WordNet

a frustum formed from a cone
replace a corner by a plane
make shorter as if by cutting off; "truncate a word"; "Erosion has truncated the ridges of the mountains" (同)cut_short
terminating abruptly by having or as if having an end or point cut off; "a truncate leaf"; "truncated volcanic mountains"; "a truncated pyramid" (同)truncated
approximate by ignoring all terms beyond a chosen one; "truncate a series"
make cone-shaped; "cone a tire"
a shape whose base is a circle and whose sides taper up to a point (同)conoid, cone shape
cone-shaped mass of ovule- or spore-bearing scales or bracts (同)strobilus, strobile
any cone-shaped artifact
a visual receptor cell in the retina that is sensitive to bright light and to color (同)cone cell, retinal cone
in opposition to a proposition, opinion, etc.; "much was written pro and con"
an argument opposed to a proposal

PrepTutorEJDIC

〈円錐・樹木など〉‘の'頭(端)を切る / 〈文章など〉‘を'切って詰める / 切り取られた / (生物の形態が)切形の(先端を切り取ったような形をしていること)
円すい,円すい形 / 円すい形のもの / 球果(松かさなど)・暴風警報球.
…'を'よく研究する
反対して / 反対投票,反対論
〈人〉'を'ペテンにかける
=convict
=confidence trick

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/01/30 23:22:14」(JST)

wiki en

In geometry, a frustum^[1] (plural: frusta or frustums) is the portion of a solid (normally a cone or pyramid) that lies between two parallel planes cutting it. A right frustum is a parallel truncation of a right pyramid.^[1]

The term is commonly used in computer graphics to describe the three-dimensional region which is visible on the screen, the "viewing frustum", which is formed by a clipped pyramid; in particular, frustum culling is a method of hidden surface determination.

In the aerospace industry, frustum is the common term for the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone. It also applies to the essential drive element of the so-far unproven Emdrive.

Elements, special cases, and related concepts

Square frustum

A regular octahedron can be augmented on 3 faces to create a triangular frustum

Each plane section is a floor or base of the frustum. Its axis if any, is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.

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 (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of the prismatoids.

Two frusta joined at their bases make a bifrustum.

Formulas

Volume

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 ca. 1850 BC.:

where a and b are the base and top side lengths of the truncated pyramid, and 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:

where B₁ is the area of one base, B₂ is the area of the other base, and h₁, h₂ are the perpendicular heights from the apex to the planes of the two bases.

Considering that

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.

By factoring the difference of two cubes ( a³ - b³ = (a-b)(a² + ab + b²) ) we get h₁−h₂ = h, the height of the frustum, and α(h₁² + h₁h₂ + h₂²)/3.

Distributing α and substituting from its definition, the Heronian mean of areas B₁ and B₂ is obtained. The alternative formula is therefore

Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary number, the square root of negative one.^[2]

In particular, the volume of a circular cone frustum is

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

where a₁ and a₂ are the sides of the two bases.

Surface area

For a right circular conical frustum^[3]

\begin{align}\text{Lateral Surface Area}&=\pi(R_1+R_2)s\\ &=\pi(R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}\end{align}

and

\begin{align}\text{Total Surface Area}&=\pi((R_1+R_2)s+R_1^2+R_2^2)\\ &=\pi((R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}+R_1^2+R_2^2)\end{align}

where R₁ and R₂ are the base and top radii respectively, and s is the slant height of the frustum.

The surface area of a right frustum whose bases are similar regular n-sided polygons is

where a₁ and a₂ are the sides of the two bases.

Examples

On the back (the reverse) of a United States one-dollar bill, a pyramidal frustum appears on the reverse of the Great Seal of the United States, surmounted by the Eye of Providence.
Certain ancient Native American mounds also form the frustum of a pyramid.
Chinese pyramids.
The John Hancock Center in Chicago, Illinois is a frustum whose bases are rectangles.
The Washington Monument is a narrow square-based pyramidal frustum topped by a small pyramid.
The viewing frustum in 3D computer graphics is a virtual photographic or video camera's usable field of view modeled as a pyramidal frustum.
In the English translation of Stanislaw Lem's short-story collection The Cyberiad, the poem Love and tensor algebra claims that "every frustum longs to be a cone".
Buckets and typical lampshades are everyday examples of conical frustums.

Notes

1.^ The term "frustum" comes from Latin frustum meaning "piece" or "crumb". The English word is often misspelled as frustrum, a different Latin word cognate to the English word "frustrate".^[4] The confusion between these two words is very old: a warning about them can be found in the Appendix Probi, and the works of Plautus include a pun on them.^[5]

References

^ William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.67
^ Nahin, Paul. "An Imaginary Tale: The story of [the square root of minus one]." Princeton University Press. 1998
^ "Mathwords.com: Frustum". Retrieved 17 July 2011.
^ Clark, John Spencer (1895), Teachers' Manual: Books I-VIII.. For Prang's complete course in form-study and drawing, Books 7-8, Prang Educational Company, p. 49 .
^ Fontaine, Michael (2010), Funny Words in Plautine Comedy, Oxford University Press, pp. 117, 154, ISBN 9780195341447 .

External links

Derivation of formula for the volume of frustums of pyramid and cone (Mathalino.com)
Weisstein, Eric W., "Pyramidal frustum", MathWorld.
Weisstein, Eric W., "Conical frustum", MathWorld.
Paper models of frustums (truncated pyramids)
Paper model of frustum (truncated cone)
Design paper models of conical frustum (truncated cones)

UpToDate Contents

全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.

1. 血小板機能検査 platelet function testing
2. 子宮頸部上皮内腫瘍：子宮頸部円錐切除術のための手順 cervical intraepithelial neoplasia procedures for cervical conization
3. 三尖弁のエプスタイン奇形 ebsteins anomaly of the tricuspid valve
4. 早期（ステージIおよびII）の頭頸部癌の治療：口腔 treatment of early stage i and ii head and neck cancer the oral cavity
5. 海洋のサンゴ、ウニ、魚類またはアカエイによる毒の注入 marine envenomations from corals sea urchins fish or stingrays

English Journal

Arm Volumetry Versus Upper Extremity Lymphedema Index: Validity of Upper Extremity Lymphedema Index for Body-Type Corrected Arm Volume Evaluation.

Yamamoto N1, Yamamoto T, Hayashi N, Hayashi A, Iida T, Koshima I.
Annals of plastic surgery.Ann Plast Surg.2016 Jun;76(6):697-9. doi: 10.1097/SAP.0000000000000259.
BACKGROUND: Volumetry, measurement of extremity volume, is a commonly used method for upper extremity lymphedema (UEL) evaluation. However, comparison between different patients with different physiques is difficult with volumetry, because body-type difference greatly affects arm volume.METHODS: Sev
PMID 25003442

Characterization of a Thin Silicon Sensor for Active Neutron Personal Dosemeters.

Takada M1, Nunomiya T2, Nakamura T3, Matsumoto T4, Masuda A4.
Radiation protection dosimetry.Radiat Prot Dosimetry.2016 May 5. [Epub ahead of print]
A thin silicon sensor has been developed for active neutron personal dosemeters for use by aircrews and first responders. This thin silicon sensor is not affected by the funneling effect, which causes detection of cosmic protons and over-response to cosmic neutrons. There are several advantages to t
PMID 27150515

Systematic study of target localization for bioluminescence tomography guided radiation therapy.

Yu J1, Zhang B2, Iordachita II3, Reyes J2, Lu Z4, Brock MV5, Patterson MS6, Wong JW2, Wang KK2.
Medical physics.Med Phys.2016 May;43(5):2619. doi: 10.1118/1.4947481.
PURPOSE: To overcome the limitation of CT/cone-beam CT (CBCT) in guiding radiation for soft tissue targets, the authors developed a spectrally resolved bioluminescence tomography (BLT) system for the small animal radiation research platform. The authors systematically assessed the performance of the
PMID 27147371

Japanese Journal

細胞刺激用マイクロデバイスに関する研究 (第1報):—導電性ポリマーによるマイクロアクチュエータ作製—

金子新,杉原達記,武田伊織
精密工学会誌 81(1), 80-85, 2015
… The fabricated PPy dot has circular truncated cone shape, in which the bottom diameter changed from approximately 10 to 30μm with electric charge. …
NAID 130004745513

Enhanced light extraction in GaN-based light-emitting diodes by evanescent wave coupling effect

Hao Guo-Dong,Jahir Ahmed,Takahashi Tokio,Shimizu Mitsuaki,Wang Xue-Lun,Kishi Hiroyuki,Hayashi Yukiko,Takeguchi Keigo
Appl. Phys. Express 7(10), 102101, 2014-09-16
… Subwavelength-sized truncated cones satisfying the requirement for evanescent wave coupling effect at the emission wavelength were fabricated on the p-side of a GaN-based blue light-emitting diode (LED). … The evanescent wave coupling effect appearing on the surface of the truncated-cone structure is believed to be the mechanism responsible for the large enhancement of light extraction. …
NAID 150000108490

Prolonged morphometric study of barnacles grown on soft substrata of hydrogels and elastomers

Ahmed Nafees,Murosaki Takayuki,Kurokawa Takayuki,Kakugo Akira,Yashima Shintaro,Nogata Yasuyuki,Gong Jian Ping
Biofouling 30(3), 271-279, 2014-03-16
… Barnacles on the PS substratum had a truncated cone shape with a flat basal plate while on soft PDMS and DN gels, barnacles had a pseudo-cylindrical shape and their basal plates showed curvature. …
NAID 120005567723