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Cylinder

From Wikipedia, the free encyclopedia

Three-dimensional solid

For other uses, seeCylinder (disambiguation).

Cylinder
A circular right cylinder of heighth and diameterd=2r
Type	Smooth surface Algebraic surface
Euler char.	2
Symmetry group	O(2)×O(1)
Surface area	2πr(r + h)
Volume	πr²h

Acylinder (from Ancient Greek κύλινδρος (kúlindros) 'roller, tumbler')^[1] has traditionally been athree-dimensional solid, one of the most basic ofcurvilinear geometricshapes. Inelementary geometry, it is considered aprism with acircle as its base.

A cylinder may also be defined as aninfinite curvilinearsurface in various modern branches of geometry andtopology. The shift in the basic meaning—solid versus surface (as in a solidball versussphere surface)—has created some ambiguity with terminology. The two concepts may be distinguished by referring tosolid cylinders andcylindrical surfaces. In the literature the unadorned term "cylinder" could refer to either of these or to an even more specialized object, theright circular cylinder.

Types

The definitions and results in this section are taken from the 1913 textPlane and Solid Geometry byGeorge A. Wentworth and David Eugene Smith (Wentworth & Smith 1913).

Acylindrical surface is asurface consisting of all the points on all the lines which areparallel to a given line and which pass through a fixedplane curve in a plane not parallel to the given line. Any line in this family of parallel lines is called anelement of the cylindrical surface. From akinematics point of view, given a plane curve, called thedirectrix, a cylindrical surface is that surface traced out by a line, called thegeneratrix, not in the plane of the directrix, moving parallel to itself and always passing through the directrix. Any particular position of the generatrix is an element of the cylindrical surface.

A right and an oblique circular cylinder

Asolid bounded by a cylindrical surface and twoparallel planes is called a (solid)cylinder. The line segments determined by an element of the cylindrical surface between the two parallel planes is called anelement of the cylinder. All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of the parallel planes is called abase of the cylinder. The two bases of a cylinder arecongruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is aright cylinder, otherwise it is called anoblique cylinder. If the bases aredisks (regions whose boundary is acircle) the cylinder is called acircular cylinder. In some elementary treatments, a cylinder always means a circular cylinder.^[2]Anopen cylinder is a cylindrical surface without the bases.

Theheight (or altitude) of a cylinder is theperpendicular distance between its bases.

The cylinder obtained by rotating aline segment about a fixed line that it is parallel to is acylinder of revolution. A cylinder of revolution is a right circular cylinder. The height of a cylinder of revolution is the length of the generating line segment. The line that the segment is revolved about is called theaxis of the cylinder and it passes through the centers of the two bases.

A right circular cylinder with radiusr and heighth

Right circular cylinders

Main article:Right circular cylinder

The bare termcylinder often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called anopen cylinder. The formulae for thesurface area and thevolume of a right circular cylinder have been known from early antiquity.

A right circular cylinder can also be thought of as thesolid of revolution generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution.^[3]

A tall and thinneedle cylinder has a height much greater than its diameter, whereas a short and widedisk cylinder has a diameter much greater than its height.

Properties

Cylindric sections

Cylindric section

A cylindric section is the intersection of a cylinder's surface with aplane. They are, in general, curves and are special types ofplane sections. The cylindric section by a plane that contains two elements of a cylinder is aparallelogram.^[4] Such a cylindric section of a right cylinder is arectangle.^[4]

A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called aright section.^[5] If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is aconic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.

Cylindric sections of a right circular cylinder

For a right circular cylinder, there are several ways in which planes can meet a cylinder. First, planes that intersect a base in at most one point. A plane is tangent to the cylinder if it meets the cylinder in a single element. The right sections are circles and all other planes intersect the cylindrical surface in anellipse.^[6] If a plane intersects a base of the cylinder in exactly two points then the line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle.

In the case of a right circular cylinder with a cylindric section that is an ellipse, theeccentricitye of the cylindric section andsemi-major axisa of the cylindric section depend on the radius of the cylinderr and the angleα between the secant plane and cylinder axis, in the following way: ${\begin{aligned}e&=\cos \alpha ,\\[1ex]a&={\frac {r}{\sin \alpha }}.\end{aligned}}$

Volume

If the base of a circular cylinder has aradiusr and the cylinder has heighth, then itsvolume is given by $V=\pi r^{2}h$

This formula holds whether or not the cylinder is a right cylinder.^[7]

This formula may be established by usingCavalieri's principle.

A solid elliptic right cylinder with the semi-axesa andb for the base ellipse and heighth

In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height. For example, an elliptic cylinder with a base havingsemi-major axisa, semi-minor axisb and heighth has a volumeV =Ah, whereA is the area of the base ellipse (=πab). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positivex-axis andA(x) =A the area of each elliptic cross-section, thus: $V=\int _{0}^{h}A(x)dx=\int _{0}^{h}\pi abdx=\pi ab\int _{0}^{h}dx=\pi abh.$

Usingcylindrical coordinates, the volume of a right circular cylinder can be calculated by integration ${\begin{aligned}V&=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{r}s\,\,ds\,d\phi \,dz\\[5mu]&=\pi \,r^{2}\,h.\end{aligned}}$

Surface area

Having radiusr and altitude (height)h, thesurface area of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:

the area of the top base:πr²
the area of the bottom base:πr²
the area of the side:2πrh

The area of the top and bottom bases is the same, and is called thebase area,B. The area of the side is known as thelateral area,L.

Anopen cylinder does not include either top or bottom elements, and therefore has surface area (lateral area) $L=2\pi rh$

The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore $A=L+2B=2\pi rh+2\pi r^{2}=2\pi r(h+r)=\pi d(r+h)$ whered = 2r is thediameter of the circular top or bottom.

For a given volume, the right circular cylinder with the smallest surface area hash = 2r. Equivalently, for a given surface area, the right circular cylinder with the largest volume hash = 2r, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).^[8]

The lateral area,L, of a circular cylinder, which need not be a right cylinder, is more generally given by $L=e\times p,$ wheree is the length of an element andp is the perimeter of a right section of the cylinder.^[9] This produces the previous formula for lateral area when the cylinder is a right circular cylinder.

Hollow cylinder

Right circular hollow cylinder (cylindrical shell)

Aright circular hollow cylinder (orcylindrical shell) is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallelannular bases perpendicular to the cylinders' common axis, as in the diagram.

Let the height beh, internal radiusr, and external radiusR. The volume is given by subtracting the volume of the inner imaginary cylinder (i.e. hollow space) from the volume of the outer cylinder: $V=\pi \left(R^{2}-r^{2}\right)h=2\pi \left({\frac {R+r}{2}}\right)h(R-r).$ Thus, the volume of a cylindrical shell equals2π × average radius × height × thickness.^[10]

The surface area, including the top and bottom, is given by $A=2\pi \left(R+r\right)h+2\pi \left(R^{2}-r^{2}\right).$ Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.^[11]

On the Sphere and Cylinder

A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases

Main article:On the Sphere and Cylinder

In the treatise by this name, writtenc. 225 BCE,Archimedes obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of asphere by exploiting the relationship between a sphere and itscircumscribed right circular cylinder of the same height anddiameter. The sphere has a volumetwo-thirds that of the circumscribed cylinder and a surface areatwo-thirds that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radiusr is⁠4/3⁠πr³ =⁠2/3⁠ (2πr³). The surface area of this sphere is4πr² =⁠2/3⁠ (6πr²). A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.

Cylindrical surfaces

In some areas of geometry and topology the termcylinder refers to what has been called acylindrical surface. A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line.^[12] Such cylinders have, at times, been referred to asgeneralized cylinders. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder.^[13] Thus, this definition may be rephrased to say that a cylinder is anyruled surface spanned by a one-parameter family of parallel lines.

A cylinder having a right section that is anellipse,parabola, orhyperbola is called anelliptic cylinder,parabolic cylinder andhyperbolic cylinder, respectively. These are degeneratequadric surfaces.^[14]

Parabolic cylinder

When the principal axes of a quadric are aligned with the reference frame (always possible for a quadric), a general equation of the quadric in three dimensions is given by $f(x,y,z)=Ax^{2}+By^{2}+Cz^{2}+Dx+Ey+Gz+H=0,$ with the coefficients beingreal numbers and not all ofA,B andC being 0. If at least one variable does not appear in the equation, then the quadric is degenerate. If one variable is missing, we may assume by an appropriaterotation of axes that the variablez does not appear and the general equation of this type of degenerate quadric can be written as^[15] $A\left(x+{\frac {D}{2A}}\right)^{2}+B\left(y+{\frac {E}{2B}}\right)^{2}=\rho ,$ where $\rho =-H+{\frac {D^{2}}{4A}}+{\frac {E^{2}}{4B}}.$

Elliptic cylinder

IfAB > 0 this is the equation of anelliptic cylinder.^[15] Further simplification can be obtained bytranslation of axes and scalar multiplication. If $\rho$ has the same sign as the coefficientsA andB, then the equation of an elliptic cylinder may be rewritten inCartesian coordinates as: $\left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1.$ This equation of an elliptic cylinder is a generalization of the equation of the ordinary,circular cylinder (a =b). Elliptic cylinders are also known ascylindroids, but that name is ambiguous, as it can also refer to thePlücker conoid.

If $\rho$ has a different sign than the coefficients, we obtain theimaginary elliptic cylinders: $\left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=-1,$ which have no real points on them. ( $\rho =0$ gives a single real point.)

Hyperbolic cylinder

IfA andB have different signs and $\rho \neq 0$ , we obtain thehyperbolic cylinders, whose equations may be rewritten as: $\left({\frac {x}{a}}\right)^{2}-\left({\frac {y}{b}}\right)^{2}=1.$

Parabolic cylinder

Finally, ifAB = 0 assume,without loss of generality, thatB = 0 andA = 1 to obtain theparabolic cylinders with equations that can be written as:^[16] $x^{2}+2ay=0.$

Inprojective geometry, a cylinder is simply a cone whoseapex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.

Projective geometry

Inprojective geometry, a cylinder is simply acone whoseapex (vertex) lies on theplane at infinity. If the cone is a quadratic cone, the plane at infinity (which passes through the vertex) can intersect the cone at two real lines, a single real line (actually a coincident pair of lines), or only at the vertex. These cases give rise to the hyperbolic, parabolic or elliptic cylinders respectively.^[17]

This concept is useful when consideringdegenerate conics, which may include the cylindrical conics.

Prisms

Tycho Brahe Planetarium building, Copenhagen, is an example of a truncated cylinder

Asolid circular cylinder can be seen as the limiting case of an-gonal prism wheren approachesinfinity. The connection is very strong and many older texts treatprisms and cylinders simultaneously. Formulas for surface area and volume are derived from the corresponding formulas for prisms by using inscribed and circumscribed prisms and then letting the number of sides of the prism increase without bound.^[18] One reason for the early emphasis (and sometimes exclusive treatment) on circular cylinders is that a circular base is the only type of geometric figure for which this technique works with the use of only elementary considerations (no appeal to calculus or more advanced mathematics). Terminology about prisms and cylinders is identical. Thus, for example, since atruncated prism is a prism whose bases do not lie in parallel planes, a solid cylinder whose bases do not lie in parallel planes would be called atruncated cylinder.

From a polyhedral viewpoint, a cylinder can also be seen as adual of abicone as an infinite-sidedbipyramid.

Family ofuniformn-gonalprisms v t e
Prism name	Digonal prism	(Trigonal) Triangular prism	(Tetragonal) Square prism	Pentagonal prism	Hexagonal prism	Heptagonal prism	Octagonal prism	Enneagonal prism	Decagonal prism	Hendecagonal prism	Dodecagonal prism	...	Apeirogonal prism
Polyhedron image												...
Spherical tiling image												Plane tiling image
Vertex config.	2.4.4	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4	...	∞.4.4
Coxeter diagram												...

See also

Lists of shapes
Steinmetz solid, the intersection of two or three perpendicular cylinders

Notes

^κύλινδρος Archived 2013-07-30 at theWayback Machine, Henry George Liddell, Robert Scott,A Greek-English Lexicon, on Perseus
^Jacobs, Harold R. (1974),Geometry, W. H. Freeman and Co., p. 607,ISBN 0-7167-0456-0
^Swokowski 1983, p. 283.
^^a ^bWentworth & Smith 1913, p. 354.
^Wentworth & Smith 1913, p. 357.
^"Cylindric section",MathWorld
^Wentworth & Smith 1913, p. 359.
^Lax, Peter D.; Terrell, Maria Shea (2013),Calculus With Applications, Springer, p. 178,ISBN 9781461479468.
^Wentworth & Smith 1913, p. 358.
^Swokowski 1983, p. 292.
^Swokowski 1983, p. 291.
^Albert 2016, p. 43.
^Albert 2016, p. 49.
^Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999),Geometry, Cambridge University Press, p. 34,ISBN 978-0-521-59787-6
^^a ^bAlbert 2016, p. 74.
^Albert 2016, p. 75.
^Pedoe, Dan (1988) [1970],Geometry a Comprehensive Course, Dover, p. 398,ISBN 0-486-65812-0
^Slaught, H.E.; Lennes, N.J. (1919),Solid Geometry with Problems and Applications(PDF) (Rev. ed.), Allyn and Bacon, pp. 79–81

References

Albert, Abraham Adrian (2016) [1949],Solid Analytic Geometry, Dover,ISBN 978-0-486-81026-3
Swokowski, Earl W. (1983),Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber & Schmidt,ISBN 0-87150-341-7
Wentworth, George; Smith, David Eugene (1913),Plane and Solid Geometry, Ginn and Co.

External links

Wikimedia Commons has media related toCylinder (geometry).

Look upcylinder in Wiktionary, the free dictionary.

Wikisource has the text of the1911Encyclopædia Britannica article "Cylinder".

v
t
e

Compact topological surfaces and their immersions in 3D

Without boundary

Orientable	Sphere (genus 0) Torus (genus 1) Number 8 (genus 2) Pretzel (genus 3) ...
Non-orientable	Real projective plane genus 1;Boy's surface Roman surface Klein bottle (genus 2) Dyck's surface (genus 3) ...

With boundary

Related
notions

Properties	Connectedness Compactness Triangulatedness orsmoothness Orientability
Characteristics	Number ofboundary components Genus Euler characteristic
Operations	Connected sum Making a hole Gluing ahandle Gluing across-cap Immersion

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