Re: Known lengths of edges of an abitrary polygon, seek for area
- From: quasi <quasi@xxxxxxxx>
- Date: Mon, 26 Dec 2005 19:15:38 -0500
On Mon, 26 Dec 2005 23:47:08 +0000 (UTC), klewis@xxxxxxxxxxxxxxx
(Keith A. Lewis) wrote:
>quasi <quasi@xxxxxxxx> writes in article <llq0r1ph9852j286sr6k942jnn40q7uv7j@xxxxxxx> dated Mon, 26 Dec 2005 17:44:36 -0500:
>>On Mon, 26 Dec 2005 22:04:04 +0000 (UTC), klewis@xxxxxxxxxxxxxxx
>>(Keith A. Lewis) wrote:
>>
>>>Right now I'm looking at the (1,1,1,1,3) pentagon. It looks like there
>>>might be 2 possible radii.
>>
>>No, there can't be 2 radii. For a given chord length, the central
>>angle for that chord decreases as the radius increases. Since the
>>central angles have to add up to 2*Pi, it follows that if there is a
>>radius, it's unique.
>
>They only add up to 2*pi if the polygon contains the center of the circle.
>Consider an obtuse triangle inscribed in a circle -- the signed central
>angles add up to 0. If you use angles in the range (0,2*pi) they must sum
>to 2*pi, hmmm.
>
>What I'm looking for is an example where you get one radius when the
>inscribed polygon contains the center and another when it doesn't.
>
>With my (1,1,1,1,3) pentagon, I'm getting the same cubic either way (with
>only 1 real root which makes r=~1.5514489). Maybe you're right about this
>and I'm just not seeing it. I was hoping to find that the area is the same
>regardless of radius. But proof that the radius is unique makes the
>conjecture more robust.
>
>>(4) Does an n-gon inscribed in a circle have the maximum area
>>over all convex polygons using the same sequence of side lengths?
>
>--Keith Lewis klewis {at} mitre.org
>The above may not (yet) represent the opinions of my employer.
Yes, I noted the typo:
In this reply you corrected (1,1,1,1,5) to (1,1,1,1,3).
.
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