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CSpeed0001
Member since Feb-19-03
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Jan-31-05, 05:44 PM (EST) |
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"shortest distance between curves"
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A friend of mine is in a calculus 4 class, and shwas asked to find the shortest distance between two curves (a line and a parabola). She asked me how I would approach it and I immediately thought Calculus of Variations. I don't have much experience with that subject, so I was unable to help her much. The approach she was using was to begin by assuming the shortest distance would have to occur on a line perpendicular to the original line. Then she differentiated the parabola, found when the slope would equal the slope of the line, and then found the distance between the two points. I was able to help her prove that the distance vector was in fact the shortest when it was perpendicular to the line (by dividing it into normal and tangential components). This approach seems to work, but I think that it has too much intuition and not enough rigor. I was wondering if anyone else had other ideas. --CS |
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CSpeed0001
Member since Feb-19-03
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Feb-02-05, 05:05 PM (EST) |
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2. "RE: shortest distance"
In response to message #1
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>The trick was in that the tangent was parallel to the line. In the case where they intersect, their slopes would not be equal. I worked it out for two arbitrary functions (versus a line and a parabola), and was hoping that it would show up as a discontinuity or something, but I didn't see it. Would the case of the functions intersecting have to be handled as a test that is done before finding the minimum distance? --CS |
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CSpeed0001
Member since Feb-19-03
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Feb-07-05, 04:37 PM (EST) |
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4. "RE: shortest distance"
In response to message #3
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Thanks for your help, I was able to figure it out. I had forgotten that values where the derivative is indefined are also considered stationary. I had been throwing out the ugly square root thing that appeared in the denominator (which works out to zero when the functions intersect). I was able to prove what I had set out to prove. Thanks --CS |
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