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ValeryOchkov24-Ruby IV
24-Ruby IV
May 14, 2021 6:59 AM UTC
Question

Parabola and catenary

There are three random points on the plane through which an arc of a parabola and a similar arc of a catenary pass. What is the probability that the length of the parabola's arc will be greater than the length of the catenary's arc??

Catenary-Parabola.png

11 replies

Fred_Kohlhepp23-Emerald I
23-Emerald I
May 14, 2021 11:44 AM UTC

Since a catenary describes a condition of minimum potential energy, I would guess that the probability is quite low.

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    LucMeekes23-Emerald V
    23-Emerald V
    May 14, 2021 3:00 PM UTC

    About 20...25 %.

    First run over 100 iterationsFirst run over 100 iterations

    No, it's more towards 30 %. The next run is over 200 iterations:

    LM_20210514_CatenaryParabole200.gif

    And finally over 1000 iterations, but showing only 1 in 5:

    LM_20210514_CatenaryParabole1000.gif

    Luc

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      ValeryOchkov24-Ruby IVAuthor
      24-Ruby IV
      May 14, 2021 5:05 PM UTC

      Is this a new math constant?

       

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      LucMeekes23-Emerald V
      23-Emerald V
      May 14, 2021 5:39 PM UTC

      Nah... don't think so.

      You would have something if you could prove that a catenary line through 3 points is always x % longer than a Parabola line through the same three points. Since that is not the case....Keep searching.

       

      I've got a puzzle for you: try to find what my select_dhm() function is... (The function I use to provide guess values for the solve block of finding the parameters for the catenary function)

       

      Success!

      Luc

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      ValeryOchkov24-Ruby IVAuthor
      24-Ruby IV
      May 15, 2021 6:36 PM UTC

      000100000100010000-0000011010000-00---110000100010000100-011001-0101101-01100000--1101--110101-010-010000101-001-010-0-0001-1-00000--0-10011101000-0-0-00-01010000-0-01-10-0001000-010100-0-00001101011100000110110111--0101-01-101010---000000001000100-000001100111000-10 (Tomorrow I will continue my research)

      1 - Sc > Sp

      0 - Sc < Sp

      - (hiccup) - no solution

      * - Sc = Sp (I am waiting for)

      It's like a hiccup. You don't know when you will hiccup next time - when next time there will be no solution of the problem of catenaries and parabolas.
      By the way, there is a very interesting study of the frequency and strength (- or ---) of hiccups.

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      ValeryOchkov24-Ruby IVAuthor
      24-Ruby IV
      May 18, 2021 6:08 AM UTC

      Catenary longer than a parabola about (a bit more) in 26 cases out of a hundred.

      p-c.png

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      ValeryOchkov24-Ruby IVAuthor
      24-Ruby IV
      May 18, 2021 8:42 AM UTC

      With prorgramming

      p-c-Prog.png

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