Which of the following surfaces are cylinders?

Figure a.

Figure b.

Figure c.

 

ALL OF THEM!

Most people would identify Figure a as a cylinder, but would not consider all three to be cylinders. The commonly-recognized cylinder in Figure a is more specifically a circular cylinder. However, if the definition of a cylinder includes all three of the above, maybe we need to ask just what exactly is a cylinder?

Definition of a Cylinder:

A cylinder is the surface traced by a straight line moving parallel to itself (called the generatrix) and intersecting a fixed curve (called the directrix). A cylinder so formed extends to infinity in both directions; however, by cutting the cylinder with two parallel planes (forming a “top” and “bottom”) the resulting space is also considered a cylinder.

When the generatrix is perpendicular to the directrix, a right cylinder (or straight cylinder) is traced; otherwise, the cylinder is oblique:

Right Circular Cylinder

Oblique Circular Cylinder

 

Cylinders can be traced from any continuous curve or curve segment. The plane depicted at the top of this page is a cylinder whose directrix is simply a straight line. To the right of the plane is a parabolic cylinder (traced from a parabola).

A curve can generally be regarded as derivable from a straight line by a reversible transformation (i.e., y = f(x) and x = f^-1(y).)

References:

1. Hughes-Hallett, et al., Single and Multivariable Calculus, 3rd ed. Wiley. 2002. p. 571.
2. The Universal Encyclopedia of Mathematics. Simon and Schuster. 1964.
3. Webster’s Third New International Dictionary. Merriam-Webster. 1993.
4. "CYLINDER." LoveToKnow 1911 Online Encyclopedia. © 2003, 2004 LoveToKnow. http://59.1911encyclopedia.org/C/CY/CYLINDER.htm
5. Kokoska, Stephen, Fifty Famous Curves, Lots of Calculus Questions, And a Few Answers. Department of Mathematics, Computer Science, and Statistics, Bloomsburg University. http://facstaff.bloomu.edu/skokoska/curves.pdf