1. ## 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!

2. ## Re: Helical Toroid Equation

Helix torus problem

posts #3 and #6 in particular

At a quick glance

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 $\omega$

3. ## Re: Helical Toroid Equation

Now I understand

Thank you romsek!

4. ## 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 ?

5. ## Re: Helical Toroid Equation

Originally Posted by kinogram
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.

6. ## 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

7. ## Re: Helical Toroid Equation

Originally Posted by kinogram
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.

8. ## 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

9. ## Re: Helical Toroid Equation

Originally Posted by kinogram
just to clarify..

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

oh.

10. ## Re: Helical Toroid Equation

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.

11. ## 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.

12. ## 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?