أسطوانة (هندسة)

Cylinder
Cylinder
	A right circular cylinder of height h and diameter d=2r
النوع	Smooth surface; Algebraic surface
خاصية أويلر	2
Symmetry group	O(2)×O(1)
المساحة السطحية	2πr(r + h)
الحجم	πr2h

هذه المقالة حول الأسطوانة كمجسم ثلاثي الأبعاد، إذا كنت تبحث عن شيء آخر انظر أسطوانة (توضيح)

الأسطوانة هو شكل هندسي ثلاثي الأبعاد ينتج من دواران كامل للمستقيم g حول مستقيم a بشرط أن يكون g و a متوازيين . في هذه الحالة يُسميان محور ومولد المخروط. أي دائرة يحصل عليها، نتيجة لذلك الدواران، تسمى قاعدة أو دليل.

ملاحظة: في حالة التقاطع بين g a يتولد المخروط كحالة إستثنائية للأسطوانة.

أسطوانة Cylindre (باللغة الفرنسية) شكل مجسم تكون قاعداته من دائرتين متطابقتين في مستويين متوازيين وسطحه الجانبي منحن.

المساحة الجانبيه = محيط القاعدة × الارتفاع
أو = ق × ط × ع
الحجم = مساحة القاعدة × الارتفاع

= نق × نق× ط × ع

المساحة الكلية = المساحة الجانبيه + مساحة القاعدتين

(ق×ط×ع) + (نق×نق×ط×2)

في الرياضيات، تعتبر الأسطوانة من المجسمات الأساسية، وهي أي مجسم يتشكل سطحه من جميع النقاط التي تبعد مسافة معينة عن قطعة مستقيمة معطاه تسمى محور الاسطوانة ويسمى الحيز المغلق بمستويين متوازيين يتعامدان مع المحور أسطوانة، ويمكن تعريفه كأي مجسم ينتج من دوران مستطيل حول أحد أضلاعه دورة كاملة، ويسمى محور الدوران بمحور الاسطوانة والضلع المقابل له يسمى بمولد أو راسم الاسطوانة. الدائرتين التي تحد المجسم من الجهتين تسمى قاعدة أو دليل، القطعة المستقيمة التي تتعامد مع القاعديتن تسمى ارتفاع الاسطوانة، إذا كان ارتفاع الاسطوانة يتعامد مع محيط قاعدتي الاسطوانة سميت اسطوانة قائمة وإلا سميت اسطوانة مائلة.
إذا قيل اسطوانة بدون تحديد فإننا نقصد الاسطوانة الدائرة القائمة.
الأسطوانة التي مقطعها العرضي هو قطع زائد أو قطع ناقص أو قطع مكافئ يسمى الاسطوانة الزائدة والاسطوانة الناقصة والاسطوانة المكافئة .ولا تنطبق عليها التعريفات السابقة .

الخصائص

المقاطع الأسطوانية

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.^[1] Such a cylindric section of a right cylinder is a rectangle.^[1]

A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section.^[2] 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 $e$ of the cylindric section and semi-major axis $a$ of the cylindric section depend on the radius of the cylinder $r$ and the angle $α$ between the secant plane and cylinder axis, in the following way: $\begin{array}{l} e & = \cos α, \\ [1 e x] a & = \frac{r}{\sin α} . \end{array}$

هذه القوانين حول الاسطوانة الدائرة القائمة

r : نصف قطر القاعدة.

h : ارتفاع الاسطوانة أو محورها.

A : مساحة القاعدة ويمكن حسابة عن طريق

A = π r^{2}

P : محيط القاعدة , ويمكن حسابة عن طريق

P = 2 π r

مساحات

المساحة الجانبيه = محيط القاعدة × الارتفاع = $P \times h$
مساحة القاعدة العليا = $π r^{2}$
مساحة القاعدة السفلى = $π r^{2}$
المساحة الكلية = $(2 \times π r^{2}) + (P \times h)$

الحجم

عنصر الحجم هو أسطوانة قائمة مساحة قاعدتها $A (w_{i})$ وحدة مربعة وسمكها $Δ_{i} x$ من الوحدات. ولذلك فإذا كان وحدات الحجم $V$ هي حجم الأسطوانية الدائرية القائمة، فحسب جموع ريمان، يكون:

V = \lim_{| | Δ \to 0 | |} \sum_{i = 1}^{n} A (w_{i}) Δ_{i} x

= \int_{0}^{h} A (y) d y

= \int_{0}^{h} π r^{2} d y

= π r^{2} h

مثال: ما المساحة الجانبيه وما الحجم لاسطوانه قطر قاعدتها 14 دسم وارتفاعها 35 دسم؟ الحل : المساحة الجانبيه = 14×22/7 × 35 = 1540 دسم2

الحجم = 7×7×22/7×35= 5390 دسم3
أي 5390 لتر أو 5 كوب و 390 لتر

يمكن ايجاد حجم الاسطوانة مثل ايجاده في المنشور :

بضرب مساحة القاعدة في الارتفاع =

A \times h

ويمكن التوصل لنفس النتيجة باعتبار الاسطوانة مجسم دوراني ينشأ عن دوران دالة ثابتة حول المحور السيني

إذن يمكن حساب الحجم عن طريق =

π \times \int_{0}^{h} [d (x)]^{2} d x

مساحة السطح

Having radius $r$ and altitude (height) $h$ , 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: $π r 2$
the area of the bottom base: $π r 2$
the area of the side: $2π rh$

The area of the top and bottom bases is the same, and is called the base area, $B$ . The area of the side is known as the lateral area, $L$ .

An open cylinder does not include either top or bottom elements, and therefore has surface area (lateral area) $L = 2 π r h$

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 $A = L + 2 B = 2 π r h + 2 π r^{2} = 2 π r (h + r) = π d (r + h)$ where $d = 2 r$ is the diameter of the circular top or bottom.

For a given volume, the right circular cylinder with the smallest surface area has $h = 2 r$ . Equivalently, for a given surface area, the right circular cylinder with the largest volume has $h = 2 r$ , that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).^[4]

The lateral area, $L$ , of a circular cylinder, which need not be a right cylinder, is more generally given by $L = e \times p,$ where $e$ is the length of an element and $p$ is the perimeter of a right section of the cylinder.^[5] 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 cylindrical shell) 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 $h$ , internal radius $r$ , and external radius $R$ . The volume is given by subtracting the volume of the inner imaginary cylinder (i.e. hollow space) from the volume of the outer cylinder: $V = π (R^{2} - r^{2}) h = 2 π (\frac{R + r}{2}) h (R - r) .$ Thus, the volume of a cylindrical shell equals 2 $π$ × average radius × height × thickness.^[6]

The surface area, including the top and bottom, is given by $A = 2 π (R + r) h + 2 π (R^{2} - r^{2}) .$ Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.^[7]

في الكرة والأسطوانة

In the treatise by this name, written 225 BCEح. 225 BCE, 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 two-thirds that of the circumscribed cylinder and a surface area two-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 radius $r$ is $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}4/3πr3 = 2/3 (2πr3)$ . The surface area of this sphere is $4 π r 2 = 2 / 3 (6 π r 2)$ . A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.

الأسطح الأسطوانية

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.^[8] Such cylinders have, at times, been referred to as generalized cylinders. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder.^[9] 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.^[10]

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 $f (x, y, z) = A x^{2} + B y^{2} + C z^{2} + D x + E y + G z + H = 0,$ with the coefficients being real numbers and not all of $A$ , $B$ and $C$ 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 $z$ does not appear and the general equation of this type of degenerate quadric can be written as^[11] $A {(x + \frac{D}{2 A})}^{2} + B {(y + \frac{E}{2 B})}^{2} = ρ,$ where $ρ = - H + \frac{D^{2}}{4 A} + \frac{E^{2}}{4 B} .$

Elliptic cylinder

If $AB > 0$ this is the equation of an elliptic cylinder.^[11] Further simplification can be obtained by translation of axes and scalar multiplication. If $ρ$ has the same sign as the coefficients $A$ and $B$ , then the equation of an elliptic cylinder may be rewritten in Cartesian coordinates as: ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = 1 .$ This equation of an elliptic cylinder is a generalization of the equation of the ordinary, circular cylinder ( $a = b$ ). Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid.

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

Hyperbolic cylinder

If $A$ and $B$ have different signs and $ρ \neq 0$ , we obtain the hyperbolic cylinders, whose equations may be rewritten as: ${(\frac{x}{a})}^{2} - {(\frac{y}{b})}^{2} = 1 .$

Parabolic cylinder

Finally, if $AB = 0$ assume, without loss of generality, that $B = 0$ and $A = 1$ to obtain the parabolic cylinders with equations that can be written as:^[12] $x^{2} + 2 a y = 0 .$

أشكال أخرى من الاسطوانة

An elliptic cylinder, or cylindroid, is a quadric surface, with the following equation in Cartesian coordinates:

{(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = 1 .

This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder ( $a = b$ ). Even more general is the generalized cylinder: the cross-section can be any curve.

The cylinder is a degenerate quadric because at least one of the coordinates (in this case $z$ ) does not appear in the equation.

An oblique cylinder has the top and bottom surfaces displaced from one another.

There are other more unusual types of cylinders. These are the imaginary elliptic cylinders:

{(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = - 1

the hyperbolic cylinder:

{(\frac{x}{a})}^{2} - {(\frac{y}{b})}^{2} = 1

and the parabolic cylinder:

x^{2} + 2 a y = 0 .

الهندسة الاسقاطية

في الهندسة الإسقاطية، الأسطوانة هي أساساً قمع رأسه في مالا نهاية.

وهو أمر مفيد في تعريف degenerate conics، التي تتطلب اعتبار cylindrical conics.

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.^[13]

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

الموشور

Tycho Brahe Planetarium in Copenhagen is an example of a truncated cylinder.

A solid circular cylinder can be regarded as the limiting case of a $n$ -gonal prism as $n$ approaches infinity. Because of this close relationship, many earlier geometry texts treat prisms and cylinders together. Standard formulas for the surface area and volume of a cylinder can be derived from the corresponding formulas for prisms by considering inscribed and circumscribed prisms and taking the limit as the number of sides increases without bound.^[14]

The focus on circular cylinders in classical treatments arises from the fact that a circular base is the only plane figure for which this limiting argument can be carried out using elementary methods, without relying on calculus. Cylinders and prisms also share parallel terminology: for example, a truncated prism is a prism whose bases do not lie in parallel planes, and by analogy a solid cylinder with non-parallel bases is called a truncated cylinder.

From a polyhedral perspective, a cylinder may also be viewed as the dual of a bicone, interpreted as an infinite-sided bipyramid.

Family of uniform n-gonal prisms 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												...