Serendipity
Dec. 10th, 2011 12:51 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowOne for the funny coincidence file:
My current research involves identifying possibly-interesting classes of polygons. (You know: like rectangles, or trapezoids, or whatever.) I've devised a systematic procedure for doing so. Unfortunately, the procedure is algebraic, so I get a description of the class of polygons in terms of an equation. Extracting geometric content from that equation is a difficult task (and one I need to improve on before I can get anything publishable from this stuff).
So. One class of quadrilaterals that looks as though it should be very interesting has been refusing to reveal its secrets to me, and I've been worrying at it for a week or so now, without making any visible progress. It involves geometrically describing a certain line (or, actually, a pair of perpendicular lines), which points in... a certain, algebraically-described direction.
This morning, as I was fixing dinner, an idea struck me. I've seen that direction before. I published a paper about three years ago, in an area which isn't really my cup of tea; I had stumbled across something, and it turned out to be publishable. Well, that very paper mentions that pair of lines! I just needed to recognize that.
I'm not quite out of the woods, though. Here's the deal. Associated with every quadrilateral is a parallelogram, which I call its "first component"; the difference between the parallelogram and the quadrilateral is a vector, the "second component". Now, if you have a parallelogram, there's an ellipse which is tangent to all four sides at their midpoints. The directions of the axes of that ellipse are the directions I'm looking for; and the geometric description is that the second component points along one of the axes of that ellipse. Short of actually constructing the first and second components and then the ellipse, though, I don't see how to relate that to the original quadrilateral - but that should be a much easier task than I was originally faced with.
(It's got to be an interesting class. I've been able to verify that it includes all parallelograms, kites, isosceles trapezoids, antiparallelograms, and "flats" - quadrilaterals whose vertices all lie along the same line.)
Serendip ho!
My current research involves identifying possibly-interesting classes of polygons. (You know: like rectangles, or trapezoids, or whatever.) I've devised a systematic procedure for doing so. Unfortunately, the procedure is algebraic, so I get a description of the class of polygons in terms of an equation. Extracting geometric content from that equation is a difficult task (and one I need to improve on before I can get anything publishable from this stuff).
So. One class of quadrilaterals that looks as though it should be very interesting has been refusing to reveal its secrets to me, and I've been worrying at it for a week or so now, without making any visible progress. It involves geometrically describing a certain line (or, actually, a pair of perpendicular lines), which points in... a certain, algebraically-described direction.
This morning, as I was fixing dinner, an idea struck me. I've seen that direction before. I published a paper about three years ago, in an area which isn't really my cup of tea; I had stumbled across something, and it turned out to be publishable. Well, that very paper mentions that pair of lines! I just needed to recognize that.
I'm not quite out of the woods, though. Here's the deal. Associated with every quadrilateral is a parallelogram, which I call its "first component"; the difference between the parallelogram and the quadrilateral is a vector, the "second component". Now, if you have a parallelogram, there's an ellipse which is tangent to all four sides at their midpoints. The directions of the axes of that ellipse are the directions I'm looking for; and the geometric description is that the second component points along one of the axes of that ellipse. Short of actually constructing the first and second components and then the ellipse, though, I don't see how to relate that to the original quadrilateral - but that should be a much easier task than I was originally faced with.
(It's got to be an interesting class. I've been able to verify that it includes all parallelograms, kites, isosceles trapezoids, antiparallelograms, and "flats" - quadrilaterals whose vertices all lie along the same line.)
Serendip ho!
