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Math Help - Helical Toroid Equation

  1. #1
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    Helical Toroid Equation

    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.


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

    for example - a helical toroid with 100 turns (3.6 pitch) and left handed chirality


    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?


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


    I have no idea where to begin, I am not a mathematician,
    I'm a designer and just need an equation for pitch and chirality functions for a design project.

    any help would be appreciated.

    Thanks!



    Last edited by kinogram; July 20th 2014 at 10:12 AM.
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  2. #2
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    Re: Helical Toroid Equation

    This thread might help.

    Helix torus problem

    posts #3 and #6 in particular

    At a quick glance

    R1 is the big radius, the radius of the toroid.

    R2 is cross radius, the width of the ribbon.

    $\beta$ controls how many turns of the ribbon in a full circle.

    For a full circle $t:[0,2\pi]$

    You will have $\beta$ turns per circle.

    R1 is equivalent to R in my mathematica sheet

    R2 I have set in code to be 1

    $\beta$ corresponds with my $\omega$
    Last edited by romsek; July 20th 2014 at 02:08 PM.
    Thanks from kinogram
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  3. #3
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    Re: Helical Toroid Equation

    Now I understand

    Thank you romsek!
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  4. #4
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    Re: Helical Toroid Equation

    One last question..

    I presume I need to add a 3rd radius (R3) if I want to make the cross section an ellipse?

    or does this happen automatically between the relationship between y and z ?
    Last edited by kinogram; July 21st 2014 at 05:26 AM.
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  5. #5
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    Re: Helical Toroid Equation

    Quote Originally Posted by kinogram View Post
    One last question..

    I presume I need to add a 3rd radius (R3) if I want to make the cross section an ellipse?

    or does this happen automatically between the relationship between y and z ?
    I'm not sure what you mean by making the cross section an ellipse. I can't visualize this.
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  6. #6
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    Re: Helical Toroid Equation

    I'm not sure what you mean by making the cross section an ellipse. I can't visualize this.

    In an elliptic torus - R2 is the first radius of an ellipse

    imagine a cross section of a ring torus, we have 2 perfect circles with a gap between,

    an elliptic ring torus, instead, has 2 ellipses with a gap between.

    a horn torus has 2 circles without a gap between

    a spindle torus has 2 intersecting circles - like a venn diagram
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  7. #7
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    Re: Helical Toroid Equation

    Quote Originally Posted by kinogram View Post
    In an elliptic torus - R2 is the first radius of an ellipse

    imagine a cross section of a ring torus, we have 2 perfect circles with a gap between,

    an elliptic ring torus, instead, has 2 ellipses with a gap between.

    a horn torus has 2 circles without a gap between

    a spindle torus has 2 intersecting circles - like a venn diagram
    oh you mean the toroid itself is elliptical.

    Yeah, you'd just have two radii and the ellipse would be in the xy plane.

    I'd use R1 and R2 for the ellipse and R3 for the width of the ribbon.
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  8. #8
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    Re: Helical Toroid Equation

    oh you mean the toroid itself is elliptical.

    just to clarify..

    I mean the big (perimeter) radius is circular and the small (cross section) radius is elliptical



    Helical Toroid Equation-ellipticaltorus_700.gif
    Last edited by kinogram; July 21st 2014 at 07:25 AM.
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  9. #9
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    Re: Helical Toroid Equation

    Quote Originally Posted by kinogram View Post
    just to clarify..

    I mean the big (perimeter) radius is circular and the small (cross section) radius is elliptical



    Click image for larger version. 

Name:	EllipticalTorus_700.gif 
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    oh.

    I'll have to think about this. My ribbon is 2 dimensional.
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  10. #10
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    Re: Helical Toroid Equation

    I'll have to think about this. My ribbon is 2 dimensional.
    Aha, okay

    In this case, all the toroids I'm describing involve a 3D tube rather than a 2D ribbon.

    And, not a solid tube, but a helix.

    Further, I need a way to describe chirality.
    Last edited by kinogram; July 21st 2014 at 08:12 AM.
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  11. #11
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    Re: Helical Toroid Equation

    I think I found the correct equation :

    x(t),y(t),z(t)⟩=⟨(R+rcos(nt))cost,(R+rcos(nt))sin(  nt),rsin(nt)


    As long as this mathematically describes a wire wrapped around a torus.
    Last edited by kinogram; July 21st 2014 at 01:39 PM.
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  12. #12
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    Re: Helical Toroid Equation

    I believe I have the correct equations for left and right-handed chirality now :


    left-handed helical torus

    x(t),y(t),z(t)⟩=⟨(R+rcos(nt))cos(t),(R+rcos(nt))si  n(t),rsin(nt)


    right-handed helical torus

    x(t),y(t),z(t)⟩=⟨(R+rcos(nt))cos(t),(R+rcos(nt))si  n(t),-rsin(nt)


    does this look mathematically right to you?
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