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I understand the graphical interpretation of the curvature of a curve in $\mathbb{R}^3$. Could you help me to understand the graphical meaning of the torsion of a curve? I know that if torsion is positive, the curve goes through the osculating plane from below upwards. Conversely, if torsion is negative, the curve goes through the osculating plane from above downwards.

Question: How can I interpret torsion geometrically? For example, the curvature is the inverse of the radius of the osculating circle. Is there a similar interpretation for torsion?

Quillo
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Mika Ike
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3 Answers3

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Here's how my differential geometry professor explained it to me.

  • Curvature measures the failure of a curve to be a line. If $\gamma$ has zero curvature, it is a line. High curvature (positive or negative corresponding to right or left) means that the curve fails to be a line quite badly, owing to the existence of sharp turns.
  • Torsion measures the failure of a curve to be planar. If $\gamma$ has zero torsion, it lies in a plane. High torsion (positive or negative corresponding to up and down) means that the curve fails to be planar quite badly, owing to it curving in various directions and through many planes.

Now for some examples.

  • $\tau = 0, \kappa = 0$: A line. Lines look very much like lines, and they are certainly planar.
  • $\tau = 0, \kappa =k > 0$: A circle. Circles don't look like lines, especially small ones. They have constant curvature. However, they do lie in a plane.
  • $\tau = c >0, \kappa = k >0$: A helix. Helixes curve like circles, failing to be lines. They also swirl upwards with constant torsion, failing to lie in a plane.
  • $\tau >0 , \kappa = k > 0$: A broken slinky. Slinkies curve like circles, failing to be lines. They generally have constant positive torsion, like helixes. But if you break them, the torsion remains positive (viewed from the bottom up), but how large the torsion is corresponds to how stretched the slinky is. A very stretched slinky has large torsion, compacted slinkies have small torsion.
Elchanan Solomon
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    Ok, Thank you. I understand what you explained me,but what i want is "what exactly is the graphical interpretation of torsion", "how torsion measure ", "If we have a torsion of 3, what this graphically means?" "What´s the graphical interpretation of kurvature=3"

    I have made a poster explaining what the curvature means You can view it here (i´m translating into english)

    https://www.dropbox.com/s/z7xglmewwl0qltb/interpretacion%20geometrica%20curvatura16%20stack.png

    I want to know what a torsion=2 means graphically

     – Mika Ike Mar 31 '13 at 14:18
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Torsion is the speed with which the osculating plane rotates around the curve in radians per unit length. Positive torsion indicates a clockwise rotation and negative torsion a counter clockwise rotation. Torsion is an infinitesimal notion, so informally it measures speed of rotation on small portions of the curve where it is an almost straight line. It is hard to see this rotation globally since the curve itself also twists through space.

Imagine a striped garden hose that follows a closed space curve. The hose ends meet with a certain rotation, which you can see by tracing the stripes along the hose. This angle is the integral of the torsion along the curve, its total torsion.

WimC
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  • ok, in general i understand but... which means a torsion of 2?

    and a torsion of -3?

     – Mika Ike Apr 01 '13 at 07:44
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    @MikaIke It means that when travelling a small distance $d$ on the curve the osculating plane rotates by approximately $2d$ and $-3d$ radians respectively (clockwise direction is positive). – WimC Apr 01 '13 at 08:06
  • @WinC i think i understand you, and i like your explanation.... thank you very much.

    But... i´m thinking any more and... will ask you 2 questions to better understanting.

    • what
     – Mika Ike Apr 01 '13 at 13:28
  • the osculating plane rotates respecto to ¿_____?

    But more simple,thinking in a standar helix

    You can view a parametrization and the curvature and torsion... torsion(t)=1/5

    for each unit we run over the spiral ¿The osculating plane rotates 1/5 radians? If so, we have in a single cycle the spiral lenght is 2PiSqrt[5] And is clear that in a cycle the osculating plane rotates 2PI,but we have that lenghttorsion=2PiSqrt[5]1/5 is different from 2Pi(the obvious solution) Could you explain a little more, beacuase i think that i´m close to understand very well :-) but still i can´t

     – Mika Ike Apr 01 '13 at 14:16
  • WimC here is the paper what i refer to

    https://www.dropbox.com/s/gcsfh2hgoj7fx2s/torsion%20graph%20interpretation%2001%20img126.jpg

     – Mika Ike Apr 01 '13 at 14:17
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    @MikaIke The axis of rotation is the curve itself. As noted, this is an infinitesimal notion. This rotation is not easily visible globally since the axis of rotation is not fixed. For one revolution in a helix the total torsion is not necessarily a multiple of $2\pi$. Again, the picture of a striped garden hose laying coiled in a helix may help. With each extra loop upward the stripes along the hose rotate only slightly outward (for a right handed helix as in your picture). – WimC Apr 01 '13 at 14:32
  • OK, @WimC THANK YOU very much – Mika Ike Apr 04 '13 at 13:11
  • Subject RESOLVE. – Mika Ike Apr 04 '13 at 13:12
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You say you understand curvature, so let's work from there.

The curvature at a given point $P$ is a measure of how fast the curve moves away from its tangent line at $P$. Saying it another way, it measures the amount on non-linearity at $P$.

Torsion is very similar: The torsion at a given point $P$ is a measure of how fast the curve moves away from its osculating plane at $P$. In other words, it measures the amount on non-planarity at $P$.

bubba
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  • Ok, Thank you. I understand what you explained me, but... – Mika Ike Mar 31 '13 at 11:29
  • You're welcome. But what ?? – bubba Mar 31 '13 at 12:50
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    bubba

    Excuse me... I understand what you explained me, i know that curvature is a measure on how the tangent is changing but... here

    https://www.dropbox.com/s/z7xglmewwl0qltb/interpretacion%20geometrica%20curvatura16%20stack.png

    you can see what a concrete curvature graphically indicates

    What i want to understand is how change a curve with torsion =1/2, how change with torsion(p)=1, =2, =4,...

    The meaning of a concrete number.

     – Mika Ike Apr 10 '13 at 12:42