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17 replies

Werner_E
Werner_E25-Diamond I
25-Diamond I
September 14, 2020

There is a long-serving regular contributor here in the forum who likes to comment on such occasions that a solution without animation would not be a solution.
You may have read it here once or twice (or was it 100 times?). 😈

ani4_OnlineConverter.gif

I apologize as this is neither a particularly beautiful nor a particularly useful animation.

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    ValeryOchkov24-Ruby IVAuthor
    24-Ruby IV
    September 14, 2020

    Yes, not each animation is informative and useful.

    I hope this is more interesting - an animation of the numeric solution

    one connected problem.

    Let's wait for other decisions, and then open our playing cards, sorry, our solutions.

    Circle-Triangle.gif

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      Werner_E
      Werner_E25-Diamond I
      25-Diamond I
      September 14, 2020

      Yes, we have already seen this animation in the past.

      Thought that the Malfatti circles would be a nice task for some kind of genetic algorithm for you to implement.

       

      It also would be nice to compare the Malfatti circles to the solution of the Malfatti problem side by side. You probably know that it was proven that the Malfatti circles never are solutions for the Malfatti problem.

      The Malfatti problem consists in finding the three inscribed and not overlapping circles with the largest area sum. It can be proven that a simple greedy algorithm does the job.

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        Werner_E
        Werner_E25-Diamond I
        25-Diamond I
        September 14, 2020

        Here you can compare the percentage of the triangle area covered by the Malfatti circles and the true solution of the Malfatti problem (maximum covered area by three circles). It demonstrates that the Malfatti circles never are solutions to the Malfatti problem.

        ani5.gif

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          ValeryOchkov24-Ruby IVAuthor
          24-Ruby IV
          September 15, 2020

          Super! Can you show Mathcad-sheets? I will do it too.

          And what about a tetrahedron and three spheres?

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          Werner_E
          Werner_E25-Diamond I
          25-Diamond I
          September 15, 2020
          @ValeryOchkov wrote:

          Super! Can you show Mathcad-sheets?

          Sure, there is nothing confidential or secret in my sheet.
          Here you are. The MC15 file is attached. Have fun!

           

          And what about a tetrahedron and three spheres?

          Why just three? Wouldn't four be more appropriate? Looking forward to see your solution 😉

          I guess that the solution for this 3D Malfatti problem (maximize the volume of the spheres) would be easier to solve (greedy algorithm) than the 3D generalization of the Malfatti circles (the spheres all being tangent to each other and to the tetrahedron sides). Proving that a greedy algorithm gives the solution for the maximum volume would be another thing. The prove for the 2D problem is 25 pages and also involves computer calculations to deal with all relevant constellations.

          In case you are really interested in a high dimensional generalization of Malfatti's problem, you may check this
          https://www.aimsciences.org/article/exportPdf?id=c026650a-cc3b-4ec0-9a6a-d729ccf92bdc
          or this
          http://www.aimsciences.org/article/id/0953c355-7831-43ea-9412-0fdcb8f9b5a7


          A classic generalization of the Malfatti circles is to change the straight triangle sides for circle arcs.

          I now found an interesting paper where the Malfatti circles were generalized (still in 2D) in a new way. The task is to fit six circles into the triangle, tangent to each other and to the triangle sides. Of course "tangent to each other" can't mean that every circle is in touch with each of the remaining ones as is the case with the original 3 Malfatti circles. While the original three Malfatti circles could be found solely by analytic methods (in my sheet I didn't as Mathcads symbolic would not be able to solve the necessary equations), the solution for the six circles needs to make heavily use of numerical methods. In case you have too much spare time (😉), here is the link: https://www.sciencedirect.com/science/article/pii/S0925772112000259

          Good luck!!

          Werner_E_0-1600160754846.png

           

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