus made by Italian humanist Iacopo da San Cassiano.Paolo d'Alessandro e Pier Daniele Napolitani, Archimede latino.Iacopo da San Cassiano e il corpus archimedeo alla metà del Quattrocento, Paris, Les Belles Lettres, 2012.
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The ribs of a fan vault are of equal curvature and rotated at equal distances around a central (vertical) axis, forming the conoid shape which gives rise to the name. In between sequences of conoids, flat central spandrels fill the space. According to Leedy (1980), the fan vault was developed in England (as opposed to France and other centres of Gothic architecture) due to the manner in which English rib vaults were normally constructed. In an English rib vault, the courses are laid perpendicular to the rib while in France they are laid perpendicular to the wall.
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The earliest experiments in Australia took place in the early 1940s and the arrival of powerful synthetic resin adhesives in the post- World War II period increased the reliability and range of applications of glue lamination. The former Burge Bros Factory in Melbourne (1945–46) is considered to be the oldest known surviving example of this technology in Australia, with semi-circular arches spanning , manufactured by Sydney-based company Ralph Symonds Ltd. Despite the technology being available, it wasn't until the 1950s, when curved shapes such as conoids and hyperbolic parabolas became fashionable, that glue laminated arches were more commonly used in Australia, particularly for church structures. Recognising the structural and aesthetic potential of this new technology, Oribin specified glue laminated timber arches for two of his earliest buildings, the Mareeba Shire Hall and Proserpine's St Paul's Church; both were designed in 1956 and completed within 5 years, making them an early use of the technology in Queensland.
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