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ValeryOchkov24-Ruby IV
24-Ruby IV
November 17, 2020
Question

Circle involutes

I am trying to build circle Involvents.

I built five involutes, and the sixth, the seventh, etc. there is not enough time and power of the computer - see attachment Prime 6.

I will be very grateful if you can give me a way to make my work.

Involute-5.png

13 replies

LucMeekes23-Emerald IV
23-Emerald IV
November 17, 2020

Hi Valery,

 

how about:

LucMeekes_0-1605607209625.png

It shouldn't be too hard to construct the next set from the previous. I see factorials in the coefficients to the various sine and cosine terms.

Here's the plot:

LucMeekes_1-1605607967300.png

Note that the task gets much more complicated if you try to do this on an ellipse (e.g. leave y(t) as is, and change x(t) to 2*cos(t))...

 

Success!
luc

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    ValeryOchkov24-Ruby IVAuthor
    24-Ruby IV
    November 17, 2020

    Thanks, Luc!

    It is Mathcad 11 with old good Maple symbolocal engine!?

    We must use the simplify operator in Prime 6.

    Can you have more than seven involutes? Слабо?

    With very interesting fractal!

    Involute-7.png

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    LucMeekes23-Emerald IV
    23-Emerald IV
    November 17, 2020

    Certainly.

    This is the set of equations:

    LucMeekes_0-1605689151107.png

    Here is the plot:

    LucMeekes_0-1605647513585.png

    Luc

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    Werner_E
    Werner_E25-Diamond I
    25-Diamond I
    November 17, 2020

    You may speed up calculations significantly if you use bold and simple numeric approximations for derivatives and integrals - working on vectors of points, not equations. Of course at the cost of accuracy (especially with higher "order" evolvents).

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      Werner_E
      Werner_E25-Diamond I
      25-Diamond I
      November 17, 2020

      Here is a first try with a numerical approach working on vectors. Its not really satisfying with other base curves, but works OK for curves like ellipses (unless the "order" gets too high). Here is the first dozen from the ellipse suggested by Luc:

      Ellipsen-Evolventen-1.gif

      Here's the construction of the first two:

      Ellipsen-Evolventen-2.gif

       

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        ValeryOchkov24-Ruby IVAuthor
        24-Ruby IV
        November 17, 2020

        Fine, Werner! Solution without atimation is not solution!

        And what about a square not a cyrcle or an ellipes?

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        DJF16-Pearl
        16-Pearl
        November 18, 2020

        It appears you try to divide by zero, so that is going to take a little while to solve.  Attached is probably no help for what you're trying to do (which I'm not sure of), but it's a little tool I've used for gear teeth.  

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        ValeryOchkov24-Ruby IVAuthor
        24-Ruby IV
        November 19, 2020

        is it possible to creat someone so: involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute - involute-evolute...

        evolute-5.png

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