Timeline for Smallest dilation of a quadrilateral?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| May 19, 2017 at 12:45 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image links broken; now fixed.
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| Aug 16, 2010 at 19:07 | comment | added | TonyK | @Joseph: Your new diagram is correct. I just ran my program again, this time outputting points B and C; if A is (0,0) and D is (1,0), then B is (g,h) and C is (1-g,h), where g = 0.38822302188... and h = 0.78404544057... This agrees with your coordinates, after scaling. | |
| Aug 16, 2010 at 13:41 | vote | accept | Joseph O'Rourke | ||
| Aug 16, 2010 at 13:41 | history | edited | Joseph O'Rourke | CC BY-SA 2.5 |
Constructed Tony's quadrilateral.
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| Aug 14, 2010 at 20:08 | comment | added | TonyK | I have realised that I can get an almost optimal (and rigorous) lower bound on the discrepancy, by combining my computer search with continuity results: under certain conditions, we can rely on the fact that if B and C each move less than a distance epsilon, then the discrepancy of the quadrilateral changes less than some linear function of epsilon. So we can rigorously delete finite regions from the search space, not just points. But the details are messy, and the end result is still just a numerical lower bound. So I probably won't see it through, unless one of you nags me about it. | |
| Aug 14, 2010 at 20:01 | comment | added | TonyK | OK: in the diagram, replace 13 by 1; 5 by cos 2*theta; and 66/13 by w (where theta is 0.5555166235227462... radians, and w is to be determined). The perimeter P is 2*(cos 2*theta + w + 1), and the height h is sin 2*theta. Now just solve for w so that the discrepancy P/2h is equal to 1.89615765267304... | |
| Aug 12, 2010 at 21:53 | comment | added | Joseph O'Rourke | @TonyK: Great work! Could you please specify the resulting quadrilateral in some definitive form? I am having some difficulty extracting exactly what is "the trapezium above." | |
| Jul 26, 2010 at 14:52 | history | edited | TonyK | CC BY-SA 2.5 |
added 65 characters in body
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| Jul 25, 2010 at 16:18 | history | edited | TonyK | CC BY-SA 2.5 |
added 1105 characters in body
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| Jul 21, 2010 at 1:06 | comment | added | Joseph O'Rourke | @TonyK: Added a drawing of your first trapezium. | |
| Jul 21, 2010 at 1:05 | history | edited | Joseph O'Rourke | CC BY-SA 2.5 |
Added trapezium
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| Jul 21, 2010 at 0:41 | history | answered | TonyK | CC BY-SA 2.5 |