# Geometry: Maximize rectangles inside overlapping circles overlapping a rectangle

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• Præmier: \$200
• Modtagne indlæg: 2
• Vinder: ivanfordjarini

## Konkurrence Instruktioner

This isn’t as hard as it sounds but not as easy as it seems.

Given H and W of the large rectangle, I want to maximize the dimensions (h,w) of the 4 inner rectangles so that they minimize the unoccupied space in the large rectangle. Each inner rectangle must stay within the limits of the outer rectangle and within its respective circle and the radius (r) of the circles should be minimized as much as possible (diagram attached). Each inner rectangle can not overlap another rectangle nor any other circle. The inner rectangles might not need to be tangent to the outer rectangle (however I guess that they will be). I think a way to solve the problem would be to fit/maximize the small rectangles into the large rectangle and then decrease their size until you can fit a circle around them. Maximizing h and w, may or may not maximize the area of the inner squares (I'm not sure about that). If that's important to the calculation then you'll have to figure that out for yourself. Deliverables should be equations for h, w, and r based on H and W. The location of the rectangles/circles (and/or d1-d4) should be able to be derived from those equations. This is a highly symmetric problem which should make some calculations easier. However, H and W are variable, therefore d1 and d2 are likely to be different, as well as d3 and d4. Also, because of the symmetry, I assumed that all the circles would be the same size (therefore only one 'r' is listed) but it may be that there might be two different circle sizes. It is a requirement that this gets solved with circles but extra consideration will be given for those who can also solve it using ellipses instead of circles (circles are special case of ellipses)--but please make sure to solve for circles first!!! (or show the solution for the special case in the ellipse).

Winner

## Offentlig Præciserings Opslagstavle

• ###### pinetree800
• 3 år siden

This problem is easy for me. I can explain about solution.

• 3 år siden
1. ###### pinetree800
• 3 år siden

I solved problem correctly for arbitrary W and H .And, in special case fro circle, solved correctly by mathematical method. It's no doubt. I am confident.

• 3 år siden
2. Konkurrenceafholder
• 3 år siden

• 3 år siden
• ###### trinhngochai1
• 3 år siden

which object do you want to optimize? To minimize the unoccupied space in the large rectangle OR minimize r? These things can not be done at the same time, otherwise your problem is a vector optimization (i.e. multi objective optimization) and this problem DOES NOT have unique solution

• 3 år siden
1. Konkurrenceafholder
• 3 år siden

The primary goal is to minimize the unoccupied space of the outer rectangle using the configuration of the internal rectangles that I show in the image. IF this can only be solved using one circle radius, then the problem is solved (I suspect this will be the case). However, if there is more than one circle radius that will allow the unoccupied space to be minimized (i.e. more than one unique minimization solution), then I want the circle with the smallest radius that also gives the minimal unoccupied space.

• 3 år siden
2. ###### pinetree800
• 3 år siden

In fact, to minimize the unoccupied space of the outer rectangle =to max the occupied space of inner rectangle .Also it sames as optimization of function of rectangle area . And it same as problem to find w so that f'(w)=0 (f(w):function of area) or to find h so that g'(h)=0 (g(h):function of area) .

• 3 år siden
• ###### MatiasPereyraArg
• 3 år siden

are the four inner rectangles the same size?

• 3 år siden
1. Konkurrenceafholder
• 3 år siden

Yes. They are the same size.

• 3 år siden
• ###### trinhngochai1
• 3 år siden

Can I suppose W>2.2*H?

• 3 år siden
1. Konkurrenceafholder
• 3 år siden

No. W and H will need to be arbitrary.

• 3 år siden
• ###### bathtubdjinn
• 3 år siden

There are a few unclear things in your description of the problem. I'm going to rephrase what I think you're asking; can you tell me if this is what you mean?

• 3 år siden
1. ###### bathtubdjinn
• 3 år siden

The reason I ask is that it seems like, based on my uderstanding of your restrictions, there is a solution that covers at least HW-ɛ for any ɛ>0. I'll post the solution soon, but it might not be what you're looking for. (Also, my first reply to my original question contains the restatement.)

• 3 år siden
2. Konkurrenceafholder
• 3 år siden

I found your rephrasing. Yes, it seems that is what I am trying to express.

• 3 år siden
• ###### trinhngochai1
• 3 år siden

are H and W abitrary? Is there any condition on them? for example W>H>0?

• 3 år siden
1. Konkurrenceafholder
• 3 år siden

Yes, H and W are arbitrary with W*H>0. W>H or H>W will give the same results with only h and w switched respectively until W=H.

• 3 år siden
• Konkurrenceafholder
• 3 år siden

As noted below, concentric circles/rectangles do *not* give the optimal solution. This means that having the circle go through the corners of the rectangle as many people have given as the solution is not the optimal solution.

• 3 år siden
• Konkurrenceafholder
• 3 år siden

Some points:
1) Maximize the internal rectangles within the larger rectangle while still being bounded by a circle without that circle being intersected by any other rectangle.
2) Maximizing a rectangle within an ellipse will maximize the rectangle in the ellipse but it doesn't guarantee that it will maximize the area that can be occupied in the outer rectangle. In the case of circles, it does not give the optimal solution.
3) Based on numerical calculations, concentric circles and rectangles do NOT give the optimal solution.

• 3 år siden
1. ###### AndyZo
• 3 år siden

So the radius of a circumscribed circle doesn't have to be as less as possible. The overlapping percentage for rectangles is more preferable than the minimal radius. Do I correctly understand this aspect of the problem?

• 3 år siden
2. ###### AndyZo
• 3 år siden

You are right. If (h/w>1.5) or (w/h>1.5) - concentric circles seem to give not optimal solution. I'm working on it...

• 3 år siden
• ###### mfpleite
• 3 år siden

Hi there, as you very well stated, this is not particularly hard, but neither is as simple as people are making it seem. In particular you need to clarify a bit better what is that you want to optimize. Because answers will differ wether you want to maximize perimeters 2(h+w) or areas h*w. Also your minimisation of the radius, is that just a sort of constraint(?), or does it add to an optimising function, say h+w-r or (h*w - pi r2)? Could you please clarify these questions? Or say if you want the solutions for all these different scenarios.

• 3 år siden
1. ###### pinetree800
• 3 år siden

I solved problem by derived function.i.e. f'(w)=0.And I posted results.

• 3 år siden
2. Konkurrenceafholder
• 3 år siden

Yes, the inner rectangles should be the same size. Thank you for your entry. I was gone yesterday (sorry) and will get back to reviewing the entries today.

• 3 år siden
• ###### aminbros
• 3 år siden

I can do it for you. please check my profile.

• 3 år siden
• ###### Zhats
• 3 år siden

• 3 år siden
1. Konkurrenceafholder
• 3 år siden

• 3 år siden
2. ###### pinetree800
• 3 år siden

I submitted all results.

• 3 år siden
• ###### uetian09ee506
• 3 år siden

hi i can solve this problem

• 3 år siden
1. Konkurrenceafholder
• 3 år siden

• 3 år siden
• ###### mattsrinc
• 3 år siden

Hi, d1 and d2 are the same, for 4 same circles, e.g. circles are the same size, right? That was not 100% clear from description and that radius is then just one variable (e.g. not 2 or more) to minimize. And an interesting problem. :-)

• 3 år siden
1. Konkurrenceafholder
• 3 år siden

I'm pretty sure d1 and d2 will not be the same. They should be derived, and will be based on H and W.

• 3 år siden
2. Konkurrenceafholder
• 3 år siden

I've updated the description again. Please review.

• 3 år siden
• 3 år siden

done

• 3 år siden
• ###### ShahidFPGA
• 3 år siden

I have recently complete a project on geometrical object recognition. I can solve this issue in short time

• 3 år siden
• ###### arbw
• 3 år siden

Hello - I assume a key point is that each rectangle must be within the bounds of only one circle, or, a corollary to that would be that the rectangle edges cannot cross any of the circle edges. That is how the picture is drawn, and if that is not a rule, then the maximum width and height are just W/2 and H/2, which is trivial, then you just draw a circle around it (r = sqrt(H^2 + W^2)/2, or something like that)

• 3 år siden
1. ###### arbw
• 3 år siden

Yes. It's also ill defined wrt what is being maximised. Ignoring the circle comments, what about the dimensions of h and w are to be maximised. h + w (i.e., the perimeter)? The area? I'm assuming it's the area.

• 3 år siden
2. Konkurrenceafholder
• 3 år siden

I have updated the description. I hope the problem is less ill defined now.

• 3 år siden
• 3 år siden

#sealed

• 3 år siden
• 3 år siden

Hi,

• 3 år siden
• 3 år siden

#sealed

• 3 år siden
• ###### uetian09ee506
• 3 år siden

the solution of this problem will give us the values of r, h, and w for given H, W which maximize the area of the inner rectangles under your constraints....................if you, let me know i can proceed further with its solution,,,,,,,,,,,,,

• 3 år siden
• ###### uetian09ee506
• 3 år siden

I have formulated this optimization problem, maximize 4hw subject to: r^2 = h^2 + w^2 Eq(1) 4hw = HW Eq.(2)

• 3 år siden
• ###### uetian09ee506
• 3 år siden

I have formulated this problem as;

• 3 år siden

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