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Rollups and Differential Geometry - INMAA Calumet
Indiana MAA Section meeting - Calumet (2009)
  • Sean A Broughton
We all know that cylinders and (frustums of) right cones can be formed by rolling up a flat strip of paper, metal, plastic, or other flexible material. In fact there are pictures of such "roll-ups" in Calculus books. However, what happens when we do not have such a standard cone shape? What region do we cut out of the paper or metal to achieve a desired cone shape? The problem started as a phone call from a local manufacturing design company who had to solve this problem. They wanted to build a specific shape but did not know what the flattened out shape would be. Since their plan was to build the part from a flattened sheet of metal, the answer to the roll-up problem was crucial. In this talk we discuss the geometry problem and show a solution using the techniques of differential geometry. The techniques are not advanced, in fact everything can be done with multi-variable Calculus and the simple separation of variables in Differential Equations. The time for the talk does not allow for a complete discussion of the "ghastly derivations" but we will discuss the formulas that allow us to solve the practical problem. The formulas can be evaluated using numerical integration (Calculus II) and we show the flattened out shape from the given problem. For those interested in full details see the technical report
  • slant cone,
  • ruled surface
Publication Date
October 17, 2009
Purdue University - Calumet
Citation Information
Sean A Broughton. "Rollups and Differential Geometry - INMAA Calumet" Indiana MAA Section meeting - Calumet (2009)
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Creative Commons License
This work is licensed under a Creative Commons CC_BY-NC-SA International License.