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24-Ruby IV
ValeryOchkov24-Ruby IV
24-Ruby IV
January 7, 2018
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Firecrackers 2018 or 5-th order curve and 20 points

  • January 7, 2018
  • 1 reply
  • 11363 views

2018-5.gif

Self-portrait of Pablo Picasso2018-5-3a.pngElephant under the dome of the circusElephant under the dome of the circusHand with a twistHand with a twistFish anglerFish anglerCupid with bowCupid with bowPotato eaterPotato eater"В Ленинграде-городе у Пяти Углов Получил по морде. Саня Соколов""В Ленинграде-городе у Пяти Углов Получил по морде. Саня Соколов"Evil personEvil person

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Best answer by twinprime

Valery: Beautiful curves!  But I'm really replying to your New Year posting about Ochkov's constant (the ellipse fraction in conics through five random points in a square), with value 0.280...  Maybe our colleagues have already provided a solution but I have spent many entertaining hours on this problem over the last month and it seems hard to me.  One has the distributions of the points and wants the distribution of the discriminant of the solved linear system; well, since your points are chosen independently, the rows of your matrix M are independent, but solving for the five coefficients tangles up the ten independent uniform variables so thoroughly that it gets very hard to see how the original uniform inputs are reflected in the first three coefficients.  My first thought was: well, let us pick four points and see what we can learn about the ellipse/hyperbola outcome when we move the fifth point around.  Through four points (let's assume the quadrilateral is convex, since if it isn't we already know that we'll get a hyperbola no matter where we put the fifth point) we can draw exactly two parabolas, and they divide our square into ellipse regions and hyperbola regions, as shown in the attached pdfs.  This is cute, but not immediately helpful.  Note that an extra layer of complexity comes from the uniform distributions which, since they are step functions, propagate annoying edges throughout the computation.  I've looked at simpler versions (let us say radially normally distributed, so the we get rid of the edges and have some symmetry to exploit) but eventually I run into the same wall, perhaps disguised as a different wall.  

Anyhow, I'm enjoying getting re-acquainted with Mathcad via version 15.   I worked at Mathsoft roughly from version 1.5 through 11 and remember that you were one of our earliest supporters, so I'm glad to see that you are still a Mathcad advocate, and I look forward to participating in the community (...once the ellipse constant problem is solved, that is).

Best from Chicago, where we just pretend to have winter.

Frank Purcell

twinprime@att.net

1 reply

V
24-Ruby IV
ValeryOchkov24-Ruby IVAuthor
24-Ruby IV
January 7, 2018

Plots with Mezentsev method

2018-5-mez.gif

T
twinprime1-VisitorAnswer
1-Visitor
February 14, 2018

Valery: Beautiful curves!  But I'm really replying to your New Year posting about Ochkov's constant (the ellipse fraction in conics through five random points in a square), with value 0.280...  Maybe our colleagues have already provided a solution but I have spent many entertaining hours on this problem over the last month and it seems hard to me.  One has the distributions of the points and wants the distribution of the discriminant of the solved linear system; well, since your points are chosen independently, the rows of your matrix M are independent, but solving for the five coefficients tangles up the ten independent uniform variables so thoroughly that it gets very hard to see how the original uniform inputs are reflected in the first three coefficients.  My first thought was: well, let us pick four points and see what we can learn about the ellipse/hyperbola outcome when we move the fifth point around.  Through four points (let's assume the quadrilateral is convex, since if it isn't we already know that we'll get a hyperbola no matter where we put the fifth point) we can draw exactly two parabolas, and they divide our square into ellipse regions and hyperbola regions, as shown in the attached pdfs.  This is cute, but not immediately helpful.  Note that an extra layer of complexity comes from the uniform distributions which, since they are step functions, propagate annoying edges throughout the computation.  I've looked at simpler versions (let us say radially normally distributed, so the we get rid of the edges and have some symmetry to exploit) but eventually I run into the same wall, perhaps disguised as a different wall.  

Anyhow, I'm enjoying getting re-acquainted with Mathcad via version 15.   I worked at Mathsoft roughly from version 1.5 through 11 and remember that you were one of our earliest supporters, so I'm glad to see that you are still a Mathcad advocate, and I look forward to participating in the community (...once the ellipse constant problem is solved, that is).

Best from Chicago, where we just pretend to have winter.

Frank Purcell

twinprime@att.net

    V
    24-Ruby IV
    ValeryOchkov24-Ruby IVAuthor
    24-Ruby IV
    February 14, 2018

    Thanks, Frank!

     

    Now I am preparing an article on this problem.

    Let do it together!

     

    Val 

     

    The begin of the article

    В стародавние времена путешествие по Свету могли позволить себе только очень богатые и физически здоровые люди. Сейчас же это удовольствие доступно очень многим, а не только избранным: сел в самолет, автомобиль или на поезд – и за короткое время с комфортом ты можешь добраться практически до любого уголка Земли.

    Такой же прогресс наблюдается в отношении и… математики. Раньше в ее интереснейшие дебри могли забираться только избранные люди – люди с особыми математическими способностями (с «математическим слухом») и получившие соответствующее математическое образование. Но в настоящее время круг таких «избранных» существенно расширился за счет появления… компьютерных математических пакетов, которые облегчают путешествие в мир математики, делают его массовым.

    И еще одно вводное замечание.

    Для чего изучают математику в школе и в вузе.

    Во-первых, для того, чтобы можно было освоить другие учебные дисциплины: физику, химию, теоретическую механику, гидрогазодинамику, сопротивление материалов, инженерную графику, экономику и т.д.

    Во-вторых, нужно всегда помнить, что математика – это лучшая гимнастика (фитнесс) для ума. Изучая математику, мы развиваем свои умственные способности, которые пригодятся нам при решении не только математических, но и разных производственных и житейских задач.

    И в-третьих, изучение математики – это само по себе интересное и увлекательное занятие, которым можно заниматься как хобби. Но без математических компьютерных пакетов это делать было почти невозможно, если, повторяем, нет особых математических талантов и соответствующего математического образования.

    In ancient times, only very rich and physically healthy people could travel around the world. Now this pleasure is available to very many, not just the elite: boarded a plane, a car or a train - and in a short time with comfort you can get to almost any corner of the earth.
    The same progress is observed in relation to and ... mathematics. Previously, in its most interesting jungles, only selected people could get in - people with special mathematical abilities (with a "mathematical ear") and received the appropriate mathematical education. But at the present time the circle of such "elect" has essentially expanded due to the appearance of ... computer mathematical packages that facilitate the journey into the world of mathematics, make it mass.
    And one more introductory remark.

    Why do we study mathematics at school and at a university.

    First, in order to be able to master other disciplines: physics, chemistry, theoretical mechanics, fluid dynamics, material resistance, engineering graphics, economics, etc.
    Secondly, you must always remember that mathematics is the best gymnastics (fitness) for the mind. Studying mathematics, we develop our mental faculties, which will be useful for us in solving not only mathematical, but also various production and everyday problems.

    And thirdly, the study of mathematics - this in itself is an interesting and fascinating occupation, which you can do as a hobby. But without mathematical computer packages, it was almost impossible to do this, if, we repeat, there are no special mathematical talents and the corresponding mathematical education.

    .

    .

     

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