frustum

English

A frustum of a decagonal pyramid

Etymology

Borrowed from Latin frustum (“morsel”).

Pronunciation

• Audio (US):

  (file)

Noun

frustum (plural frustums or frusta)

1. A cone or pyramid whose tip has been truncated by a plane parallel to its base.
   • 1742, Colin MacLaurin, A Treatise of Fluxions, volume 1, page 25:

     In a parabolic conoid this difference vaniſhes, the fruſtum being always equal to a cylinder of the ſame height upon the ſection of the conoid that biſects the altitude of the fruſtum and is parallel to its baſes.

   • 1809, William Nicholson, “FRUSTUM”, in The British Encyclopedia, or Dictionary of Arts and Sciences; […], volume III (E … I), London: Printed by C[harles] Whittingham, […]; for Longman, Hurst, Rees, and Orme, […], →OCLC:

     This theorem holds good for complete solids as well as frustums, whether right or oblique, and not only of the solids generated from the conic sections, but also of all pyramids, cones, and in short of any solid, whose parallel sections are similar figures.

   • 1974, Stanisław Lem, translated by Michael Kandel, The Cyberiad:

     Come, every frustum longs to be a cone,
     And every vector dreams of matrices.

   • 2006, Pawan Harish Nirnimesh, P. J. Narayanan, Culling an Object Hierarchy to a Frustum Hierarchy, Prem Kalra, Shmuel Peleg (editors), Computer Vision, Graphics and Image Processing: 5th Indian Conference, ICVGIP 2006, Springer, LNCS4338, page 252,

     However, when there are multiple view frustums (as in a tiled display wall), visibility culling time becomes substantial and cannot be hidden by pipelining it with other stages of rendering.

   • 2008, R. Benjamin Davis, Techniques to Assess Acoustic-structure Interaction in Liquid Rocket Engines, page 122:

     Here, the dynamics of the fluid-filled frusta of cones are considered (see Figure 5.5). The frusta are clamped at their roots and free at their ends.

2. A portion of a sphere, or in general any solid, delimited by two parallel planes.
   • 1840, James Blundell, Observations on Some of the More Important Diseases of Women, page 131:

     In some women it^{[the os uteri]} is flat, in many more tuberose, and forming, as it were, a frustum of a sphere; […] .

   • 2014, John Bird, Engineering Mathematics, page 183:

     Problem 22. Determine the volume of a frustum of a sphere of diameter 49.74 cm if the diameter^[sic] of the ends of the frustum are 24.0 and 40.0 cm, and the height of the frustum is 7.00 cm.

Usage notes

The misspelling frustrum is by incorrect analogy with frustrate, also of Latin origin.

• (portion of a sphere): The portion of the surface of a sphere delimited by parallel planes (i.e., the curved surface of a frustum) may be called a zone; however, that term is also sometimes used as a synonym of frustum.

Derived terms

• bifrustum
• frustum culling
• view frustum
• viewing frustum

Related terms

• bifrustum
• frustoconical

Translations

truncated cone or pyramid

• Bulgarian: пресечен конус m (presečen konus), пресечена пирамида f (presečena piramida)
• Catalan: tronc (ca) m
• Chinese:

  Mandarin: 平截頭體／平截头体 (píngjié tóutǐ)

• Esperanto: please add this translation if you can
• Finnish: katkaistu kartio
• French: tronc (fr) m
• Galician: tronco (gl) m
• German: (mathematics) (cone) Kegelstumpf (de) m, (mathematics) Stumpf (de) m, (computing, mathematics) (pyramid) Pyramidenstumpf (de) m
• Greek: (cone) κόλουρος κώνος m (kólouros kónos), (pyramid) κόλουρη πυραμίδα f (kólouri pyramída)
• Hindi: छिन्नक (hi) m (chinnak)
• Hungarian: csonka test (in general), csonka kúp (of a cone), csonka gúla (of a pyramid)
• Icelandic: stúfur m
• Italian: tronco (it) m
• Japanese: please add this translation if you can
• Korean: 절두체(截頭體) (jeolduche)
• Maori: motuhanga pūtake
• Russian: усечённая пирами́да f (usečónnaja piramída) pyramid, усечённый ко́нус m (usečónnyj kónus) cone
• Spanish: tronco (es) m, tronco de geometría, tronco geométrico m
• Turkish: (mathematics) kesik koni

Latin

Etymology

From Proto-Italic *frustom, from Proto-Indo-European *bʰrus-tós, from *bʰrews- (“to break up, cut”).

Pronunciation

• (Classical Latin) IPA^(key): /ˈfruːs.tum/, [ˈfruːs̠t̪ʊ̃ˑ] or IPA^(key): /ˈfrus.tum/, [ˈfrʊs̠t̪ʊ̃ˑ]
• (modern Italianate Ecclesiastical) IPA^(key): /ˈfrus.tum/, [ˈfrust̪um]

The etymology is consistent with the /u/ in the first syllable being short, and the word is shown without a macron in De Vaan's dictionary. Although Bennett 1907 says "ū acc. to the Romance", there are related words in Romance that point to short u in the descendants of the derivative *frŭstiāre, such as Old French froissier.

Noun

frū̆stum n (genitive frū̆stī); second declension

1. a piece, bit; crumb, morsel, scrap of food

Declension

Second-declension noun (neuter).

Case	Singular	Plural
Nominative	frū̆stum	frū̆sta
Genitive	frū̆stī	frū̆stōrum
Dative	frū̆stō	frū̆stīs
Accusative	frū̆stum	frū̆sta
Ablative	frū̆stō	frū̆stīs
Vocative	frū̆stum	frū̆sta

Synonyms

• (piece): fragmentum, segmentum
• (crumb): mīca

Derived terms

• frustulentus

Descendants

• English: frustum
• Italian: frusta
• Spanish: frusto, frusta
• Vulgar Latin: *frustiāre (see there for further descendants)

Further reading

• “frustum”, in Charlton T. Lewis and Charles Short (1879) A Latin Dictionary, Oxford: Clarendon Press
• “frustum”, in Charlton T. Lewis (1891) An Elementary Latin Dictionary, New York: Harper & Brothers
• frustum in Charles du Fresne du Cange’s Glossarium Mediæ et Infimæ Latinitatis (augmented edition with additions by D. P. Carpenterius, Adelungius and others, edited by Léopold Favre, 1883–1887)