Hi Prof. Crane. Thanks for your videos -- I really love your visualizations. There is a small typo around 46:26: you should have T(s) = (\cos\theta(s), \sin\theta(s)) instead of (\cos s, \sin s). Just wanted to point this out because it may confuse your students who are seeing this for the first time. Thanks again for your great work on this course!
The animations of the curves are so much fun you should consider sending them on Twitter on an animated GIF to play on a loop if you haven't yet. Thanks again for this body of work. Absolutely out of this world!
At 47:35, there is a mistake that tangents should be cosine of thelta of s and sine of thelta of s rather than cosine of s and sine of s, unless thelta of s is equal to s like subsequent example. I don't if I understand these concept correctly or not. Please correct my reply if I'm wrong.
Around 47:00, it's a little odd to refer to constants of integration when definite integrals are displayed. It might be clearer to just write/say that the first definite integral is θ(s)-θ(0) and similarly for the second.
At 1:20 [28 seconds] I think there is light error-not dx/dz, rather dz/ds. However, at this point I would like to express my gratitude for these great videos.
The explanation at 1:20:00 was great, but I was wondering if the coordinate functions necessarily be independent with respect to s? So if y was defined as a function of x along s would it be valid to apply the chain rule when finding the partial for y? It feels like that should be the case but I'm having a hard time visualizing if that would be acceptable in any case
Great lecture series. The animations in this video in particular are amazing and really useful. One comment: the Clifford Torus is a flat torus in 4D and cannot be embedded in 3D. The Borrelli Torus (2012) is a flat torus in 3D. It is C^1 and fractal-like.
Thanks-this is indeed a bit of abbreviation. People who work on Willmore flow, for instance, sometimes talk about the Clifford torus as a surface in R^3, because Willmore energy is Möbius invariant in R^3 (and hence really better thought of as an energy for surfaces in S^3…). So, any stereographic projection of the flat Clifford torus in 4D to a torus embedded in 3D will be a Willmore minimizer.
A short note: I come from machine and deep learning and went through your course specifically to understand what talking about manifolds in higher dimensions is about. Kind of a disappointing moment to see your statements at minute 8. 😂
It occurs to me that because curvature is really a 2D phenomenon (the twisting of the tangent vector in a certain plane) it is more intuitive to conceptualize it as a bivector instead of the dot product between N and dT/ds it would be something like κ = T ∧ dT/ds, which should have the same magnitude but also gives you the plane of the curvature. And then torsion is the twisting happening orthogonal to this plane which would be like a trivector, maybe something like τ = κ ∧ d²T/ds² ? I don't know, haven't done the derivation yet to see what the equivalent expression would be.
Hi Prof. Crane. Thanks for your videos -- I really love your visualizations. There is a small typo around 46:26: you should have T(s) = (\cos\theta(s), \sin\theta(s)) instead of (\cos s, \sin s). Just wanted to point this out because it may confuse your students who are seeing this for the first time. Thanks again for your great work on this course!
The animations of the curves are so much fun you should consider sending them on Twitter on an animated GIF to play on a loop if you haven't yet. Thanks again for this body of work. Absolutely out of this world!
Thank you so much for doing these, I am just getting into 3d deep learning so these videos are perfect.
At 47:35, there is a mistake that tangents should be cosine of thelta of s and sine of thelta of s rather than cosine of s and sine of s, unless thelta of s is equal to s like subsequent example. I don't if I understand these concept correctly or not. Please correct my reply if I'm wrong.
Around 47:00, it's a little odd to refer to constants of integration when definite integrals are displayed. It might be clearer to just write/say that the first definite integral is θ(s)-θ(0) and similarly for the second.
Osculations to prof for this excellent lecture, top quality graphics...!
Happy to see new lecture online!
At 1:20 [28 seconds] I think there is light error-not dx/dz, rather dz/ds. However, at this point I would like to express my gratitude for these great videos.
The explanation at 1:20:00 was great, but I was wondering if the coordinate functions necessarily be independent with respect to s? So if y was defined as a function of x along s would it be valid to apply the chain rule when finding the partial for y? It feels like that should be the case but I'm having a hard time visualizing if that would be acceptable in any case
Great lecture series. The animations in this video in particular are amazing and really useful. One comment: the Clifford Torus is a flat torus in 4D and cannot be embedded in 3D. The Borrelli Torus (2012) is a flat torus in 3D. It is C^1 and fractal-like.
Thanks-this is indeed a bit of abbreviation. People who work on Willmore flow, for instance, sometimes talk about the Clifford torus as a surface in R^3, because Willmore energy is Möbius invariant in R^3 (and hence really better thought of as an energy for surfaces in S^3…). So, any stereographic projection of the flat Clifford torus in 4D to a torus embedded in 3D will be a Willmore minimizer.
I think there is a typo in d gamma(X) in 1:20:50
I'm excited for the discrete version. I'm thrilled that we're covering so much coordinate-free stuff.
A short note: I come from machine and deep learning and went through your course specifically to understand what talking about manifolds in higher dimensions is about. Kind of a disappointing moment to see your statements at minute 8. 😂
It occurs to me that because curvature is really a 2D phenomenon (the twisting of the tangent vector in a certain plane) it is more intuitive to conceptualize it as a bivector instead of the dot product between N and dT/ds it would be something like κ = T ∧ dT/ds, which should have the same magnitude but also gives you the plane of the curvature. And then torsion is the twisting happening orthogonal to this plane which would be like a trivector, maybe something like τ = κ ∧ d²T/ds² ?
I don't know, haven't done the derivation yet to see what the equivalent expression would be.
Am surprised the view number is so low, the quality of the course is superb!
Best course I have seen on differential geometry - period.
Amazing lectures! FYI, the slide at 45:00 should have "turn right" for the -k(s) axis (it says "turn left" for both +k(s) and -k(s))
At around 49:50, how to get that last integration of the vector?
Your visualizations are off the charts. Thanks a bunch :)
For 2D curves, how is Winding Number defined for points on the curve?
It seems that the winding number is not well-defined for points on the circle. At the definition shown at 56:30, the denominator goes to zero.
54:21
Please provide me the simple defination of Curve along with Examples