Determine quadrilateral with given sides and minimal difference between interior angles
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Define the "spread" of a planar quadrilateral as the difference between its largest and smallest angles. Given positive numbers $a,b,c,d,$ find the quadrilateral of minimal spread and sides $a,b,c,d$ in cyclic order. I believe that the two largest angles of the optimum quadrilateral will be equal and adjacent, but I have no idea how to prove it.
This question is inspired by this one. The OP says I guessed the solution, but it was an accident. I thought he was talking about the above problem, but apparently not. I'm still not sure what problem he actually meant. I guessed that the quadrilateral would be cyclic. The OP's solution was just an assertion that this was correct, and I decided to try to prove that my guess was correct for the problem I have given above. It turns out to be sadly wrong.
I couldn't think of any way to attack the problem with calculus. How do you handle the minimum? I thought about analyzing the problem on the assumption that near optimality, once would always be looking at the difference of the same two variable angles, but even so, the formulas look unpleasant. One can calculate the angles with
$arccos(Acdot BVert AVert Vert BVert)$ but that doesn't account for reflex angles. Again, perhaps one can prove that the optimum quadrilateral must be convex, so that this complication can be ignored.
Anyway, before trying to solve the problem rigorously, I wrote an interactive script that allowed me to drag the vertices of the quadrilateral around. The program calculates the spread and keeps track of the minimum. In all the experiments I have performed, it turn out that the two largest angles are equal and adjacent. Here are a couple of examples:
My guess about cyclic quadrilaterals was sadly mistaken
If this result about the two equal angles is known, can you give me a reference? If it is not known, do you have any suggestions about how to go about proving (or disproving) it?
geometry optimization computational-geometry
asked Aug 26 at 13:09
saulspatz
11.3k21324
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Define the "spread" of a planar quadrilateral as the difference between its largest and smallest angles. Given positive numbers $a,b,c,d,$ find the quadrilateral of minimal spread and sides $a,b,c,d$ in cyclic order. I believe that the two largest angles of the optimum quadrilateral will be equal and adjacent, but I have no idea how to prove it.
This question is inspired by this one. The OP says I guessed the solution, but it was an accident. I thought he was talking about the above problem, but apparently not. I'm still not sure what problem he actually meant. I guessed that the quadrilateral would be cyclic. The OP's solution was just an assertion that this was correct, and I decided to try to prove that my guess was correct for the problem I have given above. It turns out to be sadly wrong.
I couldn't think of any way to attack the problem with calculus. How do you handle the minimum? I thought about analyzing the problem on the assumption that near optimality, once would always be looking at the difference of the same two variable angles, but even so, the formulas look unpleasant. One can calculate the angles with
$arccos(Acdot BVert AVert Vert BVert)$ but that doesn't account for reflex angles. Again, perhaps one can prove that the optimum quadrilateral must be convex, so that this complication can be ignored.
Anyway, before trying to solve the problem rigorously, I wrote an interactive script that allowed me to drag the vertices of the quadrilateral around. The program calculates the spread and keeps track of the minimum. In all the experiments I have performed, it turn out that the two largest angles are equal and adjacent. Here are a couple of examples:
My guess about cyclic quadrilaterals was sadly mistaken
If this result about the two equal angles is known, can you give me a reference? If it is not known, do you have any suggestions about how to go about proving (or disproving) it?
geometry optimization computational-geometry
asked Aug 26 at 13:09
saulspatz
11.3k21324
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Define the "spread" of a planar quadrilateral as the difference between its largest and smallest angles. Given positive numbers $a,b,c,d,$ find the quadrilateral of minimal spread and sides $a,b,c,d$ in cyclic order. I believe that the two largest angles of the optimum quadrilateral will be equal and adjacent, but I have no idea how to prove it.
This question is inspired by this one. The OP says I guessed the solution, but it was an accident. I thought he was talking about the above problem, but apparently not. I'm still not sure what problem he actually meant. I guessed that the quadrilateral would be cyclic. The OP's solution was just an assertion that this was correct, and I decided to try to prove that my guess was correct for the problem I have given above. It turns out to be sadly wrong.
I couldn't think of any way to attack the problem with calculus. How do you handle the minimum? I thought about analyzing the problem on the assumption that near optimality, once would always be looking at the difference of the same two variable angles, but even so, the formulas look unpleasant. One can calculate the angles with
$arccos(Acdot BVert AVert Vert BVert)$ but that doesn't account for reflex angles. Again, perhaps one can prove that the optimum quadrilateral must be convex, so that this complication can be ignored.
Anyway, before trying to solve the problem rigorously, I wrote an interactive script that allowed me to drag the vertices of the quadrilateral around. The program calculates the spread and keeps track of the minimum. In all the experiments I have performed, it turn out that the two largest angles are equal and adjacent. Here are a couple of examples:
My guess about cyclic quadrilaterals was sadly mistaken
If this result about the two equal angles is known, can you give me a reference? If it is not known, do you have any suggestions about how to go about proving (or disproving) it?
geometry optimization computational-geometry
asked Aug 26 at 13:09
saulspatz
11.3k21324
Define the "spread" of a planar quadrilateral as the difference between its largest and smallest angles. Given positive numbers $a,b,c,d,$ find the quadrilateral of minimal spread and sides $a,b,c,d$ in cyclic order. I believe that the two largest angles of the optimum quadrilateral will be equal and adjacent, but I have no idea how to prove it.
This question is inspired by this one. The OP says I guessed the solution, but it was an accident. I thought he was talking about the above problem, but apparently not. I'm still not sure what problem he actually meant. I guessed that the quadrilateral would be cyclic. The OP's solution was just an assertion that this was correct, and I decided to try to prove that my guess was correct for the problem I have given above. It turns out to be sadly wrong.
I couldn't think of any way to attack the problem with calculus. How do you handle the minimum? I thought about analyzing the problem on the assumption that near optimality, once would always be looking at the difference of the same two variable angles, but even so, the formulas look unpleasant. One can calculate the angles with
$arccos(Acdot BVert AVert Vert BVert)$ but that doesn't account for reflex angles. Again, perhaps one can prove that the optimum quadrilateral must be convex, so that this complication can be ignored.
Anyway, before trying to solve the problem rigorously, I wrote an interactive script that allowed me to drag the vertices of the quadrilateral around. The program calculates the spread and keeps track of the minimum. In all the experiments I have performed, it turn out that the two largest angles are equal and adjacent. Here are a couple of examples:
My guess about cyclic quadrilaterals was sadly mistaken
If this result about the two equal angles is known, can you give me a reference? If it is not known, do you have any suggestions about how to go about proving (or disproving) it?
geometry optimization computational-geometry
asked Aug 26 at 13:09
saulspatz
11.3k21324
asked Aug 26 at 13:09
saulspatz
11.3k21324
asked Aug 26 at 13:09
saulspatz
11.3k21324
asked Aug 26 at 13:09
saulspatz
11.3k21324
11.3k21324
add a comment |Â
add a comment |Â
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