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Helical Toroid Equation

  1. Jul 20, 2014 #1
    1. The problem statement, all variables and given/known data

    The equation below describes a helical toroid

    I need a way to define pitch and chirality, if someone can please help me with these functions.


    I found this equation on the internet, but it's greek to me

    2. Relevant equations


    <cos(t)(R1+R2 cos(βt)),sin(t)(R1+R2 cos(βt)),R2 sin(βt)>



    3. The attempt at a solution

    I assume R1 is the radius of the torus and R2 is the radius of the helical cross section?

    or.. the other way around - I don't know, do you know? :confused:


    I could not find any explanation for (t) or (βt) either


    I have no idea where to begin
    I am not a mathematician and have no desire to become a mathematician

    I'm a designer and just need an equation for pitch and chirality functions
    any help would be appreciated.

    Thanks



    I am not a student, and this is not a homework assigment

    I'm working on a design project that involves toroid geometry
     
    Last edited: Jul 20, 2014
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  3. Jul 20, 2014 #2

    HallsofIvy

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    In the xy-plane [itex]x= r cos(\theta)[/itex], [itex]y= r sin(\theta)[/itex] is a circle with center at (0, 0) and radius r. Ignoring the z- component, here you have [itex]x= [R_1+ R_2 cos(\beta t)]cos(t)[/itex], [itex]y= [R_1+ R_2 cos(\beta t)]sin(t)[/itex]. It looks to me like the radius of the "helix" depends upon t so this is NOT true "helix".
     
  4. Jul 20, 2014 #3


    Then I seem to have the wrong expression

    So, what would be the correct expression to decribe a "true" helix?
    can you also please define t and βt?

    Thanks HallsofIvy
     
  5. Jul 20, 2014 #4

    LCKurtz

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    @kinogram: Perhaps your boss should put someone on the project that understands a little mathematics and knows what a helical toroid, and its pitch and chirality are. If the equations you found on the internet are "Greek" to you and you don't even know if they are what you want, why are you working on this project?

    [Edit, added] I don't know if this is any use to you but look here:
    http://math.stackexchange.com/quest...equations-create-a-helix-wrapped-into-a-torus
     
    Last edited: Jul 20, 2014
  6. Jul 20, 2014 #5
    I'm working on a personal design project (artwork) - I am my own boss.

    I know perfectly what a helical toroid, and its pitch and chirality are in physical terms (in reality)

    I need to describe a helical toroid in mathematical terms
    and learn which expressions describe pitch and chirality
    so I can adjust these parameters as needed.

    I am an artist, and this is a visual design project (a line graphic in pen and ink )
    I've created a line drawing of a helical toroid on illustration board
    below the torus image will be an image of the mathematical expression of the toroid
    (also in pen and ink), as part of the artwork.

    I like the visual look of mathematical expressions, and plan to create a series of works
    of geometric forms - with mathematical expressions that describe them.


    I thought it would be a simple problem for mathematicians

    but, maybe not (?)



    so according to the discussion at the link..

    You need two radii to decribe a torus. R1 and R2

    Then the parametric equations of the torus are:

    x = (R2+R2 cosu) cos v
    y = (R1+R2 cosu) sin v
    z = R2 sin u

    Then, to get a helical curve, set v = ku, where k << 1

    R1 = 3, R2 = 1, k = 0.05:


    so how do I put this together into a single expression?
     
    Last edited by a moderator: May 6, 2017
  7. Jul 20, 2014 #6

    LCKurtz

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    You don't. Space curves are usually given in parametric form just as in that example. Did you read farther down that page where they explained how the parameters affect the shape and showed some plot printouts? Plot packages use the parametric equations as above and as shown in that link.
     
  8. Jul 21, 2014 #7
    According to explanations I've found on the internet

    in this expression :

    <cos(t)(R1+R2 cos(βt)),sin(t)(R1+R2 cos(βt)),R2 sin(βt)>


    R1 is the radius of the toroid.

    R2 is the cross section radius

    β controls the number of turns in the torus.

    For a full torus, then t:[0,2π]


    is this correct?





    but, can I arrange the expression like this?


    xyz = (R + r cos (nt)) cos (t) = (R + r cos (nt)) sin (t) = r sin (nt)


    if HelixPlot[6, 5, 20] produces the helical torus below with 20 turns

    then z = R2 sin u defines the number of turns?
     

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    Last edited: Jul 21, 2014
  9. Jul 21, 2014 #8

    LCKurtz

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    Yes. But here you are using the notation with ##R_1, R_2, \beta## and below you are using ##R, r, n##. Pick one or the other.

    That is what I would call artistic license. It doesn't make any sense mathematically to write it that way.
     
  10. Jul 21, 2014 #9

    HallsofIvy says the first equation does not describe a true helix,

    So, I need the equation which correctly describes a true helix,
    that is - a 3D tube rather than a 2D ribbon.

    not a solid tube, but a helix.

    Further, I need a way to describe chirality.



    Does it make sense mathematically to write the equation as :


    xyz = <(R + r cos(nt))cos(t), (R + r cos(nt))sin(t), r sin(nt)>

    or does it have to be :

    x = (R + r cos(nt))cos(t)
    y = (R + r cos(nt))sin(t)
    z = r sin(nt)

    graphically I'm looking for an equation which can be written in a single line.

    .
     
    Last edited: Jul 21, 2014
  11. Jul 21, 2014 #10

    LCKurtz

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    You asked for a toroidal helix which looks like a wire wrapped around a torus. A true helix looks like a wire wrapping around a cylindrical tube as it climbs. Of course a toroidal helix is not a true helix, but you didn't ask for a true helix. Which do you really want?

    Writing it like this would be mathematically correct$$
    \langle x(t),y(t),z(t)\rangle = \langle (R+r\cos(nt))\cos t,(R + r\cos(nt))\sin(nt),r\sin(nt)\rangle$$
     
  12. Jul 21, 2014 #11

    Exactly!

    As the title of the topic suggests - I need a helical toroid,
    which is represented by a wire wrapped around a torus.




    Perfect

    Thank you LCKurtz!



    Lastly - I just need to know which parameters of the equation control chirality.
     
    Last edited: Jul 21, 2014
  13. Jul 21, 2014 #12

    LCKurtz

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    If you change the sign of any one of the three components, it will reverse the chirality. For example change the last component to ##-r\sin(nt)##.
     
    Last edited: Jul 21, 2014
  14. Jul 22, 2014 #13



    have I got it right?

    ⟨x(t), y(t), z(t)⟩ = ⟨(R + r sin(nt))sin t,(R + r sin(nt))cos(nt),r cos(nt)⟩



    incidentally.. does the un-reversed equation describe right-handed chirality?

    also, in every instance of n, n controls the number of turns?

    .
     
    Last edited: Jul 22, 2014
  15. Jul 22, 2014 #14

    LCKurtz

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    In post #10 I wrote$$
    \langle x(t),y(t),z(t)\rangle = \langle (R+r\cos(nt))\cos t,(R + r\cos(nt))\sin(nt),r\sin(nt)\rangle$$which had a typo. It should have been$$
    \langle x(t),y(t),z(t)\rangle = \langle (R+r\cos(nt))\cos t,(R + r\cos(nt))\color{red}{\sin(t)},r\sin(nt)\rangle$$This equation is what is given in that link I gave you in my first post. You have changed it.


    Look at the link again. It says n controls the number of turns. Also, the pictures look like left handed chirality to me, if I understand chirality correctly for a torus.
     
  16. Jul 22, 2014 #15
    Here is the original equation from the link :


    [itex]x=(a+bcosu)cosv[/itex]
    [itex]y=(a+bcosu)sinv[/itex]
    [itex]z=bsinu[/itex]


    I have changed [itex]a + b[/itex] to [itex]R + r[/itex] and [itex]u[/itex] and [itex]v[/itex] to [itex](nt)[/itex] and [itex](t)[/itex] respectively, which I assume have the same meaning.

    left-handed helix

    [itex]⟨x(t),y(t),z(t)⟩=⟨(R+rcos(nt))cos(t),(R+rcos(nt))sin(t),rsin(nt)⟩[/itex]


    right-handed helix

    [itex]⟨x(t),y(t),z(t)⟩=⟨(R+rcos(nt))cos(t),(R+rcos(nt))sin(t),-rsin(nt)⟩[/itex]


    If I understood you correctly, these are the correct equations for left and right chirality.


    Yes, I just wanted to make sure I read it correctly.

    The helix in the picture is indeed left-handed.
     

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    Last edited: Jul 22, 2014
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