A school warm-up exercise to occupy your time - page 2
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Algorithmically, it's a simple search, take an angle, identify the limits of change, search - and then recursively, selecting the maximum area. The accuracy and duration depends on the choice of angle at each step.
But the total duration is rather long to put it mildly.
If you shove it into some optimizer, it should converge faster.
Just look for the radius R of the circumcircle. Express the angle Ai between radii to the ends of i-th side of the cubicle by R and its length Li. The sum of all Ai must be equal to 2*Pi. We obtain the equation for R.
1) It turns out that the order of sides is unimportant.
2) The area of the mnc is easily expressed by Ai and R
For an N-facet with fixed side lengths, you also need to know the angles between the N-3 sides. Then we can find the area of a particular figure. But the maximal possible area (for: sides known, angles arbitrary) is the only one
The angle will be variable. The formula must have three variables.
Or you can take not the angle as a variable, but the third side of a triangle formed by two adjacent sides.
Simply find the radius R of the circumcircle. Express the angle Ai between the radii to the ends of the i-th side of the cubit by R and the length of this side Li. The sum of all Ai must be equal to 2*Pi. We obtain the equation for R.
Then the problem splits in 2 - find radius of minimal circumcircle (because there are a lot of circumcircles) and then what ?
somehow change angles between sides so that R is minimal... we can also say that if sum of angles->max then area->max, but this does not make algorithmic search (or formula output) for maximal area easier.
Then the problem is split into 2 - find the radius of the smallest circumcircle (because there are many circumcircles) and then what ?
somehow change angles between sides to minimize R... we can also say that if sum of angles->max, area->max, but this doesn't help in algorithmic search (or in formulae) for maximal area
Ai = 2*arcsin(Li/(2*R))
A1+A2+A3+A4 = 2*Pi - the equation for finding R, which will have to be solved numerically (e.g. by dichotomy)
Maybe you should start by looking through the reference books, maybe there is a solution already?
There is a theorem (Cramer's, I think) which says that the area of a polygon with given sides will be maximal when its vertices lie on a circle.
How do you prove it? I can't think of a simple way.
____
There is a theorem (Cramer's, I think) which says that the area of a polygon with given sides will be maximal when its vertices lie on a circle.
saw already when wrote
How do you prove it? I can't think of a simple way.
____
I saw it when I wrote it.
I have to think about it, but I'm too lazy for some reason)
Ancient task
There is 100 roubles.
solve with nested cyclesHow many bulls, cows and calves can you buy with all this money,
if the fee for a bull is 10 roubles,
for a cow is 5 roubles,
for a calf is 0.5 roubles
and 100 cattle must be bought?
An old problem
There are 100 100 roubles.How many bulls, cows and calves can be bought with all this money, if the fee for a bull is 10 10 roubles, for a cow - 5 5 roubles, for a calf - 0.5 0.5 roubles and you have to buy 100 100 cattle?
"per calf - 0.5 0.5 roubles"?
how do you understand that?