Torsion: How curves twist in space, and the TNB or Frenet Frame
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- เผยแพร่เมื่อ 14 ต.ค. 2024
- If you have a curve through space, torsion measures the degree to which the curve "twists". This is separate from how the curve "curves", which we saw was given by the curvature formula. In this video we will study the TNB or Frenet Frame given by three vectors, the Tangent, the principle Normal, and the new vector called the BiNormal vector. The Tangent and Normal Vectors define a plane and what Torsion measures is how that plane changes as you move along the curve.
TYPO: Near the beginning, the Normal Vector should be 1/k dT/ds. I forgot the 1/k, where k is the curvature.
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The normal vector N is not dT/ds. Rather it is dT/ds normalized, that is, dT/ds divided by its norm. dT/ds = curvature times N. Another way of saying it is that N is dT/ds divided by the curvature. He has this right later on, but not at the beginning.
@@DrTrefor Thanks for the nice video. I was looking for clips to show my multivariable calculus class to show visualizations of torsion and the Frenet-Serret frame.
Someone points out an error, and Dr. Bazett pins the comment. He is far more concerned with making sure everyone understands the math correctly than with appearing to be perfect. He is my role model. I want to grow up to be Trefor Bazett. (I'm older than he is, so that might not work.)
Yes, I too got a little confused due to this, but spending few minutes deriving it understood this small imperfection...thanks for the great video
Thanks for pointing this out, and thanks to Dr. Bazett for fixing it and pinning this comment
@@ConceptualCalculusYes, unlike the "Professor Dave Explains" guy who is arrogant and does not take criticism well.
I’m taking differential geometry now nearly 30 years after vector calculus and multivariable calculus. Your videos are saving me massive amounts of time getting reacquainted with topics.
What's your passion or doing job or study math till
The quality of this content is unbelievable!
One of the best examples of 'How complexity gets simplified'... Probably now I won't forget it
Thank you sir
I am a first year electrical engineering student , thanks for your well define explanation as well as animation too. LOVE FROM INDIA
Your animation explains the concept better than the equations alone.
For a self learning student this amazing video will be a boon. Thank you sir.
I'm searching for torsion lately but I didn't get any proper idea about torsion. But suddenly i see you video totally, it's awesome and i got idea properly. Thanks 👍
I have been watching a lot of videos on youtube, your this video is the only one that tells me the underneath meaning of the binormal vector. All other videos just give the mathematical formula which have no added value than reading textbooks. You did help me a lot in my self study.
Very Great Work ! Thank you so much !
Jack Lam from Hong Kong
Brilliant! You were able to explain in little less than 15 minutes what my teacher struggled to teach me for, what, two weeks already? I'm so incredibly glad I found your channel. You just got yourself a new sub.
Wow, I have just discovered this channel and I'm really excited. I'm going to watch all of the differential geometry videos and others. Thank you! Subscribed!
i felt some difficult , but i always appreciated how you get us your information
no one like you , you are so special ❤
I love it so much! Can you share which software you use to draw these pictures?
Stucked at these terms months sir, thank you for this great explanation. Won't forget for life
I can't express how thankful I am for your videos!
Glad you like them!
Thank you for your hard work for explaining such a beautiful concept!!
Just had a revisit of differential geometry and it now became must easier and less daunting than it used to be. The animation/visualisation of the frame in this video is very helpful. Thanks for sharing. One more like.
Good morning Dr,
Your videos are amazing to understand the non linear Euler Bernouilli beam theory as we use transformation matrices and curvatures rates to recover section deflections and their rates !
Amazing work !
Cool!
Thank you for answering our homework. ❤️
Interesting. I'm studying w/Wolfram U's new course on multi-variable calculus. Wolfram Lecturer Tim mentioned the torsional component of curves. I don't believe he called it a BiNormal Vector; your explanation noting it was the cross product of the other 2 vectors was highly useful to understand.
Torque with Viviani's Curve had gotten my attention before I understood any of the mathematics. There's a rope-movement exercise technique called RMT Ropes (AKA "Flow Rope") where the rope's centerline approximates Viviani's Curve. When up to speed, there is a significant amount of torque when the rope hits the ground in front of the body: rope-over-the-top. There's also rope-under-the-bottom torque when the rope hits the ground behind the body. These can be rather significant; the rope will make a noticeable bounce at these 2 positions. Significant energy is required to propel the rope over the head; this reactance can provide a significant fraction of that energy. I do not know about torques elsewhere in the cycle; I'll have to plot the torque of the curve...
In any case, I highly recommend checking out this exercise/movement technique. We rarely access our pronators/supinators directly, and all the spiraling lines in the body (arm, torso, and legs) are intimately connected. This could bring some lively homework to a math class!
Benditos sean los su títulos y a este maravilloso Dr. que comparte sus conocimientos. Solamente puedo decir Gracias, explica de maravilla un tema que me parece complicado y el cual a mi profesor de Dinámica no le entiendo nada.
Seeing these visuals helps me so much more when understanding what I’m doing
Couldn't find best then this to explain such topics. Tq for explaining it. 🙌
Incredible! Thank you so much for producing this video. This video helped me understand tangent, normal, and binormal components for my Dynamics class.
Take care. God bless. Jesus loves you.
Amazing work the visuals made the concept much easier to grasp
Thank you so much for this amazing explanation! Your videos keep blowing me away!
Thank you for this masterpiece, I really wish I found your content last year before maybe I wouldn't have failed
Thank you, sorry to hear that!
Your way of speech always got me nice video man !
Great videos! Won't you mind sharing the software or applications that you're using to create these amazing graphics/animations?TIA
It is all done in MATLAB
Amazing video. Thank you for putting your hardwork in explaining and making visualisation easier.
Sir the graphics are very helpful in understanding the concept .. thanku😄😄
Watched two times and saved it clearly in my minds. No need to racall.
Thank you sir, your teaching is great. Please sir, make a video on TORSORS.
Very well explained professor!
Thanks. It helped me a lot! Have a beautiful maths day.
Thanks a LOT ....there's only FEW english tutorials on TH-cam ...helped a lot ..tomorrow is my finals ...
helps with the roller coaster rides at Canada’s Wonderland …and perhaps with the making of Funnel Cake which they sell at Wonderland. …or even better, Churro! Churro has a delicious texture which indicates torsion, but i believe that same delicious texture is apparent in Twister Licorices.
Best explanation.
What a greatly explaination of torsion
Thanks for such wonderful explanation
I was having a hard time visualizing the Plain that T and N are in. in other words, I could not understand what is the B vector. I thought the B vector WAS the plain. but now I understand that B itself is actually the normal vector of the plain that consists of T and N vectors. Thank you so much for the animation and your simple explanation.
your students are very lucky to have you as their professor
loved your explanation and representation sir,new subscriber and love from India
Thanks and welcome!
Beautifully explained! Great!
Glad you think so!
the best lesson I have seen ever, thanks
Thank you!!
Beautiful explanation... Thank you!
Thank you so much sir it aids my understanding of topic
This video shows how beautiful Mathematics really is.
P.S. Thanks Dr. Trefor
Dear Dr Bazett, Thank you for the great explanation. I wanted to ask…say a particle with constant velocity is travelling in a straight line (not turning in the plane) but it is twisting on its axis (along the tangent T unit vector) how is the binormal calculated?….since if T is constant and the particle is not turning which means the normal N = 0 as well…and since the binormal is based on T and N, how is B (and hence the rate of particle twist) calculated in this case? Thanks a lot
Excellent visual explanation ,, please do more explanation with helix
@Trefor Bazett dr/ds is Unit Tanget Vector rather than Tangent vector. Similary dT/ds is Unit Normal Vector rather than Normal vector. It looks like complaining but being specific helps a lot of time for viewers.
Nice video! I will recommend it to my students as a complement to our class.
Cool, glad you think it will be helpful!
How you got such a deep insight? I am amazed! A great teacher!!!!!!!
thank you!
I love this video, Dr. Bazett is the goat
I didn't go to college today because I have watched your video. You brought the college in my room.
This is the benefit of video ,...
Excellent!!
Hi Dr Trefor, in which video did you go over the concept discussed at 5:59 about the unit vector multiplication with its derrivative?
Idts he ever mentioned that. But here's the thing:
If |r(t)|=1, i.e., r(t) is a Unit Vector, then
|r(t)|²=1
r(t) • r(t) = 1
Differentiate,
r'(t) • r(t) + r(t) • r'(t) = 0
2 r'(t) • r(t) = 0
Hence,
r'(t) • r(t) = 0 if |r(t)| = 1
doctor please explain dynamic systems analysis
hello, I found this video and I feel very thankful to you. And I have a question of what I saw in Wikipedia. I search “Frenet-Serret frame” description, and I cannot understand “suppose that the observer carries an (inertial) top (or gyroscope) with them along the curve. If the axis of the top points along the tangent to the curve, then it will be observed to rotate about its axis with angular velocity -τ relative to the observer's non-inertial coordinate system.” And, “If the top points in the direction of the binormal, then by conservation of angular momentum it must rotate in the opposite direction of the circular motion. In the limiting case when the curvature vanishes, the observer's normal precesses about the tangent vector, and similarly the top will rotate in the opposite direction of this precession.” I think it’s hard to understand because I didn’t studied calculas 3.I hope you answer me. I apologize if the nuances of my comment are little bit not polite, because I am not very familiar with the nuances of English...
Sir..As we khown for the figure Tangent ,Normal and Binormal are perpendicular to each other and also khown B=T×N this is Binormal...sir This statement is ture N=T×B..from the figure...
I wish you had a lecture on elementary differential geometry.
Really good!
Thank you for the clear explanation, but you need to buy a better microphone.
Sir your video is really helpful to us in understanding thing in more visual way.thanku😁
So glad it helped!
Thank you sir
thank you very much for the wonderful explanation. please are the curvature and torsion of a regular curve thesame all over the curve?
@@DrTrefor ok , does that mean the curvature and torsion are the same at all points on the curve? and what will happen to its TNB s, are they also the same at all points? thank you .
wonderful explanation
Glad it was helpful!
Thank you !!
How did you animate those curves and those moving vector in powerpoint? Its amazing, can you please tell how you did it?
all MATLAB tricks:)
@@DrTrefor thanks al lot
Hi! I like the video a lot. It's very helpful. I have a question like if the dB/ds is parallel with unit vector N, so they are parallel in the same direction or opposite direction? As N always points inwards, would dB/ds also point inwards? Thank you!
Thanks for the video and clear explanation. I wanted to know how to transform TNB to Cartesian coordinate system.It is useful in the path planning of vehicles.
This is probably a stupid question but i have to ask. If we make a copy of the circle in the video, and squish it a bit so that we have a somewhat deformed circle, how can we measure the distortion introduced by our deforming of this second circle? Assuming we have a ortogonal frame of non-normalized vectors on both circles, can we just calculate the volume of the parallipiped defined by these 3 vectors on each circle? If so then what? Is the defornation just a ratio of the two volumes? Assuming we know nothing about the deformation applied to the 2nd circle. Sorry if it's idiotic but now i'm intrigued. The circle in the videos are in R^3, so this could apply to a sphere as well i suppose
Excellent, as usual.
i just feel each of my brain cells waking up
Excellent visual explanation. However, I find the choice of notation (T= tangnet ,N= normal, B=binormal) is perhaps a bit unfortunate, especially when you are explaining the concept to learners. "Normal" usually means a vector perpendicular to the surface of a plane rather than a vector perpendicular to another vector. Would it not be better to use, for illustration purposes, v=velocity (your T), a=accelearation (your N), n=normal (your B)? So, the normal vector n is perpendicular to the plane formed by the velocity and acceleration vectors, and the derivative of (v x a) is, simply by definition of the cross product, a (torque) vector perpendicular to the plane formed by v and a, namely, in the direction of n.
Muito bom (do Brasil)! Very good (from Brasil)!
Great work!!!
Thank you so much sir❤
underrated
Greetings, i am a math student, and i do not know how to contact you, SO i will ask in comments... May i translate this video to spanish and upload it to my channel ? This video is great and helped me in differential geometry, and now i want to use it for calculus ii... Would You allow it ?
I'm sorry, but I don't allow reuploads (I have had a few negative experiences in the past; you don't sound like that type of person at all, but I would prefer to just be safer).
@@DrTrefor thank You very much anyways :") and thanks again for the video. I understand the copyright stuff ^^
Jesus you also have videos on this, I've now been saved from my professors terrible lecture notes an non existent explanation
haha that sounds horrible, glad at least I could help!
Fantastic explanation
Thx u for this ur work was superb thx
Glad you liked it:)
Nice explanation.Great work
That was really amazing
Thank you so much 🙌🏻
Does the concept of Binormal Vector exist in n-dimensional space?
For example, if we have an n-dimensional curve defined by an n-dimensional real vector-valued function.
Or may be, it has something important to analyze a surface in n-dimensional space?
Cross products are only defined in 3 dimensions, so no.
@@replicaacliper so that's also what I understand so far. Thanks
Again, excellent video. My professor ( from my engineering dynamic class ) switches from Cartesian coordinate system to the Frenet frame. I do know what's the difference between those 2, but I don't know why he switches. May you explain ?
Think of a helicopter flying around in the general vicinity of an air-traffic control tower.
The air-traffic controllers in the tower think of the helicopter's position in terms of a fixed coordinate system with themselves at the origin. The helicopter is x meters east of them, y meters north of them, and z meters above them. This is the Cartesian coordinate system.
The pilot in the helicopter thinks in terms of a moving coordinate system with herself at the origin. A bird is 70 meters in front of her, along her T axis. Another helicopter is 800 meters to her left, along the N axis. As she moves around, the direction that is in front of her moves with her. Her T axis is always directly in front of her, which is sort of constant from her point of view, but that changes from the perspective of the tower.
There is a great idea! For the dark side of the Universe - suppose that it consists of short-term interactions in long-lived fractal networks, the smallest quantum operators in energy, spherical rosebuds, consisting of a large set; 1 - rolled into a sphere, 2 - half collapsed into a sphere and 3 - flat, vibrating quantum membranes relative to their working centers in the sphere
So how do we generalize this into multiple (3+) dimensions?
This play list serve me
Gd mrng ....
If the direction of tangent and binormal is fixed then what will happen???
Definitely the best !
Awesome , keep it up.
At 4 min45 is it the same explanation if we take the formula of the normal vector which is N= (1/K) (dT/ds)
and then multiply both side by kappa ? KN=(K/K)(dT/ds)
If not i am really confuse why we are multplying the K by a normal vector>
This video was really cool. But unfortunately it did not answered my question: Why does the third derivative appears when calculating torsion? What does the third derivative mean in this scenario? Can I have a non continuous third degree curve when calculation torsion? I would really appreciate if you could help me. Thanks!
Anyone help with this?
u made everything clear,thanks
Thank you for clarification !
I have a doubt. What exactly tangent vector is? In previous video u used dr/dt to calculate tangent vector. And in this video u r using dr/ds. Are both dr/dt and dr/ds same?