IngentaConnect Unwrapping Curves from Cylinders and Cones

Authors: Apostol, Tom M.; Mnatsakanian, Mamikon A.

Source: American Mathematical Monthly, Volume 114, Number 5, May 2007 , pp. 388-416(29)

Publisher: Mathematical Association of America

Abstract:

What happens to the shape of a curve lying on the surface of a circular cylinder when the cylinder is unwrapped onto a plane? Conversely, draw a plane curve on transparent plastic, and roll it into cylinders of different radii. What shapes does the curve take on these cylinders? How do they appear when viewed from different directions? Similar questions are investigated for space curves unwrapped from the surface of a right circular cone, including conic sections, spirals, and geodesics. Unwrapped conic sections produce a new class of plane curves called generalized conics.

This paper formulates these somewhat vague questions in terms of equations, and analyzes them with surprisingly simple two-dimensional geometric transformations that lead to many unexpected results. The methods for analyzing cones and cylinders differ substantially, but both use the fact that unwrapping a developable surface preserves arclength. Applications are given to diverse fields such as descriptive geometry, computer graphics, sheet metal construction, and educational hands-on activities.

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