Really enjoyed the graphics and explanations in 20:10 - 27:18. Thinking of principal curvatures as the axes of a small ellipse projected on the surface (and its relation to the weingarten map) is a very useful mental model, and illuminated a fair bit of intuition.
This is an amazing presentation with a profound excellency , thank you for your great efforts, I'm sure these videos will become an iconic guide for all researchers for many years to come. God bless you
I believe to have found a possible misinterpretation for the saddle(pringle) shape: the mean curvature is not zero. It can be zero at some point, but is definitely not zero at every point. H =/= 0, otherwise it would be considered as a minimal surface. Using r(x, y) = (x, y, xy)*, one can find that H = $1/2 (k1 + k2) = 8xy / (1 + 8(x^2 + y^2)^3/2 $ =/= 0. QED *which is the same as using r(u,v), by linear change of variables u = x+y, v = x-y. = (1/2(u+v), 1/2(u-v) , u^2 - v^2).
He hasn't made them, but is considering to do so. See his reply in Lecture 13, it reads "As mentioned by @jrkirby, I started recording these videos in the middle of the semester, since CMU switched to remote teaching. I may go back and record 1-12 after the semester is over." thank you for making the lectures public.
This is likely a silly question, but why here is the unit normal to a unit sphere the negative of its position? I'm usually lead to think it is equal to the position as point (1,0,0) would have the normal (1,0,0). Why is the negative orientation necessary in this case? So more generally, what determines which "direction" you define the surface normal? i.e. Why do you define it in the "Concave" direction here, while 3D models used in games usually use the "Convex", or "outwards" direction?
40:20, So what are the domains and codomains of the weingarten map and the shape operator? Is it safe to say that the weingarten map takes tangent vectors on the tangent plane (at a point on the surface) and outputs tangent vectors on the tangent plane, while the shape operator takes a direction on the parametrization domain and outputs a direction on the parametrization domain?
I got confused by the formula at 31:04, is the domain of gauss map the parametrization space, or the embedded surface given by f(R^2)? Taking the inner product of vectors that belong to different tangent spaces seems inconvenient. There must be more elegant formulation.
At 23:30, aren't dN(X1) and dN(X2) the "axes" of the ellipse on the unit sphere and not X1 and X2? Edit: Nevermind, I think i understand: Although X1 and X2 aren't drawn on the surface, they are indeed the axes of the ellipses :)
Hello, I was wondering why for your kappa curvature equation, on wikipedia's and in my book it's the magnitude of 2nd derivative dot 1st derivative of gamma over 1st derivative to the 3rd power. How come in your equation, it's to the 2nd power?
If you're referring to the formula κ(X) = /|df(X)|² (for instance at 31:17), it's likely because N is a unit normal, which means it already incorporates a normalization factor.
I know this comment is a year old, but just in case someone comes along and is also confused: df is basically the Jacobian of f and so can be written as a 3x2 matrix. You can see the matrix form of df written on the screen (it's the top-left most matrix). X_1 and X_2 are vectors within the 2D domain. So to apply df to them, simply multiply them by the matrix.
This is great! Thank you! Can you clarify why why @59:41 there is a deficit in the area on the surface patches, as opposed to a gain as in here: www.ams.org/publicoutreach/feature-column/fcarc-sphericon2
Never mind, the radius of the disc on the flat surface, r, will follow a geodesic along the deformed plane, and the area will decrease: If the disc is on a sphere of radius R, the area will be: 2 pi R^2 (1 - cos(r/R)), as opposed to pi r^2, as in here: math.stackexchange.com/a/1300712/152225.
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
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This is an incredible resource that's very well made. Thank you for posting.
Really enjoyed the graphics and explanations in 20:10 - 27:18. Thinking of principal curvatures as the axes of a small ellipse projected on the surface (and its relation to the weingarten map) is a very useful mental model, and illuminated a fair bit of intuition.
This is an amazing presentation with a profound excellency , thank you for your great efforts, I'm sure these videos will become an iconic guide for all researchers for many years to come.
God bless you
To support mathematicians I am subscribing. Thank you for contribution to math science.
Incredible, thank you so much
Thank you for this amazing lecture series!!!
Excellent!
Excellent, very organized video.
Amazing!
I believe to have found a possible misinterpretation for the saddle(pringle) shape:
the mean curvature is not zero.
It can be zero at some point, but is definitely not zero at every point.
H =/= 0, otherwise it would be considered as a minimal surface.
Using r(x, y) = (x, y, xy)*, one can find that H = $1/2 (k1 + k2) = 8xy / (1 + 8(x^2 + y^2)^3/2 $ =/= 0. QED
*which is the same as using r(u,v), by linear change of variables u = x+y, v = x-y.
= (1/2(u+v), 1/2(u-v) , u^2 - v^2).
Request you to please record and post lectures before Lecture 14. :)
He probably hasn't made them in this format (i guess this is the format for online teaching due to covid restrictions) :-(
He hasn't made them, but is considering to do so. See his reply in Lecture 13, it reads "As mentioned by @jrkirby, I started recording these videos in the middle of the semester, since CMU switched to remote teaching. I may go back and record 1-12 after the semester is over."
thank you for making the lectures public.
This is likely a silly question, but why here is the unit normal to a unit sphere the negative of its position?
I'm usually lead to think it is equal to the position as point (1,0,0) would have the normal (1,0,0). Why is the negative orientation necessary in this case?
So more generally, what determines which "direction" you define the surface normal?
i.e.
Why do you define it in the "Concave" direction here, while 3D models used in games usually use the "Convex", or "outwards" direction?
I was wondering too. did a little digging and found the reference is to this spot in lecture 13 -> ruclips.net/video/FRvhgkGKfSM/видео.html
40:20, So what are the domains and codomains of the weingarten map and the shape operator? Is it safe to say that the weingarten map takes tangent vectors on the tangent plane (at a point on the surface) and outputs tangent vectors on the tangent plane, while the shape operator takes a direction on the parametrization domain and outputs a direction on the parametrization domain?
I got confused by the formula at 31:04, is the domain of gauss map the parametrization space, or the embedded surface given by f(R^2)?
Taking the inner product of vectors that belong to different tangent spaces seems inconvenient. There must be more elegant formulation.
At 23:30, aren't dN(X1) and dN(X2) the "axes" of the ellipse on the unit sphere and not X1 and X2?
Edit: Nevermind, I think i understand: Although X1 and X2 aren't drawn on the surface, they are indeed the axes of the ellipses :)
Hello, I was wondering why for your kappa curvature equation, on wikipedia's and in my book it's the magnitude of 2nd derivative dot 1st derivative of gamma over 1st derivative to the 3rd power.
How come in your equation, it's to the 2nd power?
If you're referring to the formula κ(X) = /|df(X)|² (for instance at 31:17), it's likely because N is a unit normal, which means it already incorporates a normalization factor.
min 46:37. Can someone tell the formula he uses to apply df(.) on X1, and X2? are Xi components (du, dv)? thanks :)
I know this comment is a year old, but just in case someone comes along and is also confused:
df is basically the Jacobian of f and so can be written as a 3x2 matrix. You can see the matrix form of df written on the screen (it's the top-left most matrix).
X_1 and X_2 are vectors within the 2D domain. So to apply df to them, simply multiply them by the matrix.
This is great! Thank you! Can you clarify why why @59:41 there is a deficit in the area on the surface patches, as opposed to a gain as in here: www.ams.org/publicoutreach/feature-column/fcarc-sphericon2
Never mind, the radius of the disc on the flat surface, r, will follow a geodesic along the deformed plane, and the area will decrease: If the disc is on a sphere of radius R, the area will be: 2 pi R^2 (1 - cos(r/R)), as opposed to pi r^2, as in here: math.stackexchange.com/a/1300712/152225.
12:30
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