i am not your student, but interresting in this topic, because it is rare that someone with so much expertise, shares such well structured knowledge from ground up. Why am i interested in this ? (i know you didn't ask 😂) It is like you mentioned: you need it for almost everything now, engineering, AI, architecture, graphics and so on. Thank's for sharing this content for free 👍 This is Science ! Awesome !
This is the soul of Teachers. To pass on information in order for our society to advance in many areas ( in our case, it is Technology ) Thank you very much.
0:38 Why we might want to study geometry? 1:48 Applications of Discrete Differential Geometry 6:09 What will we Learn in This Class? 8:06 What won't we learn in this class? 9:38 Assignments 11:26 What is Differential Geometry? 13:17 What is Discrete Differential Geometry? 15:32 Grand Vision 16:31 How can we get there? (Game) 18:53 Example: Discrete Curvature of Plane Curves 19:55 Curvature of a Curve - Motivation 20:54 Curves in the Plane 21:45 Example 22:16 Discrete Curves in the Plane 23:17 Example 23:50 Tangent of a Curve 25:22 Example 26:27 Normal of a Curve 27:37 Example 29:29 Curvature of a Plane Curve 32:46 Curvature: From Smooth to Discrete 34:57 When is a Discrete Definition "Good"? 36:36 Playing the Game 37:46 Turning Angle 39:04 Integrated Curvature 41:15 Discrete Curvature (Turning Angle) 43:03 Length Variation 45:14 Gradient of Length for a Line Segment 46:46 Gradient of Length for a Discrete Curve 48:03 Discrete Curvature (Length Variation) 49:19 A Tale of Two Curvatures 50:52 Steiner Formula 51:56 Discrete Normal Offsets 54:24 Discrete Curvature (Steiner Formula) 55:51 Osculating Circle 56:51 Discrete Curvature (Osculating Circle) 57:27 A Tale of Four Curvatures 58:04 Pick the Right Tool for the Job! 58:55 Curvature Flow 59:53 Toy Example: Curve Shortening Flow 1:01:10 Discrete Curvature Flow - No Free Lunch 1:03:25 No Free Lunch - Other Examples 1:04:56 Course Roadmap 1:07:26 Applications & Hands-On Exercises
This is great. It's refreshing to see computation treated as a first class citizen instead of an afterthought like other math lectures. My cold programmer heart grew three sizes today, which I can compute with a discrete curve lengthening flow.
Sometimes when I come across resources like this that are free of cost it truly makes me happy to live in today's day and age! How amazing is this?! Thank you
I have rarely seen such a perfect pedagogy on youtube, everything is perfect, the examples, the breaks to think... And on a fascinating subject! Keep it up!
Some courses are in huge demands but little in supplies. It has 15k views for 5 months since it was uploaded (a super impressive number for a pure academic course). Been looking everywhere last year for an introductory differential geometry courses for engineering, thank you for making this freely available online professor!
Wow, the idea that discrete differential geometry is the language of a new world really fascinates me... and the GAME of searching for different perspectives of translating between smooth and discrete geometry seems really interesting. Thank you, sir!
Thank you so much! I'd taken a classic differential geometry course in university, and this is great stuff! Really clearing up a lot about how to apply the things I learned in a computational setting!
Thanks a lot Professor Crane. I'm not your student and have no knowledge about differential geometry (continuous and discrete) but geometry of curves and surfaces is my absolute favourite topic to think about in mathematics. I'm totally looking forward to your videos. Thanks a lot again for sharing your expertise for free!!! :)
Hey Keenan, great video. This reminds me of a simple interview question I sometimes ask: if you have a rope going around the equator, and you increase it by 1m, by how much can you raise the rope above the ground? Most people think it's going to be microscopic (because 1m is very little compared to the circumference of the earth), when in fact it's 1m/(2*pi). This relates to some of the concepts in your video. When I try to explain why the intuition of people is wrong, I resort to imagining a 2D equivalent, a rope around the border of a circular lake, and then thinking of a rope around a square lake - and how much the rope needs to be extended if you want to 'lift' the rope, i.e. bring it a bit further away from the border of the lake. If you try that, you'll notice that the places where you need more rope are the corners, i.e. the places where there is curvature (as seen by the fact that the normal turns there). I always saw that as a 'Dirac' distribution of curvature, but your lecture explains how this can be seen in various different ways. Looking forward to watching the rest (btw I got a PhD in diff. geometry 20y ago, but went to work in an applied field since).
I love how you are using illustrations to explain your maths. Graphics help me commit things to memory. Thank you for making tutorials. I hope you will make more videos.
Just started my research on this topic and my advisor suggested one of your articles, came here by chance and I must say I'm not disappointed! Awesome :) thank you for the classes!
It was evident after listening for only a minute or two that you are a confident and experienced lecturer. It is rare to encounter a speaker who so deftly combines informality and polish. You've made a complex topic understandable and interesting. Thank you so much for sharing your expertise freely with everyone. You are a credit to the best traditions of scholarship. Subscribed!
Thank you so much for sharing this course! This is really helpful for students who not only require understandings from pure mathematics point of view but also from a more applied prospective, for building intuitions.
@keenan crane The inner product does not only preserve the sign. We also want to extract the normal component of the derivative so that we can negate any change in the speed at which we are traveling along the curve, of course this is 0 in the case of arc length parametrized curve.I really like the lectures so far. Thank you for sharing them with the world and putting the effort in!
Thank you so much Prof. Keenan Crane for generously sharing your knowledge and wisdom to help people gain knowledge and wisdom to improve humanity all together. You and many other teachers who generously shared your wisdom and knowledge inspired me to share my knowledge and wisdom to people in the future when I become an expert in some areas. Thank you so much again!
I find it interesting that as theta approaches zero (meaning we approach a smooth function), theta = 2 sin(theta/2) = 2 tan(theta/2). At 50:08 Dr. Crane mentioned the small angle approximation for sin(x) but did not mention it holds for tan(x) as well. As x->0, sin(x)->x and cos(x) -> 1 so tan(x) -> x/1.
Thanks for sharing the class materials! I'm really excited to get started here. Could you share what tool you used to generate the beautiful purple and black cell-shaded plots? They are wonderful!
thanks for this. Im interested in topological data analysis, and algebraic topology. This course is a gem. Anyone wants to create a discord group to discuss ideas and assignments?
Hi Prof. Crane, thank you for the amazing lectures! I had a question about minimizing discrete curvature flow at 1:02:27 I think earlier in the lecture you mentioned that the normal is not defined at the vertices. Then how do we move the vertices in the direction of the normal N_i? Or is N_i here the same as the N_i that we computed in the case of Length Variation curvature, i.e., the direction of the perpendicular bisector of the angle at \gamma_i. Apologies if I missed something obvious.
Your slides are pretty neat and intuitive !! Thank you sir for sharing this lecture series !!! How did you make these slides? LaTeX and TikZ? Just for curiosity...
I use Apple's Keynote; most equations are typeset in TeX via LaTeXiT! and imported as images. I don't use TikZ for images-you can find out more here: www.cs.cmu.edu/~kmcrane/faq.html#figures
Hi Keenan, I was wondering if the coding assignments across the course would also be available for us RUclips students? :) Thanks again for this invaluable contribution to making geometry processing research intuitive, exciting and accessible!
Yes, absolutely. Everything will be posted online at geometry.cs.cmu.edu/ddg as the course progresses. In fact, you can already find all the same material from last year's course at brickisland.net/DDGSpring2020/
When defining curvature using the angle, shouldn't you divide by length somehow? Couldn't you define length at a vertex as the the sum of the lengths from the vertex to the midpoint of adjacent edges. Mirroring what you do when using the Hodge star using the dual mesh.
At 31:50 why can we not simply say the curvature is the second derivative, that way you dont lose the directional info in the first place, what does the inner product with the norm do?
Check out this lecture on curves, which discusses the definition of the normal for space curves, and more generally the Frenet frame: ruclips.net/video/seLcPBax3OI/видео.html
If I'm interested in Differential geometry specifically for theoretical physics, you know, much more in the technical and rigorous maths side, would this course still be useful for me? As you may know, the more mathematical Differential geometry courses at CMU are rarely, if at all, offered anymore, and they require quite mathematically intensive prereqs. Obviously I understand this class probably won't be a substitute, but if I still plan on learning Diff Geo myself, would it help to take this class?
Yes, this would be a great intro if you haven't taken any course in differential geometry before. A lot of the motivation comes from algorithms and applications in geometry processing, but the core tools (and intuition) should serve you well for any further study. In particular, the course puts an emphasis on differential forms, which are fundamental in modern mathematical physics-a good companion book for the course if you want to go deeper is "Manifolds, Tensor Analysis, and Applications" by Abraham, Marsden, and Ratiu. I learned differential forms from Marsden, and his book (and others) served as inspiration for that part of the course.
@@keenancrane Very helpful reply! Thank you, I hope you continue offering this course in the coming years and I can take it with you. And even if my schedule doesn't work out, I'll be sure to go through these videos.
I'm crocheter and I'm watching in order to understand pattern design, esp for 3D non-symmetrical objects (think more crochet realism than amigurimi) 😂.
Yeah, also why in 43:19 we are talking about decrease of length of curve? Didn't we increase a length of the curve after transformation by eta? Is it related to choose normal as JT, and not -JT?
Hi Prof Crane, thanks for sharing the lecture videos! I had a clarification question on the example of circle as a parameterized curve at time @22:05. You expressed the constraint gamma(0) = gamma(2*pi), but the domain of gamma is [0,2*pi), so not defined on 2*pi.
Yes, that is true; I should be more careful here. The reason for not simply defining the map over the closed interval [0,2π] (or the whole real line) is that we will later use this example to understand the concept of homeomorphism, which captures the notion of the "topology" of a shape. Specifically, this example will make it clear that a continuous injective map does not always have a continuous inverse.
Sometimes it's called "calculus on manifolds" (in fact, this is the title of a classic differential geometry textbook by Michael Spivak). Basically how do you apply calculus on spaces that are topologically different from R^n. A basic tool is indeed to apply ordinary calculus in local coordinate charts on R^n. But this is just the means, rather than the end-differential geometry is all about discovering the amazing things that can happen on spaces beyond ordinary Euclidean R^n…
Наука
i am not your student, but interresting in this topic, because it is rare that someone with so much expertise, shares such well structured knowledge from ground up.
Why am i interested in this ? (i know you didn't ask 😂)
It is like you mentioned: you need it for almost everything now, engineering, AI, architecture, graphics and so on.
Thank's for sharing this content for free 👍 This is Science ! Awesome !
Thank you very much Prof. Keenan Crane for being generous to share the course content. Looking forward to more courses from you. Thank you!
This is the soul of Teachers. To pass on information in order for our society to advance in many areas ( in our case, it is Technology ) Thank you very much.
0:38 Why we might want to study geometry?
1:48 Applications of Discrete Differential Geometry
6:09 What will we Learn in This Class?
8:06 What won't we learn in this class?
9:38 Assignments
11:26 What is Differential Geometry?
13:17 What is Discrete Differential Geometry?
15:32 Grand Vision
16:31 How can we get there? (Game)
18:53 Example: Discrete Curvature of Plane Curves
19:55 Curvature of a Curve - Motivation
20:54 Curves in the Plane
21:45 Example
22:16 Discrete Curves in the Plane
23:17 Example
23:50 Tangent of a Curve
25:22 Example
26:27 Normal of a Curve
27:37 Example
29:29 Curvature of a Plane Curve
32:46 Curvature: From Smooth to Discrete
34:57 When is a Discrete Definition "Good"?
36:36 Playing the Game
37:46 Turning Angle
39:04 Integrated Curvature
41:15 Discrete Curvature (Turning Angle)
43:03 Length Variation
45:14 Gradient of Length for a Line Segment
46:46 Gradient of Length for a Discrete Curve
48:03 Discrete Curvature (Length Variation)
49:19 A Tale of Two Curvatures
50:52 Steiner Formula
51:56 Discrete Normal Offsets
54:24 Discrete Curvature (Steiner Formula)
55:51 Osculating Circle
56:51 Discrete Curvature (Osculating Circle)
57:27 A Tale of Four Curvatures
58:04 Pick the Right Tool for the Job!
58:55 Curvature Flow
59:53 Toy Example: Curve Shortening Flow
1:01:10 Discrete Curvature Flow - No Free Lunch
1:03:25 No Free Lunch - Other Examples
1:04:56 Course Roadmap
1:07:26 Applications & Hands-On Exercises
Thanks for the index!
thank you 😇
thanks
This is great. It's refreshing to see computation treated as a first class citizen instead of an afterthought like other math lectures. My cold programmer heart grew three sizes today, which I can compute with a discrete curve lengthening flow.
Amazing course, really well structured and more entertaining than Netflix.
This is a teaching masterpiece!
This is true education, and its meaning. Thank you so much!
Sometimes when I come across resources like this that are free of cost it truly makes me happy to live in today's day and age! How amazing is this?! Thank you
This was excellent. I really enjoy this “game”. In general in mathematics this sort of generalization in all directions is always such a thrill.
OMG this is gold. Thanks for this playlist.
Wow, absolutely high-end presentation.
Thank you!
I have been waiting for this. Thanks a lot for sharing!
These lectures are absolute Gold....Thank you Prof.
I have rarely seen such a perfect pedagogy on youtube, everything is perfect, the examples, the breaks to think... And on a fascinating subject! Keep it up!
I just stumbled on this course after reading one of your papers and i must say it's really awesome, thank you for sharing it!
Thank you for this, 10/10 ❤️. Looking forward to seeing more advanced topics!
I love your teaching style. Thank You
Very brief and deep course. this is art!
Thank you so much for recording/uploading the complete set of lectures!
Excellent class!! Thanks so much for sharing the knowledge so clearly.
Some courses are in huge demands but little in supplies. It has 15k views for 5 months since it was uploaded (a super impressive number for a pure academic course). Been looking everywhere last year for an introductory differential geometry courses for engineering, thank you for making this freely available online professor!
Wow, the idea that discrete differential geometry is the language of a new world really fascinates me... and the GAME of searching for different perspectives of translating between smooth and discrete geometry seems really interesting. Thank you, sir!
Thank you so much! I'd taken a classic differential geometry course in university, and this is great stuff! Really clearing up a lot about how to apply the things I learned in a computational setting!
Great presentation of a subject that I want to learn. Thanks for making them available!
Thanks a lot Professor Crane. I'm not your student and have no knowledge about differential geometry (continuous and discrete) but geometry of curves and surfaces is my absolute favourite topic to think about in mathematics. I'm totally looking forward to your videos. Thanks a lot again for sharing your expertise for free!!! :)
Hey Keenan, great video.
This reminds me of a simple interview question I sometimes ask: if you have a rope going around the equator, and you increase it by 1m, by how much can you raise the rope above the ground? Most people think it's going to be microscopic (because 1m is very little compared to the circumference of the earth), when in fact it's 1m/(2*pi). This relates to some of the concepts in your video.
When I try to explain why the intuition of people is wrong, I resort to imagining a 2D equivalent, a rope around the border of a circular lake, and then thinking of a rope around a square lake - and how much the rope needs to be extended if you want to 'lift' the rope, i.e. bring it a bit further away from the border of the lake. If you try that, you'll notice that the places where you need more rope are the corners, i.e. the places where there is curvature (as seen by the fact that the normal turns there). I always saw that as a 'Dirac' distribution of curvature, but your lecture explains how this can be seen in various different ways.
Looking forward to watching the rest (btw I got a PhD in diff. geometry 20y ago, but went to work in an applied field since).
I love how you are using illustrations to explain your maths. Graphics help me commit things to memory.
Thank you for making tutorials. I hope you will make more videos.
Just started my research on this topic and my advisor suggested one of your articles, came here by chance and I must say I'm not disappointed! Awesome :) thank you for the classes!
It was evident after listening for only a minute or two that you are a confident and experienced lecturer. It is rare to encounter a speaker who so deftly combines informality and polish. You've made a complex topic understandable and interesting. Thank you so much for sharing your expertise freely with everyone. You are a credit to the best traditions of scholarship. Subscribed!
Thank you so much for sharing this course! This is really helpful for students who not only require understandings from pure mathematics point of view but also from a more applied prospective, for building intuitions.
Amazing course, thank you!
Good professor and great content!
Thanks Keenan. The course has been extremely helpful..
Really nice and intuitive explaination
I'm starting this curse a little late, but thank you so much for uploading all the material. This looks really interesting.
Thank you for sharing this great stuff !!
It's a very interesting lecture!
Thank you for sharing these valuable lecture series!
Amazing videos, I am learning so many useful concepts and deep understandings towards to advanced topology and algebra!
@keenan crane
The inner product does not only preserve the sign. We also want to extract the normal component of the derivative so that we can negate any change in the speed at which we are traveling along the curve, of course this is 0 in the case of arc length parametrized curve.I really like the lectures so far. Thank you for sharing them with the world and putting the effort in!
thank you!
I like your lectures!
great course, thx for sharing
Finally here!
Thank you so much Prof. Keenan Crane for generously sharing your knowledge and wisdom to help people gain knowledge and wisdom to improve humanity all together. You and many other teachers who generously shared your wisdom and knowledge inspired me to share my knowledge and wisdom to people in the future when I become an expert in some areas. Thank you so much again!
Thank you for sharing those lectures.
Excellent lecture. Thank you sir.
THANKS A LOT FOR EXCELLENT VIDEO!!
Thank you very much
Thank you so much!
38:35 shouldn't curvature be given by a second derivative? I think there is a derivative missing there
I feel extremely privileged to have access to content of this quality for free. Thank you
Correction: @38:12 - It should be the second derivate of the curve, not the first derivative in the Curvature formula.
fabulous! I was tirsty for this subject in my college curriculum
Yesss TYSM!
I find it interesting that as theta approaches zero (meaning we approach a smooth function), theta = 2 sin(theta/2) = 2 tan(theta/2). At 50:08 Dr. Crane mentioned the small angle approximation for sin(x) but did not mention it holds for tan(x) as well. As x->0, sin(x)->x and cos(x) -> 1 so tan(x) -> x/1.
thank you
Thanks for sharing the class materials! I'm really excited to get started here.
Could you share what tool you used to generate the beautiful purple and black cell-shaded plots? They are wonderful!
thanks for this. Im interested in topological data analysis, and algebraic topology. This course is a gem. Anyone wants to create a discord group to discuss ideas and assignments?
Discrete diffrential dream geometry.
Hi Prof. Crane, thank you for the amazing lectures! I had a question about minimizing discrete curvature flow at 1:02:27 I think earlier in the lecture you mentioned that the normal is not defined at the vertices. Then how do we move the vertices in the direction of the normal N_i? Or is N_i here the same as the N_i that we computed in the case of Length Variation curvature, i.e., the direction of the perpendicular bisector of the angle at \gamma_i. Apologies if I missed something obvious.
Your slides are pretty neat and intuitive !! Thank you sir for sharing this lecture series !!!
How did you make these slides? LaTeX and TikZ? Just for curiosity...
I use Apple's Keynote; most equations are typeset in TeX via LaTeXiT! and imported as images. I don't use TikZ for images-you can find out more here: www.cs.cmu.edu/~kmcrane/faq.html#figures
Hello Dr. Crane, what are your thoughts on geometric algebra and geometric calculus and its relationship with differential geometry?
Hey Prof. I was wondering what are you using to produce your images? Thank you for the lectures!
Hi Keenan,
I was wondering if the coding assignments across the course would also be available for us RUclips students? :)
Thanks again for this invaluable contribution to making geometry processing research intuitive, exciting and accessible!
Yes, absolutely. Everything will be posted online at geometry.cs.cmu.edu/ddg as the course progresses.
In fact, you can already find all the same material from last year's course at brickisland.net/DDGSpring2020/
I want to do coding in this subject. Can you show.... How to put algorithms into the programming
When deriving the length variation formula. Aren't you using the half of the integral of the norm square of the derivative (i.e "elastic energy")?
Sorry couldn't find your discord server. And I am stock in some problems . Is the discord server down?
where can i find the derivation of Discrete curvature (osculating circle) k = 2sin(theta)/w
When defining curvature using the angle, shouldn't you divide by length somehow? Couldn't you define length at a vertex as the the sum of the lengths from the vertex to the midpoint of adjacent edges. Mirroring what you do when using the Hodge star using the dual mesh.
At 31:50 why can we not simply say the curvature is the second derivative, that way you dont lose the directional info in the first place, what does the inner product with the norm do?
Quite good. It is very useful. (Does anyone research using Discrete Element Method? How can DDG be applied in that?)
Isn't the norm of the tangent at 25:58 supposed to be in square root?
26:55 how do you determine the normal direction in 3d? unlike in 2d, you don't have a fixed rotation axis in 3d.
Check out this lecture on curves, which discusses the definition of the normal for space curves, and more generally the Frenet frame: ruclips.net/video/seLcPBax3OI/видео.html
If I'm interested in Differential geometry specifically for theoretical physics, you know, much more in the technical and rigorous maths side, would this course still be useful for me? As you may know, the more mathematical Differential geometry courses at CMU are rarely, if at all, offered anymore, and they require quite mathematically intensive prereqs. Obviously I understand this class probably won't be a substitute, but if I still plan on learning Diff Geo myself, would it help to take this class?
Yes, this would be a great intro if you haven't taken any course in differential geometry before. A lot of the motivation comes from algorithms and applications in geometry processing, but the core tools (and intuition) should serve you well for any further study. In particular, the course puts an emphasis on differential forms, which are fundamental in modern mathematical physics-a good companion book for the course if you want to go deeper is "Manifolds, Tensor Analysis, and Applications" by Abraham, Marsden, and Ratiu. I learned differential forms from Marsden, and his book (and others) served as inspiration for that part of the course.
@@keenancrane Very helpful reply! Thank you, I hope you continue offering this course in the coming years and I can take it with you. And even if my schedule doesn't work out, I'll be sure to go through these videos.
I'm crocheter and I'm watching in order to understand pattern design, esp for 3D non-symmetrical objects (think more crochet realism than amigurimi) 😂.
can I have the name of the book
at 54:21 how is the new length is smaller than the old length, shouldnt it be bigger
Yeah, also why in 43:19 we are talking about decrease of length of curve? Didn't we increase a length of the curve after transformation by eta? Is it related to choose normal as JT, and not -JT?
May Allah give you Hidaya. Aameen!
Hi Prof Crane, thanks for sharing the lecture videos! I had a clarification question on the example of circle as a parameterized curve at time @22:05. You expressed the constraint gamma(0) = gamma(2*pi), but the domain of gamma is [0,2*pi), so not defined on 2*pi.
Yes, that is true; I should be more careful here. The reason for not simply defining the map over the closed interval [0,2π] (or the whole real line) is that we will later use this example to understand the concept of homeomorphism, which captures the notion of the "topology" of a shape. Specifically, this example will make it clear that a continuous injective map does not always have a continuous inverse.
👍🏼
wooooooooooow
Why "discrete" in the name???
5:36
Думаю автор удивится, откуда тут столько русскоязычных. Так вот: мы от New Deal
Have you looked at John Gabriel's new calculus? It may help here.
that is quite literally crankery, buddy
Так! Я не понял. А почему не по по-русски?
Why call it "differential geometry" when in fact it's nothing but plain calculus?
Sometimes it's called "calculus on manifolds" (in fact, this is the title of a classic differential geometry textbook by Michael Spivak). Basically how do you apply calculus on spaces that are topologically different from R^n. A basic tool is indeed to apply ordinary calculus in local coordinate charts on R^n. But this is just the means, rather than the end-differential geometry is all about discovering the amazing things that can happen on spaces beyond ordinary Euclidean R^n…