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Breakthrough
Feb. 1st, 2007 09:30 pm:bounce bounce:
As I've mentioned from time to time, my current research interest has to do with parametrizing polygons - coming up with quantities which somehow characterize them, distinguishing different (for various values of the word) polygons from each other. My main focus for the last while has been the classification of quadrilaterals up to affine equivalence. (Two quadrilaterals are "affine equivalent" if you can turn one into the other by applying a matrix and/or a translation.) I solved that problem, in one sense, quite a while ago; I have two parameters which do the job completely. (Two quadrilaterals give the same values for those parameters if and only if they're affine equivalent.) But, though I could compute the parameters for any quadrilateral, I had no clear idea of their meaning. Other lines of research gave tantalizing hints, but nothing I could really get a grip on.
The tide finally came in last week: I came up with solid and interesting geometric interpretations of both parameters. (One measures the failure of the diagonals to bisect each other; the other is determined by, on the one hand, the failure of the diagonals to bisect the quadrilateral, and on the other by the failure of the midlines - the lines connecting the midpoints of opposite sides - to do so.)
Tonight, a new thought struck me, and about forty minutes of computation gave me a second, completely different and equally interesting interpretation. Now I've got a link between two different properties of quadrilaterals.
Throw in those tantalizing hints I mentioned before, and I've got meat enough for my next paper.
:bounce bounce:
As I've mentioned from time to time, my current research interest has to do with parametrizing polygons - coming up with quantities which somehow characterize them, distinguishing different (for various values of the word) polygons from each other. My main focus for the last while has been the classification of quadrilaterals up to affine equivalence. (Two quadrilaterals are "affine equivalent" if you can turn one into the other by applying a matrix and/or a translation.) I solved that problem, in one sense, quite a while ago; I have two parameters which do the job completely. (Two quadrilaterals give the same values for those parameters if and only if they're affine equivalent.) But, though I could compute the parameters for any quadrilateral, I had no clear idea of their meaning. Other lines of research gave tantalizing hints, but nothing I could really get a grip on.
The tide finally came in last week: I came up with solid and interesting geometric interpretations of both parameters. (One measures the failure of the diagonals to bisect each other; the other is determined by, on the one hand, the failure of the diagonals to bisect the quadrilateral, and on the other by the failure of the midlines - the lines connecting the midpoints of opposite sides - to do so.)
Tonight, a new thought struck me, and about forty minutes of computation gave me a second, completely different and equally interesting interpretation. Now I've got a link between two different properties of quadrilaterals.
Throw in those tantalizing hints I mentioned before, and I've got meat enough for my next paper.
:bounce bounce:
