How Do You Determine the Axis of a Helix?

[ziggy]

I'm trying to understand how to find the axis of a helix but so far I seem to be hitting a blank. How for instance would I go about finding the axis of the helix r(t)=(e^t)costi+(e^t)sin(t)j+(e^t)k ? I would be very gratefull for any and all assistance.

 

[pkleinod]

The equation describes a vector r(t) that spirals around the k-axis, increasing its distance from the axis k as t increases. Strictly speaking, is this a helix?

 

[HallsofIvy]

That is NOT helix since the radius increases with height above the xy-plane. Projecting down into the xy-plane gives, not a circle, but a spiral.

In any case, it should be clear that the axis is the z-axis.

 

[ObsessiveMathsFreak]

First, check to see if you can find three points p1,p2,p3 for which the three tangent vectors t1,t2,t3 are linearly independent.

If you can find three linearly independent tangent vectors, then use the following. If d is the unit vector in the direction of the axis, then d makes a constant angle with all tangent vectors. i.e.

$$\mathbf{t}\cdot\mathbf{d}=\cos\left(\alpha_0\right)=|\mathbf{d}| k$$

But this means

$$\mathbf{t}_1\cdot\mathbf{d}=\mathbf{t}_2\cdot\mathbf{d}=\mathbf{t}_3\cdot\mathbf{d}=k$$

Performing some subtractions we get
$$\left(\mathbf{t}_1-\mathbf{t}_2\right)\cdot\mathbf{d}=0$$
$$\left(\mathbf{t}_2-\mathbf{t}_3\right)\cdot\mathbf{d}=0$$

So d is orthogonal to both t1-t2 and t2-t3

So, simply take $$\mathbf{d}=\left(\mathbf{t}_1-\mathbf{t}_2\right)\times\left(\mathbf{t}_2-\mathbf{t}_3\right)=\mathbf{t}_1\times\mathbf{t}_2+\mathbf{t}_2\times\mathbf{t}_3+\mathbf{t}_3\times\mathbf{t}_1$$
as the axis of the general helix.

I hope that's right.

 

[ziggy]

First, check to see if you can find three points p1,p2,p3 for which the three tangent vectors t1,t2,t3 are linearly independent.

If you can find three linearly independent tangent vectors, then use the following. If d is the unit vector in the direction of the axis, then d makes a constant angle with all tangent vectors. i.e.

$$\mathbf{t}\cdot\mathbf{d}=\cos\left(\alpha_0\right)=|\mathbf{d}| k$$

But this means

$$\mathbf{t}_1\cdot\mathbf{d}=\mathbf{t}_2\cdot\mathbf{d}=\mathbf{t}_3\cdot\mathbf{d}=k$$

Performing some subtractions we get
$$\left(\mathbf{t}_1-\mathbf{t}_2\right)\cdot\mathbf{d}=0$$
$$\left(\mathbf{t}_2-\mathbf{t}_3\right)\cdot\mathbf{d}=0$$

So d is orthogonal to both t1-t2 and t2-t3

So, simply take $$\mathbf{d}=\left(\mathbf{t}_1-\mathbf{t}_2\right)\times\left(\mathbf{t}_2-\mathbf{t}_3\right)=\mathbf{t}_1\times\mathbf{t}_2+\mathbf{t}_2\times\mathbf{t}_3+\mathbf{t}_3\times\mathbf{t}_1$$
as the axis of the general helix.

I hope that's right.

Thanks for the effort you put into your reply, that gives me a lot to be thinking about. Can
I ask, in your reply is k the curvature of the general helix?

 

[ObsessiveMathsFreak]

No. k is just the cosine of the angle between the tangents and the axis.

In a general helix, the curvature may not necessarily be constant.

 

[TomyDuby]

Let me add an explanation:
(1) A general helix is a curve in 3D space that at every point its tangent forms a constant angle with a constant line in this space.
(2) A circular helix of radius a and pitch b, given by equation

x = a cos(t),
y = a sin(t),
z = bt

this is simple: the tangent is (-a sin(t), a cos(t), b). The angle between this tangent and (0,0,1) direction is constant.

(3) Your curve is, as it was mentioned above, a general helix.

 

[ziggy]

Let me add an explanation:
(1) A general helix is a curve in 3D space that at every point its tangent forms a constant angle with a constant line in this space.
(2) A circular helix of radius a and pitch b, given by equation

x = a cos(t),
y = a sin(t),
z = bt

this is simple: the tangent is (-a sin(t), a cos(t), b). The angle between this tangent and (0,0,1) direction is constant.

(3) Your curve is, as it was mentioned above, a general helix.

Thanks man, I've been waiting for a reply that makes sense for months (you're the first one!). However, my degree is over and I now understand how to find the axis and pitch of a general helix. So thanks again for taking the time to answer my query.

 

[Usaf Moji]

Hi guys, I had an unrelated helix question (thought I'd throw it in here since you guys were talking about helixes).

I know that if you draw a bunch of lines, all of the same length, at various points tangential to a circle, then connect the outer ends of all those tangential lines, the resulting shape will simply be a larger circle.

But, what if you did that for a helix? I.e. if one were to draw a bunch of lines, all of the same length, at various points tangential to a circular helix, then connect the outer ends of all those tangential lines, would you just get a fatter helix with the same pitch? Or would you get a bigger helix that has the same ratio of pitch to radius as the original?

I've been trying to visualize the thing in my mind, but can't. Any help appreciated.

 

[DeaconJohn]

Hi guys, I had an unrelated helix question (thought I'd throw it in here since you guys were talking about helixes).

I know that if you draw a bunch of lines, all of the same length, at various points tangential to a circle, then connect the outer ends of all those tangential lines, the resulting shape will simply be a larger circle.

Not necessairly. Needs more constraints. What did you have in mind?

 

[Usaf Moji]

Not necessairly. Needs more constraints. What did you have in mind?

Yeah, to be more precise, let's say that I'm drawing these tangential lines with equal arc distances along the circle between their start points. Then I connect the outer ends of each tangential line with those immediately neighbouring it. What I'll end up with is (roughly) a circle that's just bigger (the more lines you draw, the closer the thing will resemble a circle).

Maybe I still didn't explain it right, but I hope you understand what I'm getting at.

So, I'm wondering what I would get if I did the analogous thing to a helix.

 

[DeaconJohn]

Yeah, to be more precise, let's say that I'm drawing these tangential lines with equal arc distances along the circle between their start points. Then I connect the outer ends of each tangential line with those immediately neighbouring it. What I'll end up with is (roughly) a circle that's just bigger (the more lines you draw, the closer the thing will resemble a circle).

Maybe I still didn't explain it right, but I hope you understand what I'm getting at.

So, I'm wondering what I would get if I did the analogous thing to a helix.

You needn't be more precise. I get it now.

I don't know what happens for the helix.

Let's see.

a heliz with z-axis as axis is periodic in the z direction.

therefore, the mythical curve would be pedriodic in the z-direction.

a helix is invariant under rotations in the x-y plane, rotations that leave the z-azis fixed.

Therefore, the mythical curve would also be invariant under those rotations.

It's beginning to sound more and more like just a bigger helix all the time.

I think I need one more constraint to really lock it down. and I think I have some idea what that consraint would be. But I've got to leave something for somebody else, right?

 

[TomyDuby]

Usaf Moji,

Let me try to answer your question using some equtions. They may help you to understand the answer.

Assume we have the following helix:

x = a cos(t)
y = a sin(t)
z = bt

where a is the radius and b is the pitch of the helix. Fix your t to a certain value, say t_0. Find the tangent for this t. Extend its length as required. Assume the end of this vector is point A with coordinates (x_A, y_A, z_A). Now repeat the same calculation for t = t_0 + 2*pi. You get point B with coordinates (x_B, y_B, z_B). It is easy to see that

x_B = x_A
y_B = x_A.

What about z_B? Well, because the pitch of the helix is b and you did one full rotation

z_B = z_A + b.

Clear?

What about other points? Well we have done it as the choice of t_0 was arbitrary.

To conclude: the radius of the new helix will increase but its pitch will remain unchanged.

I hope that this helps.

 

[Usaf Moji]

Thanks for the replys DeaconJohn and TomyDuby.

I caved and just did it the lumberjack way - printing out 2D projections of a helix and drawing tangents to it. I think you're both correct, in a sense.

If you look at the helix from above, it looks like a cirlce, and we're back to the circle hypothetical that I posed originally. If you look at the helix from the side, it looks like a sine wave. I just googled "sine wave", printed it out, and started drawing tangents to it (yeah, real sophisticated, I know).

If you only consider one small portion of arc length - let's say 1/8 of a rotation - the corresponding portion on the new structure will be larger in both diameter and pitch. But if you go a full rotation, the new structure stops looking like a helix altogether. It looks instead like bits and pieces of a helix sort of grafted onto each other; and in totality, the "pitch" of this weird new structure ends up being the same as the original helix.

Uggh, it's confusing anyway.

 

[maze]

If you look at the helix from above, it looks like a cirlce

In the more general definition, a helix doesn't nexessairily have to look like a circle when looked down on from above. Like TomyDuby said, that would be a circular helix which is a special case.

If you add on tangent lines to create a new curve, the curve will get fatter where the curvature is large, and basically stay the same where the curvature is small (when the curvature is small it looks like a straight line so the tangent follows the curve). A circle of course has constant curvature, so it uniformly gets pushed out.

 

[DeaconJohn]

Thanks for the replys DeaconJohn and TomyDuby.

I caved and just did it the lumberjack way - printing out 2D projections of a helix and drawing tangents to it. I think you're both correct, in a sense.

If you look at the helix from above, it looks like a cirlce, and we're back to the circle hypothetical that I posed originally. If you look at the helix from the side, it looks like a sine wave. I just googled "sine wave", printed it out, and started drawing tangents to it (yeah, real sophisticated, I know).

If you only consider one small portion of arc length - let's say 1/8 of a rotation - the corresponding portion on the new structure will be larger in both diameter and pitch. But if you go a full rotation, the new structure stops looking like a helix altogether. It looks instead like bits and pieces of a helix sort of grafted onto each other; and in totality, the "pitch" of this weird new structure ends up being the same as the original helix.

Uggh, it's confusing anyway.

Are you sure you don't have a bug in your program? If not, you've got a new theory. One well worth working out. Ask your professors (teachers, friends, buddies, whatever) DJ