Cylinder (EntityTopic, 14)
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- | {{Shape| | + | <[#ontology [kind topic] [cats 3D Curved Rotatope] [alt [[freebase:03h_4m]] [[wikipedia:Cylinder_(geometry)]]]]> |
- | + | {{STS Shape | |
- | A '''cylinder''' is a special case of a [[prism]] where the base is a [[circle]]. | + | | image=<[#embed [hash BQMXW9YSFVDCTRCX8119RZYAJV] [width 180]]> |
+ | | dim=3 | ||
+ | | elements=2 [[circle]]s, 1 hose, 2 circular edges, 0 | ||
+ | | genus=0 | ||
+ | | ssc=[(xy)z] | ||
+ | | ssc2=+T2 | ||
+ | | pv_circle=~0.3934 | ||
+ | | pv_square=π⁄4 ≈ 0.7854 | ||
+ | | extra={{STS Matrix| | ||
+ | <i>2 2</i> <s>0 0</s> | ||
+ | 0 0 0 0 | ||
+ | 2 1 2 1}}{{STS Tapertope | ||
+ | | order=2, 0 | ||
+ | | notation=21 | ||
+ | | index=6 | ||
+ | }}{{STS Toratope | ||
+ | | expand=[[Cylinder|21]] | ||
+ | | notation=(II)I | ||
+ | | index=3a | ||
+ | }}{{STS Bracketope | ||
+ | | index=7 | ||
+ | | notation=[(ii)i]}}}} | ||
+ | A '''cylinder''' is a special case of a [[prism]] where the base is a [[circle]]. it has two circles at the ends connected by a surface, called a hose. | ||
- | + | == Equations == | |
*Variables: | *Variables: | ||
<blockquote>''r'' ⇒ radius of cylinder<br> | <blockquote>''r'' ⇒ radius of cylinder<br> | ||
Line 27: | Line 49: | ||
<blockquote>[!x,!y] ⇒ [[rectangle]] with width (2''r''cos(π''n''/2)), height (''h'')<br> | <blockquote>[!x,!y] ⇒ [[rectangle]] with width (2''r''cos(π''n''/2)), height (''h'')<br> | ||
[!z] ⇒ [[circle]] of radius (''r'')</blockquote> | [!z] ⇒ [[circle]] of radius (''r'')</blockquote> | ||
+ | |||
+ | == Homology groups == | ||
+ | All homology groups are zero unless stated. Here X is the shape in the given frame, and nZ is the direct sum of n copies of the group of integers Z. | ||
+ | |||
+ | ;1-frame (two circles):H<sub>0</sub>X = 2ℤ, H<sub>1</sub>X = 2ℤ | ||
+ | ;2-frame (2 disks and a tube):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = 0, H<sub>2</sub>X = ℤ | ||
+ | ;3-frame (solid cylinder):H<sub>0</sub>X = ℤ | ||
+ | |||
+ | == Cylindrogram == | ||
+ | A ''cylindrogram'' is the [[surface of revolution]] of a [[parallelogram]], just as a [[cylinder]] is the surface of revolution of a [[rectangle]]. It can also be thought of as a cylinder with a cone removed from one end and placed on the other. As such, this shape has the same [[volume]] as a cylinder with the same radius and height. | ||
<br clear="all"><br> | <br clear="all"><br> | ||
- | {{ | + | {{Trishapes}} |
- | {{ | + | {{Tapertope Nav|5|6|7|3<br>Sphere|21<br>Cylinder|111<br>Cube|hedra}} |
- | {{Bracketope Nav|6|7|8| | + | {{Toratope Nav A|2|3|4|III<br>Cube|(III)<br>Sphere|(II)I<br>Cylinder|((II)I)<br>Torus|IIII<br>Tesseract|(IIII)<br>Glome|hedra}} |
+ | {{Bracketope Nav|6|7|8|(III)<br>Sphere|[(II)I]<br>Cylinder|<(II)I><br>Bicone|hedra}} |
Latest revision as of 16:46, 25 March 2017
A cylinder is a special case of a prism where the base is a circle. it has two circles at the ends connected by a surface, called a hose.
Equations
- Variables:
r ⇒ radius of cylinder
h ⇒ height of cylinder
- All points (x, y, z) that lie on the surface of a cylinder will satisfy the following equations:
x2 + y2 = r2
abs(z) ≤ h/2
-- or --
x2 + y2 < r2
abs(z) = h/2
- All points (x, y, z) that lie on the edges of a cylinder will satisfy the following equations:
x2 + y2 = r2
abs(z) = h/2
- The hypervolumes of a cylinder are given by:
total edge length = 4πr
surface area = 2πr(r+h)
volume = πr2h
- The planar cross-sections (n) of a cylinder are:
[!x,!y] ⇒ rectangle with width (2rcos(πn/2)), height (h)
[!z] ⇒ circle of radius (r)
Homology groups
All homology groups are zero unless stated. Here X is the shape in the given frame, and nZ is the direct sum of n copies of the group of integers Z.
- 1-frame (two circles)
- H0X = 2ℤ, H1X = 2ℤ
- 2-frame (2 disks and a tube)
- H0X = ℤ, H1X = 0, H2X = ℤ
- 3-frame (solid cylinder)
- H0X = ℤ
Cylindrogram
A cylindrogram is the surface of revolution of a parallelogram, just as a cylinder is the surface of revolution of a rectangle. It can also be thought of as a cylinder with a cone removed from one end and placed on the other. As such, this shape has the same volume as a cylinder with the same radius and height.
Notable Trishapes | |
Regular: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
Direct truncates: | tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |
Mesotruncates: | stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |
Snubs: | snub staurohedron • snub rhodohedron |
Curved: | sphere • torus • cylinder • cone • frustum • crind |
5. 3 Sphere | 6. 21 Cylinder | 7. 111 Cube |
List of tapertopes |
2a. III Cube | 2b. (III) Sphere | 3a. (II)I Cylinder | 3b. ((II)I) Torus | 4a. IIII Tesseract | 4b. (IIII) Glome |
List of toratopes |
6. (III) Sphere | 7. [(II)I] Cylinder | 8. <(II)I> Bicone |
List of bracketopes |