Cylinder

The term "cylinder" has a number of related meanings. In its most general usage, the word "cylinder" refers to a solid bounded by a closed generalized cylinder (a.k.a. cylindrical surface) and two parallel planes (Kern and Bland 1948, p. 32; Harris and Stocker 1998, p. 102). A cylinder of this sort having a polygonal base is therefore a prism (Zwillinger 1995, p. 308). Harris and Stocker (1998, p. 103) use the term "general cylinder" to refer to the solid bounded a closed generalized cylinder.

Unfortunately, the term "cylinder" is commonly used not only to refer to the solid bounded by a cylindrical surface, but to the cylindrical surface itself (Zwillinger 1995, p. 311). To make matters worse, according to topologists, a cylindrical surface is not even a true surface, but rather a so-called surface with boundary (Henle 1994, pp. 110 and 129).

As if this were not confusing enough, the term "cylinder" when used without qualification commonly refers to the particular case of a solid of circular cross section in which the centers of the circles all lie on a single line (i.e., a circular cylinder). A cylinder is called a right cylinder if it is "straight" in the sense that its cross sections lie directly on top of each other; otherwise, the cylinder is said to be oblique. The unqualified term "cylinder" is also commonly used to refer to a right circular cylinder (Zwillinger 1995, p. 312), and this is the usage followed in this work.

The right cylinder of radius with axis given by the line segment with endpoints and is implemented in the Wolfram Language as Cylinder[x1, y1, z1, x2, y2, z2, r].

The illustrations above show a circular right cylinder of height and radius .

If a plane inclined with respect to the caps of a right circular cylinder intersects a cylinder, it does so in an ellipse. The cylinder was extensively studied by Archimedes in his two-volume work On the Sphere and Cylinder in ca. 225 BC.

As illustrated above, a cylinder can be described topologically as a square in which top and bottom edges are given parallel orientations and the left and right edges are joined to place the arrow heads and tails into coincidence (Gray 1997, pp. 322-323). The cylindrical surface of a circular cylinder has Euler characteristic 0 (Alexandroff 1998, p. 99).

The lateral surface of a cylinder of height and radius can be described parametrically by

			(1)
			(2)
			(3)

for and .

These are the basis for cylindrical coordinates. The lateral surface area and volume of the cylinder of height and radius are

			(4)
			(5)

The formula for the volume of a cylinder leads to the mathematical joke: "What is the volume of a pizza of thickness and radius ?" Answer: pi z z a. This result is sometimes known as the second pizza theorem.

If the top and bottom caps are added, the total surface area of a cylinder is given by

			(6)
			(7)

The interior of the cylinder of radius , height , and mass has moment of inertia tensor about its centroid is

(8)

The volume-to-total surface area ratio for a cylindrical solid is

(9)

which is related to the harmonic mean of the radius and height . The fact that

(10)

was known to Archimedes (Steinhaus 1999, p. 223).

Using the parametrization

			(11)
			(12)
			(13)

gives coefficients of the first fundamental form

			(14)
			(15)
			(16)

the coefficients of the second fundamental form

			(17)
			(18)
			(19)

area element

(20)

Gaussian curvature

(21)

mean curvature

(22)

and principal curvatures

			(23)
			(24)

It is possible to arrange seven finite cylinders so that each is tangent to the other six, as illustrated above.