How Do You Calculate Pitch and Chirality in a Helical Toroid Equation?

[kinogram]

Homework Statement

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

Homework Equations

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

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?


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

 

[HallsofIvy]

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".

 

[kinogram]

It looks to me like the radius of the "helix" depends upon t so this is NOT true "helix

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

 

[LCKurtz]

@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

 

[kinogram]

why are you working on this project?

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 (?)

I don't know if this is any use to you but look here:
http://math.stackexchange.com/questi...d-into-a-torus

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?

 

[LCKurtz]

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?

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.

 

[kinogram]

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?

You don't. Space curves are usually given in parametric form

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?

 

[LCKurtz]

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?

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.

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?

That is what I would call artistic license. It doesn't make any sense mathematically to write it that way.

 

[kinogram]

Yes. But here you are using the notation with R1,R2,β and below you are using R,r,n. Pick one or the other.

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.

That is what I would call artistic license. It doesn't make any sense mathematically to write it that way.

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.

.

 

[LCKurtz]

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.

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?

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)>

Writing it like this would be mathematically correct$$
\langle x(t),y(t),z(t)\rangle = \langle (R+r\cos(nt))\cos t,(R + r\cos(nt))\sin(nt),r\sin(nt)\rangle$$

 

[kinogram]

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?

Exactly!

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

Writing it like this would be mathematically correct
⟨x(t),y(t),z(t)⟩=⟨(R+rcos(nt))cost,(R+rcos(nt))sin(nt),rsin(nt)⟩

Perfect

Thank you LCKurtz!Lastly - I just need to know which parameters of the equation control chirality.

 

[LCKurtz]

Lastly - I just need to know which parameters of the equation control chirality.

If you change the sign of anyone of the three components, it will reverse the chirality. For example change the last component to ##-r\sin(nt)##.

 

[kinogram]

If you change the sign of anyone of the three components, it will reverse the chirality. For example change the last component to −rsin(nt).

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?

.

 

[LCKurtz]

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)⟩

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.

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

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

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.

 

[kinogram]

This equation is what is given in that link I gave you in my first post. You have changed it.

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.

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.

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

The helix in the picture is indeed left-handed.