[math-fun] regular _elliptical_ n-gons?
In playing with the Marden Theorem, I started thinking about approximations to ellipses, and was wondering about regular elliptical n-gons. I'm not talking about a regular n-gon stretched/squashed or squashed in the X or Y direction, because for stretched/squashed n-gons the lengths of those sides are no longer equal. Since the perimeter of an ellipse isn't easily calculated, I would imagine that this will be a difficult problem. What I'm really looking for is a kind of inverse to the ellipse perimeter problem: given n sides of length 1 and a major/minor radius ratio (alternately an eccentricity), how to draw the ellipse itself. Which major/minor ratios and/or eccentricities can actually be represented this way? I would assume that someone has looked into this sort of problem, but I wouldn't even know where to begin to look.
Is what you want related to Poncelet's Porism? http://mathworld.wolfram.com/PonceletsPorism.html On Wed, Nov 17, 2010 at 11:39 PM, Henry Baker <hbaker1@pipeline.com> wrote:
In playing with the Marden Theorem, I started thinking about approximations to ellipses, and was wondering about regular elliptical n-gons. I'm not talking about a regular n-gon stretched/squashed or squashed in the X or Y direction, because for stretched/squashed n-gons the lengths of those sides are no longer equal.
Since the perimeter of an ellipse isn't easily calculated, I would imagine that this will be a difficult problem.
What I'm really looking for is a kind of inverse to the ellipse perimeter problem: given n sides of length 1 and a major/minor radius ratio (alternately an eccentricity), how to draw the ellipse itself. Which major/minor ratios and/or eccentricities can actually be represented this way?
I would assume that someone has looked into this sort of problem, but I wouldn't even know where to begin to look.
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Rita the dog asked a more specific but related question on Y!A a while back, with links to a couple of pictures in the additional details. See here: http://answers.yahoo.com/question/index;_ylt=As6rCggttnM1wrVEcxQE_VXty6IX;_y... On Wed, Nov 17, 2010 at 10:39 PM, Henry Baker <hbaker1@pipeline.com> wrote:
In playing with the Marden Theorem, I started thinking about approximations to ellipses, and was wondering about regular elliptical n-gons. I'm not talking about a regular n-gon stretched/squashed or squashed in the X or Y direction, because for stretched/squashed n-gons the lengths of those sides are no longer equal.
Since the perimeter of an ellipse isn't easily calculated, I would imagine that this will be a difficult problem.
What I'm really looking for is a kind of inverse to the ellipse perimeter problem: given n sides of length 1 and a major/minor radius ratio (alternately an eccentricity), how to draw the ellipse itself. Which major/minor ratios and/or eccentricities can actually be represented this way?
I would assume that someone has looked into this sort of problem, but I wouldn't even know where to begin to look.
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participants (3)
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Henry Baker -
James Buddenhagen -
Victor Miller