What a fascinating excercise in geometry....You probably didn't realize it, but if you'll look up the Wankel Rotary engine, you'll discover that the interior shape outlined in your drawing has some very special (and unexpected) properties.
Also, the process of taking a set of equally-spaced points along two lines and then connecting them in the way you've shown has interesting properties of its own, I think. For instance, if the two lines are at 90 degrees to each other, the connecting lines will all be tangent to one quadrant of a circle with infinite radius. Once the angle between the two lines drops below 90, however, I think the connecting lines all become tangent to some hyperbolic curve, with its diagonal asymptotes being equal to the lines the original points were drawn along.
(I should clarify that I was talking about the third, colored drawing.)
And upon further consideration, I'd like to retract my previous statement. If the series of points were continued indefinitely along the two original lines, the definition for the interior shape breaks down (though there would be exactly one point on the interior shape at precisely 1/infinity along the line bisecting the angle made between the two original lines (which we all know is simply pathological)).
So! We must revise our definition. For a finite set of equally-spaced points along two intersecting, non-parallel lines, if lines are drawn connecting every opposite pair of points such that the points closest to the vertex along one line are connected to the points furthest from the vertex of the other, these connecting lines will all be tangent to a single hyperbolic curve.
Yes, yes I think that's valid. I leave a rigorous proof to all those dorky white guys. pshh
any curved lines in the last picture is due to water warping the paper
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Most everything is just empty space...Mmm...
What a fascinating excercise in geometry....You probably didn't realize it, but if you'll look up the Wankel Rotary engine, you'll discover that the interior shape outlined in your drawing has some very special (and unexpected) properties.
Also, the process of taking a set of equally-spaced points along two lines and then connecting them in the way you've shown has interesting properties of its own, I think. For instance, if the two lines are at 90 degrees to each other, the connecting lines will all be tangent to one quadrant of a circle with infinite radius. Once the angle between the two lines drops below 90, however, I think the connecting lines all become tangent to some hyperbolic curve, with its diagonal asymptotes being equal to the lines the original points were drawn along.
Pretty sweet, eh?
(I should clarify that I was talking about the third, colored drawing.)
And upon further consideration, I'd like to retract my previous statement. If the series of points were continued indefinitely along the two original lines, the definition for the interior shape breaks down (though there would be exactly one point on the interior shape at precisely 1/infinity along the line bisecting the angle made between the two original lines (which we all know is simply pathological)).
So! We must revise our definition. For a finite set of equally-spaced points along two intersecting, non-parallel lines, if lines are drawn connecting every opposite pair of points such that the points closest to the vertex along one line are connected to the points furthest from the vertex of the other, these connecting lines will all be tangent to a single hyperbolic curve.
Yes, yes I think that's valid. I leave a rigorous proof to all those dorky white guys. pshh
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