I just wanted to say; keep doing what you're doing because I love your videos! I hope one day many people discover this channel. I shared one of your videos to a friend and it helped them a lot too. The animation is beautiful and I love how you explain difficult concepts with ease :D
@leventeabraham6906 Wow, thank you so much for your nice words, Levente. You have no idea how much it means to us! We will definitely keep going and increasing the quality even more of the videos, as well as the quantity of videos we publish per week. Please, let us know how we can help you, i.e. what kind of content would be useful to you 😎
@@dibeos I have all my respect for you two making these visual videos! In my free time I like to learn new math topics and this channel is the motivation. I think a video on Laplace transform would be interesting. I am an undergraduate mechatronics engineer and we use this all the time for system controls and solving differential equations. I think it would help others engineers too 😊
@@Prof_Michael yes, we thought about putting “differential geometry” in the title, but “Gaussian curvature” seemed more appropriate. Anyway, this week we will publish another one going deeper on differential geometry
@@dibeos very good… I’m also working on starting my channel and solving Mathematics related problems from Calculus I & II (because I Love Sir Isaac Newton so much, hopefully you’ll make a video devoted to Newton), Laplace Transform, Fourier Transform, Z Transform, Matrix Algebra, Probability Theories like Bivariate Normal distributions etc..
@@daniel_77. I remember watching some video of Andrew university where the question was answered. It was one of many in a playlist about the local theory of space curves (or something like that? I don't remember exactly) I'll reply again with more details.
I believe there is a theorem which states that the curvature and torsion of a curve will determine a curve up to translation and direction. For a plane curve, the torsion is 0. So all curves with a fixed constant curvature are the same (up to translation and direction traversed), i.e., they are all circles with radius 1/k.
Channel name: Math at Andrew university. Playlist: differential geometry I couldn't find which video it was, sorry 😅 . Your question is also related to a bigger question: can we uniquely determine the shape of a curve just from its curvature and torsion? And the answer is yes! This is called the fundamental theorem of space curves.
It would be nice if you guys motivated your content. Might be a bit less jumping into it but then I always learn better when I understand why I need a solution/method then just how to go about it
I'm wondering how do intrinsic curvatures add up in 3-D? Let's say we have a transformation or process (F1) that generates curvature k1 at point P in 3-D. Another process F2 would generate k2 at the same point. Now, what is the curvature K at P if we apply both transformation in succession? What is curvature at P after F1(F2(P))? Do they commute? F1(F2(P)) = F2(F1(P)) ? Is K = k1 + k2 ?
@@tonibat59 F2 output a curvature and then you put it inside F1 ? isn't the point P the thing that you're supposed to put in F1 and F2? Composing two transformations only make sense if the output space of the one you apply 1st is a subset of the input space of the one you apply 2nd.
@@UA-cam_username_not_found F2 is an operation (transformation) that takes a 3D space and transforms it. When applied to flat space, F2 generates curv k2. What curvature do you get when you apply F2 to a curved space with curvature k1?
@@tonibat59 In your comment you said F1 and F2 outputs the curvature, that's why I didn't get your idea. Thanks that you clarified your thoughts. As far as I can see, the question can't be answered because you only know that F1 transforms a flat region into a region with curvature K1. You don't know how it will transform an arbitrary region with curvature K or hoe much the new curvature is. I am not an expert on differential geometry so I can't say for certain that the given information is insufficient.
@@UA-cam_username_not_found Thanks for trying. I am also unsure if the question is well posed and has an answer, but there's a practical application for the topic of gravitational potential that looks in some way related.
@@tonibat59 Your question is definitely interesting, for its own right and also for its possible applications. I suppose you're talking about general relativity when you mention gravitational potential. I hope you find the answer to your question someday.
@gewinnste “Loci” is the Latin plural of “locus”. I just looked up and the pronunciation varies, but in English, it is usually pronounced as “low-sai” (/ˈloʊsaɪ/). So, yeah we pronounced it wrong in the video. Sorry.
@@ianfowler9340 I know (I had 5 years of Latin in school), it was the old, now recognized as wrong, way - but when these Latin words found their way into the English language, the old ways were still taught, so most of the time they're still pronounced with a soft c (and considered correct, despite actually being wrong), e.g. "acid", "celestial", "tacit" etc.
@@gewinnste Thanks. I didn't think of other words like tacit. I guess the rule, in English, is to soften the c if a vowel is the next letter. I always pronounced foci with a hard c - too old school to change my ways. lol. Anyway, I appreciate the info.
@@lordlouckster2315Pronunciations of English words are not always the same as their Latin ancestors. Loci is pronounced “LO-sigh” according to multiple English dictionaries and I don’t see any source saying it’s pronounce “LO-chee” or “LAW-chee” or “LO-kai”.
I just wanted to say; keep doing what you're doing because I love your videos! I hope one day many people discover this channel. I shared one of your videos to a friend and it helped them a lot too. The animation is beautiful and I love how you explain difficult concepts with ease :D
@leventeabraham6906 Wow, thank you so much for your nice words, Levente. You have no idea how much it means to us! We will definitely keep going and increasing the quality even more of the videos, as well as the quantity of videos we publish per week. Please, let us know how we can help you, i.e. what kind of content would be useful to you 😎
@@dibeos I have all my respect for you two making these visual videos! In my free time I like to learn new math topics and this channel is the motivation. I think a video on Laplace transform would be interesting. I am an undergraduate mechatronics engineer and we use this all the time for system controls and solving differential equations. I think it would help others engineers too 😊
Thanks for the explanation.
I like the transparency that you shared the resources used for making this video.
You, such a great video, another level each time!)
@@SobTim-eu3xu thanks for the amazing comment of encouragement (as always!!!) 😎
@@dibeos yea, bc you do such a good videos
(as always!!!) 😎
Congratulations, this video was really good, great content!
@@Luan-bs1vp obrigado pelas palavras de encorajamento, Luan! 😎
Good stuff!
This is Differential Geometry on a beautiful Visualisation
@@Prof_Michael yes, we thought about putting “differential geometry” in the title, but “Gaussian curvature” seemed more appropriate. Anyway, this week we will publish another one going deeper on differential geometry
@@dibeos very good…
I’m also working on starting my channel and solving Mathematics related problems from Calculus I & II (because I Love Sir Isaac Newton so much, hopefully you’ll make a video devoted to Newton), Laplace Transform, Fourier Transform, Z Transform, Matrix Algebra, Probability Theories like Bivariate Normal distributions etc..
@@Prof_Michael woooow, i love these subjects as well, really. Please let me know when you post videos about them, I want to watch
Awesome!
I wonder:is the circle the only object with constant curvature at every single point?
There's the reverse circle which has negative curvature at every point. It looks like a cone.
@@daniel_77.
I remember watching some video of Andrew university where the question was answered. It was one of many in a playlist about the local theory of space curves (or something like that? I don't remember exactly)
I'll reply again with more details.
@@adiaphoros6842 Don't you mean the inverse sphere?
I believe there is a theorem which states that the curvature and torsion of a curve will determine a curve up to translation and direction. For a plane curve, the torsion is 0. So all curves with a fixed constant curvature are the same (up to translation and direction traversed), i.e., they are all circles with radius 1/k.
Channel name: Math at Andrew university.
Playlist: differential geometry
I couldn't find which video it was, sorry 😅 .
Your question is also related to a bigger question: can we uniquely determine the shape of a curve just from its curvature and torsion?
And the answer is yes! This is called the fundamental theorem of space curves.
It would be nice if you guys motivated your content. Might be a bit less jumping into it but then I always learn better when I understand why I need a solution/method then just how to go about it
I'm wondering how do intrinsic curvatures add up in 3-D?
Let's say we have a transformation or process (F1) that generates curvature k1 at point P in 3-D. Another process F2 would generate k2 at the same point. Now, what is the curvature K at P if we apply both transformation in succession?
What is curvature at P after F1(F2(P))?
Do they commute?
F1(F2(P)) = F2(F1(P)) ?
Is K = k1 + k2 ?
@@tonibat59
F2 output a curvature and then you put it inside F1 ? isn't the point P the thing that you're supposed to put in F1 and F2?
Composing two transformations only make sense if the output space of the one you apply 1st is a subset of the input space of the one you apply 2nd.
@@UA-cam_username_not_found F2 is an operation (transformation) that takes a 3D space and transforms it. When applied to flat space, F2 generates curv k2.
What curvature do you get when you apply F2 to a curved space with curvature k1?
@@tonibat59 In your comment you said F1 and F2 outputs the curvature, that's why I didn't get your idea. Thanks that you clarified your thoughts.
As far as I can see, the question can't be answered because you only know that F1 transforms a flat region into a region with curvature K1. You don't know how it will transform an arbitrary region with curvature K or hoe much the new curvature is.
I am not an expert on differential geometry so I can't say for certain that the given information is insufficient.
@@UA-cam_username_not_found Thanks for trying. I am also unsure if the question is well posed and has an answer, but there's a practical application for the topic of gravitational potential that looks in some way related.
@@tonibat59 Your question is definitely interesting, for its own right and also for its possible applications. I suppose you're talking about general relativity when you mention gravitational potential.
I hope you find the answer to your question someday.
The link to Garman & Bonnie's paper is broken. Please fix.
Fixed it, their link was broken but we found another one 👍🏻
@@dibeos Thanks, prima!
@tonibat59 prego ;)
how does yt read my mind, i was literally just thinking of this
YT didn’t. Sofia and I read your mind! 👀
Why are you using the plural of 'locus'? Plus, Isn't it pronounced "lossaaii" instead of 'lotchie"?
@gewinnste “Loci” is the Latin plural of “locus”. I just looked up and the pronunciation varies, but in English, it is usually pronounced as “low-sai” (/ˈloʊsaɪ/). So, yeah we pronounced it wrong in the video. Sorry.
@@dibeos No worries :)
A hard c is also acceptable. There is no soft c in classical Latin.
@@ianfowler9340 I know (I had 5 years of Latin in school), it was the old, now recognized as wrong, way - but when these Latin words found their way into the English language, the old ways were still taught, so most of the time they're still pronounced with a soft c (and considered correct, despite actually being wrong), e.g. "acid", "celestial", "tacit" etc.
@@gewinnste Thanks. I didn't think of other words like tacit. I guess the rule, in English, is to soften the c if a vowel is the next letter. I always pronounced foci with a hard c - too old school to change my ways. lol. Anyway, I appreciate the info.
The word "loci" is pronounced as LOH-sigh
A hard c is also acceptable. In classical Latin there is no soft c.
Modern Italianate Ecclesiastical: 'lo.tʃi
Or lockai.
@@lordlouckster2315Pronunciations of English words are not always the same as their Latin ancestors. Loci is pronounced “LO-sigh” according to multiple English dictionaries and I don’t see any source saying it’s pronounce “LO-chee” or “LAW-chee” or “LO-kai”.
COol
@@jammasound please, let us know what kind of content you’d like to see in the channel 😎
@@dibeos geometry, algebra, calculus, probability
@@jammasound straight to the point! 👌🏻thanks! 😎
Attributing the evolute to Huygens?!?!? Nah. Not even watching the rest, clearly not researched and not understood.
@@tinkeringtim7999 please enlighten us