Английская Википедия:Cylinder
Шаблон:Short description Шаблон:Other uses Шаблон:Infobox polyhedron
A cylinder (Шаблон:Etymology)[1] has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the right circular cylinder.
Types
The definitions and results in this section are taken from the 1913 text Plane and Solid Geometry by George A. Wentworth and David Eugene Smith Шаблон:Harv.
A Шаблон:Dfn is 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. Any line in this family of parallel lines is called an element of the cylindrical surface. From a kinematics point of view, given a plane curve, called the directrix, a cylindrical surface is that surface traced out by a line, called the generatrix, 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 solid bounded by a cylindrical surface and two parallel planes is called a (solid) Шаблон:Dfn. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an element 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 a Шаблон:Dfn of the cylinder. The two bases of a cylinder are congruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a Шаблон:Dfn, otherwise it is called an Шаблон:Dfn. If the bases are disks (regions whose boundary is a circle) the cylinder is called a Шаблон:Dfn. In some elementary treatments, a cylinder always means a circular cylinder.[2]
The Шаблон:Dfn (or altitude) of a cylinder is the perpendicular distance between its bases.
The cylinder obtained by rotating a line segment about a fixed line that it is parallel to is a Шаблон:Dfn. 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 the Шаблон:Dfn of the cylinder and it passes through the centers of the two bases.
Right circular cylinders
Шаблон:Main The bare term cylinder 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 an Шаблон:Dfn. The formulae for the surface area and the volume of a right circular cylinder have been known from early antiquity.
A right circular cylinder can also be thought of as the solid 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.Шаблон:Sfn
A tall and thin needle cylinder has a height much greater than its diameter, whereas a short and wide disk cylinder has a diameter much greater than its height.
Properties
Cylindric sections
A cylindric section is the intersection of a cylinder's surface with a plane. They are, in general, curves and are special types of plane sections. The cylindric section by a plane that contains two elements of a cylinder is a parallelogram.Шаблон:Sfn Such a cylindric section of a right cylinder is a rectangle.Шаблон:Sfn
A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a Шаблон:Dfn.Шаблон:Sfn 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 a conic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.
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 an ellipse.[3] 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, the eccentricity Шаблон:Math of the cylindric section and semi-major axis Шаблон:Math of the cylindric section depend on the radius of the cylinder Шаблон:Math and the angle Шаблон:Math between the secant plane and cylinder axis, in the following way:
<math display="block">\begin{align} e &=\cos\alpha, \\[1ex] a &=\frac{r}{\sin\alpha}. \end{align}</math>
Volume
If the base of a circular cylinder has a radius Шаблон:Math and the cylinder has height Шаблон:Mvar, then its volume is given by
<math display=block>V = \pi r^2h</math>
This formula holds whether or not the cylinder is a right cylinder.Шаблон:Sfn
This formula may be established by using Cavalieri's principle.
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 having semi-major axis Шаблон:Mvar, semi-minor axis Шаблон:Mvar and height Шаблон:Mvar has a volume Шаблон:Math, where Шаблон:Mvar is the area of the base ellipse (= Шаблон:Math). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive Шаблон:Mvar-axis and Шаблон:Math the area of each elliptic cross-section, thus:
<math display=block>V=\int_0^h A(x) dx = \int_0^h \pi ab dx = \pi ab \int_0^h dx = \pi abh.</math>
Using cylindrical coordinates, the volume of a right circular cylinder can be calculated by integration
<math display=block>\begin{align} V &=\int_{0}^{h} \int_{0}^{2\pi} \int_{0}^{r} s \,\, ds \, d\phi \, dz \\[5mu] &=\pi\,r^2\,h. \end{align}</math>
Surface area
Having radius Шаблон:Math and altitude (height) Шаблон:Mvar, the surface area of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:
- the area of the top base: Шаблон:Math
- the area of the bottom base: Шаблон:Math
- the area of the side: Шаблон:Math
The area of the top and bottom bases is the same, and is called the base area, Шаблон:Math. The area of the side is known as the Шаблон:Dfn, Шаблон:Math.
An open cylinder does not include either top or bottom elements, and therefore has surface area (lateral area)
<math display=block>L = 2 \pi rh</math>
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
<math display=block>A = L + 2B = 2\pi rh + 2\pi r^2 = 2 \pi r (h + r) = \pi d (r + h)</math>
where Шаблон:Math is the diameter of the circular top or bottom.
For a given volume, the right circular cylinder with the smallest surface area has Шаблон:Math. Equivalently, for a given surface area, the right circular cylinder with the largest volume has Шаблон:Math, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).[4]
The lateral area, Шаблон:Mvar, of a circular cylinder, which need not be a right cylinder, is more generally given by
<math display=block>L = e \times p,</math>
where Шаблон:Mvar is the length of an element and Шаблон:Mvar is the perimeter of a right section of the cylinder.Шаблон:Sfn This produces the previous formula for lateral area when the cylinder is a right circular cylinder.
Right circular hollow cylinder (cylindrical shell)
A right circular hollow cylinder (or Шаблон:Dfn) is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders' common axis, as in the diagram.
Let the height be Шаблон:Math, internal radius Шаблон:Math, and external radius Шаблон:Math. The volume is given by
<math display=block> V = \pi ( R ^{2} - r ^{2} ) h = 2\pi \left ( \frac{R + r}{2} \right) h (R - r). </math>
Thus, the volume of a cylindrical shell equals 2Шаблон:Pi × average radius × altitude × thickness.Шаблон:Sfn
The surface area, including the top and bottom, is given by
<math display=block> A = 2 \pi ( R + r ) h + 2 \pi ( R^2 - r^2 ). </math>
Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.Шаблон:Sfn
On the Sphere and Cylinder
Шаблон:Main In the treatise by this name, written Шаблон:Circa, Archimedes obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a sphere by exploiting the relationship between a sphere and its circumscribed right circular cylinder of the same height and diameter. The sphere has a volume Шаблон:Nowrap that of the circumscribed cylinder and a surface area Шаблон:Nowrap 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 radius Шаблон:Mvar is Шаблон:Math. The surface area of this sphere is Шаблон:Math. A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.
Cylindrical surfaces
Шаблон:Anchor In some areas of geometry and topology the term cylinder refers to what has been called a cylindrical 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.Шаблон:Sfn Such cylinders have, at times, been referred to as Шаблон:Dfn. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder.Шаблон:Sfn Thus, this definition may be rephrased to say that a cylinder is any ruled surface spanned by a one-parameter family of parallel lines.
A cylinder having a right section that is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder and hyperbolic cylinder, respectively. These are degenerate quadric surfaces.[5]
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
<math display=block>f(x,y,z)=Ax^2 + By^2 + Cz^2 + Dx + Ey + Gz + H = 0,</math>
with the coefficients being real numbers and not all of Шаблон:Mvar, Шаблон:Mvar and Шаблон:Mvar 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 appropriate rotation of axes that the variable Шаблон:Mvar does not appear and the general equation of this type of degenerate quadric can be written asШаблон:Sfn
<math display=block>A \left ( x + \frac{D}{2A} \right )^2 + B \left(y + \frac{E}{2B} \right)^2 = \rho,</math>
where
<math display=block>\rho = -H + \frac{D^2}{4A} + \frac{E^2}{4B}.</math>
Elliptic cylinder
If Шаблон:Math this is the equation of an elliptic cylinder.Шаблон:Sfn Further simplification can be obtained by translation of axes and scalar multiplication. If <math>\rho</math> has the same sign as the coefficients Шаблон:Mvar and Шаблон:Mvar, then the equation of an elliptic cylinder may be rewritten in Cartesian coordinates as:
<math display=block>\left(\frac{x}{a}\right)^2+ \left(\frac{y}{b}\right)^2 = 1.</math>
This equation of an elliptic cylinder is a generalization of the equation of the ordinary, circular cylinder (Шаблон:Math). Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid.
If <math>\rho</math> has a different sign than the coefficients, we obtain the imaginary elliptic cylinders:
<math display=block>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = -1,</math>
which have no real points on them. (<math>\rho = 0</math> gives a single real point.)
Hyperbolic cylinder
If Шаблон:Mvar and Шаблон:Mvar have different signs and <math>\rho \neq 0</math>, we obtain the hyperbolic cylinders, whose equations may be rewritten as:
<math display=block>\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = 1.</math>
Parabolic cylinder
Finally, if Шаблон:Math assume, without loss of generality, that Шаблон:Math and Шаблон:Math to obtain the parabolic cylinders with equations that can be written as:Шаблон:Sfn
<math display=block> {x}^2+2a{y}=0 .</math>
Projective geometry
In projective geometry, a cylinder is simply a cone whose apex (vertex) lies on the plane 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.[6]
This concept is useful when considering degenerate conics, which may include the cylindrical conics.
Prisms
A solid circular cylinder can be seen as the limiting case of a [[regular polygon|Шаблон:Mvar-gonal]] prism where Шаблон:Math approaches infinity. The connection is very strong and many older texts treat prisms 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.[7] 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 a truncated 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 a truncated cylinder.
From a polyhedral viewpoint, a cylinder can also be seen as a dual of a bicone as an infinite-sided bipyramid.
See also
- List of shapes
- Steinmetz solid, the intersection of two or three perpendicular cylinders
Notes
References
External links
Шаблон:Commons Шаблон:Wiktionary Шаблон:EB1911 poster
- Шаблон:MathWorld
- Surface area of a cylinder at MATHguide
- Volume of a cylinder at MATHguide
Шаблон:Compact topological surfaces Шаблон:Authority control
- ↑ κύλινδρος Шаблон:Webarchive, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
- ↑ Шаблон:Citation
- ↑ Шаблон:Cite web
- ↑ Шаблон:Citation.
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
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