Unwrapping Curves from Cylinders and Cones

Abstract

Now imagine the elliptical cross section replaced by any curve lying on the surface of a right circular cylinder. What happens to this curve when the cylinder is unwrapped? Consider also the inverse problem, which you can experiment with by yourself: Start with a plane curve (line, circle, parabola, sine curve, etc.) drawn with a felt pen on a rectangular sheet of transparent plastic, and roll the sheet into cylinders of different radii. What shapes does the curve take on these cylinders? How do they appear when viewed from different directions? A few trials reveal an enormous number of possibilities, even for the simple case of a circle. This paper formulates these somewhat vague questions more precisely, in terms of equations, and shows that they can be answered with surprisingly simple twodimensional geometric transformations, even when the cylinder is not circular. For a circular cylinder, a sinusoidal influence is always present, as exhibited in Figures