Unwrapping Curves

In Mathematical Snapshots, Hugo Steinhaus described the following method for generating an interesting curve. Wrap a strip of paper around a cylindrical candle. Using a sharp knife, slice the wrapped candle into two pieces, making the cut at an oblique angle. The candle's cross section at the location of the cut is an ellipse. Unwinding each of the resulting paper scraps reveals a sinusoidal edge where the paper was cut.


An elliptical cross section of a cylinder becomes sinusoidal when unwrapped.

A number of years ago, Mamikon A. Mnatsakanian showed that you can get the same result when you dip a cylindrical paint roller at an angle into a container of paint. When you roll the applicator on a flat surface, you also see a sinusoidal wave pattern.


A paint roller can be used to print sinusoidal waves on a flat surface.

Now, Mnatsakanian and Tom M. Apostol have extended Steinhaus's original demonstration further to include a wide variety of cross sections. "Imagine the elliptical cross section replaced by any curve lying on the surface of a right circular cylinder," they write in the May American Mathematical Monthly. "What happens to this curve when the cylinder is unwrapped?"

Apostol and Mnatsakanian also consider the inverse problem. What happens to curves on flat surfaces that are then rolled into cylinders? You can experiment with such transformations yourself simply by drawing curves (lines, circles, parabolas, and so on) on rectangular sheets of transparent plastic, then seeing what happens when these sheets are rolled into cylinders of different radii.

"A few trials reveal an enormous number of possibilities, even for the simple case of a circle," the mathematicians note.


Rolling a circle drawn on a sheet of transparent plastic into cylinders of various radii produces strikingly different geometric patterns. The dashed curves are on the rear half of the cylinder.

Working with just a straight line segment drawn on a transparency produces striking visual evidence of the nature of geodesics on a cylinder. A line segment on a flat sheet is the shortest path between the line's endpoints. Because distances are preserved when a cylinder is unwrapped, such a line segment, when rolled into a cylinder, becomes a geodesic (or the shortest path) on the cylinder. You see, no matter what the cylinder's radius, that such a geodesic on a circular cylinder is always part of a circular helix. Viewed from the side, the pattern looks like a sine wave.

For circular cylinders, Apostol and Mnatsakanian note, a sinusoidal influence is always present.

They go on to investigate the curves that result from drilling holes into cylinders and from unwrapping curves from a right circular cone.

A plane, for example, cuts a cone along a conic section. "We can analyze the shape of the corresponding unwrapped conic," Apostol and Mnatsakanian report. "This leads to a remarkable family of periodic plane curves that apparently have not been previously investigated."

References:

Apostol, T.M., and M.A. Mnatsakanian. 2007. Unwrapping curves from cylinders and cones. American Mathematical Monthly 114(May):388-416.

Steinhaus, H. 1969. Mathematical Snapshots, 3rd. ed. Oxford University Press.