Cylinder (EntityTopic, 14)

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Revision as of 20:52, 8 February 2014

A cylinder is a special case of a prism where the base is a circle.

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.