Geometry: Maximize rectangles inside overlapping circles overlapping a rectangle

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  pinetree800

  • 8 år siden

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

  • 8 år siden
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  1. 

     pinetree800

     • 8 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.

     • 8 år siden
  2. 
     Konkurrenceafholder

     • 8 år siden

     Have you posted your solution?

     • 8 år siden
• 

  trinhngochai1

  • 8 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

  • 8 år siden
  1. 
     Konkurrenceafholder

     • 8 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.

     • 8 år siden
  2. 

     pinetree800

     • 8 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) .

     • 8 år siden
• 

  MatiasPereyraArg

  • 8 år siden

  are the four inner rectangles the same size?

  • 8 år siden
  1. 
     Konkurrenceafholder

     • 8 år siden

     Yes. They are the same size.

     • 8 år siden
• 

  trinhngochai1

  • 8 år siden

  Can I suppose W>2.2*H?

  • 8 år siden
  1. 
     Konkurrenceafholder

     • 8 år siden

     No. W and H will need to be arbitrary.

     • 8 år siden
• 

  bathtubdjinn

  • 8 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?

  • 8 år siden
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  1. 

     bathtubdjinn

     • 8 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.)

     • 8 år siden
  2. 
     Konkurrenceafholder

     • 8 år siden

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

     • 8 år siden
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  trinhngochai1

  • 8 år siden

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

  • 8 år siden
  1. 
     Konkurrenceafholder

     • 8 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.

     • 8 år siden
• 
  Konkurrenceafholder

  • 8 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.

  • 8 år siden
• 
  Konkurrenceafholder

  • 8 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.

  • 8 år siden
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  1. 

     AndyZo

     • 8 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?

     • 8 år siden
  2. 

     AndyZo

     • 8 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...

     • 8 år siden
• 

  mfpleite

  • 8 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.

  • 8 år siden
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  1. 

     pinetree800

     • 8 år siden

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

     • 8 år siden
  2. 
     Konkurrenceafholder

     • 8 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.

     • 8 år siden
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  aminbros

  • 8 år siden

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

  • 8 år siden
• 

  Zhats

  • 8 år siden

  hi. I can manage your task. Please check my profile for details.

  • 8 år siden
  1. 
     Konkurrenceafholder

     • 8 år siden

     I've updated the description. Please review and submit your solution.

     • 8 år siden
  2. 

     pinetree800

     • 8 år siden

     I submitted all results.

     • 8 år siden
• 

  uetian09ee506

  • 8 år siden

  hi i can solve this problem

  • 8 år siden
  1. 
     Konkurrenceafholder

     • 8 år siden

     I've updated the description. Please review and submit your solution.

     • 8 år siden
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  mattsrinc

  • 8 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. :-)

  • 8 år siden
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  1. 
     Konkurrenceafholder

     • 8 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.

     • 8 år siden
  2. 
     Konkurrenceafholder

     • 8 år siden

     I've updated the description again. Please review.

     • 8 år siden
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  ivanfordjarini

  • 8 år siden

  done

  • 8 år siden
• 

  ShahidFPGA

  • 8 år siden

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

  • 8 år siden
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  arbw

  • 8 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)

  • 8 år siden
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  1. 

     arbw

     • 8 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.

     • 8 år siden
  2. 
     Konkurrenceafholder

     • 8 år siden

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

     • 8 år siden
• 

  kristtyflores

  • 8 år siden

  #sealed

  • 8 år siden
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  pinetree800

  • 8 år siden

  Hi,

  • 8 år siden
• 

  adityapathania

  • 8 år siden

  #sealed

  • 8 år siden
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  uetian09ee506

  • 8 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,,,,,,,,,,,,,

  • 8 år siden
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  uetian09ee506

  • 8 år siden

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

  • 8 år siden
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  uetian09ee506

  • 8 år siden

  I have formulated this problem as;

  • 8 år siden

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