Putting Algebraic Curves in Perspective
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- čas přidán 9. 12. 2019
- Ever wonder what happens when you combine graphing algebraic curves with drawing in perspective? The result uncovers some beautiful relationships between seemingly different shapes, and all because of what happens when you include infinity through projective geometry.
This video was a project for MA 721 - Projective Geometry, as part of the Master of Science program in Mathematics at Emporia State University.
Special thanks to Kevin Turner for assisting with post-production!
References:
* Ash, Avner, and Robert Gross. Elliptic Tales: Curves, Counting, and Number Theory. Princeton, NJ: Princeton University Press, 2014.
* Brady, Zarathustra Elezar. “Cross Ratios.” MIT Mathematics. Accessed November 24, 2019. math.mit.edu/~notzeb/cross.pdf.
* Coxeter, H. S. M. Projective Geometry. New York: Springer, 2003.
“Cross Ratio.” from Wolfram MathWorld. Accessed November 24, 2019. mathworld.wolfram.com/CrossRat....
* Hisel, Jordan. “Addition Law on Elliptic Curves." 2014.
* Leykin, Anton. “Systems of Polynomial Equations.” Lecture notes from MATH 4803: Introduction to Algebraic Computation. Accessed November 24, 2019. people.math.gatech.edu/~aleyki....
* “Projectively Extended Real Numbers.” from Wolfram MathWorld. Accessed November 24, 2019. mathworld.wolfram.com/Projecti....
Image credits:
* Albert Durer - Public Domain
* Charles Rex Arbogast/AP, CC BY 2.0, www.flickr.com/photos/2392540...
* Claudio Rocchini - Own work, CC BY-SA 3.0, commons.wikimedia.org/wiki/Fi...
* Hans Vredeman de Vries - Public Domain
* www.cashadvance6online.com/the...
* Mikael Hvidtfeldt Christensen - Own work, CC BY 2.0, www.flickr.com/photos/syntopi...
* new 1lluminati - Own work, CC BY 2.0, www.flickr.com/photos/6719472...
* Theon - Own work, CC BY-SA 3.0, commons.wikimedia.org/w/index...
Music: DM Ashura vs. Enoch - Chaotic White
This is what the Internet is for.
Well, this and sending missile launch codes between silos.
@@williamchamberlain2263 the main task of the internet was to see cats from anywhere on the world
@@williamchamberlain2263 oh wow, this video got recommended to everyone I guess🌟
What about cat memes 😔
@@williamchamberlain2263 boring
"wraps around at infinity", that BLEW MY MIND, and I think nothing in my life will ever blow my mind as much as that again.
i've accepted that the number line loops around as a fact for years now
@@vindi167 real question is , does that interfere logically with any other results ? if its not inconsistent i dont see a reason it is not to be acceptable
Chill
@@whannabi cool*
@@vindi167 Same, since I was in middle school I got convinced myself that the numerical plane is actually a sphere
As a student of arithmetic geometry, this is one of the best videos on algebraic geometry on YT, especially as an introduction. This is criminally underrated.
"criminally underrated." By whom please??
@@azzteke No need to be pedantic, he clearly means this should have more views.
I hear prison food is pretty good...
So no "Like" from me! LOL
I've heard the phrase "Point at Infinity" so many times before in math, but this video made me finally understand what exactly it meant
I remember combing CZcams for videos on Projective Geometry a few months back and wishing there was a good introductory single video. Now there is a great one! 10/10.
Yeah the same thing happened to me!! Finally got a clear grasp on homogenous coordinates :D
@AP.17 It means I was watching many videos and trying to find a good one. The use of "combing" refers to running a comb through hair to try to find something within.
17:03 I'm surprised you didn't mention the best part!
In computer graphics when you create a function which projects 3D space down to a plane, you divide by the Z component of the camera's vision. You never want Z to be negative, however if you allow that to happen anyways (i.e. not clipping the world behind you). Everything that isn't normally visible actually shows up ABOVE the horizon, and flipped 180°. For the case of the hyperbola, this means the rest of the ellipse image actually continues perfectly as expected, which is awesome! :)
I had created a Desmos graph last year which demonstrated exactly that, unfortunately youtube has a field day when links are posted in the comments so I cant share right now, oh well
Hey i'm curious to see the video, do you think you could upload it to your channel or something like that?
@@Bankosek I added the link to my channel description. Have fun!
@@NonTwinBrothers Thanks a lot!
(this is the case in 3D) You usually divide by W, the homogenous coordinate, into "NDC coordinates" where Z is then used to write to the depth buffer. Depth testing wouldn't work if you were to divide by Z. Check out the full parameterization of a projection matrix for the rest of the info on how it works.
You can break up the link with spaces to still post it
I do want to look at it
(Never mind just saw your channel but in case of future cases you can try breaking the link up
Btw the desmos graph is AWESOME
A parabola stretched to infinity being an ellipse is so cool to me, because in my dynamics class we have been studying orbits, and they have four shapes: circle, ellipse, parabola, and hyperbola, in different perspectives, these are all ellipses!
Honestly, ellipses are the key to the future of science and mathematics, just as circles and triangles have been for millenia.
there is a great reason for learning conic sections in pre-calculus, applies to way more than people would think.
@@lookupverazhou8599 Please, can you elaborate more about ?
Your videos inspired me to pursue higher mathematics back in 2016. I just finished my MSc degree in math. I thank you from the bottom of my heart.
As an Art teacher, I taught this to my students, except I humanized it by usi g the 60° Cone of Vision, to find the intersections in the first instance. Did it work? Surprisingly well!
Robert.
This was during the 1980's.
As an art student I know what the cone of vision is but what is the first instance?
How pretty... Done my thesis on projective geometry also and this guy has made an incredible good and easy to understand explanation of this beautiful field of mathematics
projective algebra is the coolest thing ive ever seen
I watched your introductory series on maths for the first time 5 years ago and I recently just graduated, I can say your videos gave me the insight I needed to get here in the first place, thank you for dedication, I'm especially glad to see a new video coming out!
One of the best explanations I've ever seen for Algebraic Geometry and the other branches touched in this video. Congrats!
Outstanding. I remember loving projective geometry when a student eons ago but this is several levels above and even more awesome. Well done.
I remember reading a book on analytic conics and hearing you talk about the concepts I read in that book really helps them click. Thanks a lot!
This is the best video on that topic so far, your visualizations are extremely helpful
Last quarter of this video was legitimately mind-blowing. Thanks for inspiring me and no doubt many others!
This video is severely underrated. Incredibly well done explanation!
I can't believe it took me this long to stumble on this video. I was learning how to draw and found perspective very interesting. Projective Geometry was exactly what I was looking for.
Great video. Most probably the best I ve watched in projective geometry!
It really let me... see things from a new angle!
This is a ridiculously good video. Just crazy.
This video is gold! Having studied this stuff some years ago at uni, I was able to recover all the lost knowledge in just 20 minutes!
I took a course on projective geometry, but we never made it as far as homogeneous coordinates and there wasn't a lot of perspective (pun intended) on how to view these things or how it all comes together. It really was one of my favourite geometry courses (we covered inversive geometry as well), but this video really helped to put a neat little bow on it. Thanks so much, and please keep up the great work!
It's impossible to do projective geometry without homogenous coordinates. You cannot do computations.
@@MultiAndAnd actually there is a lot you can do with just the cross ratio. This course was proof-oriented rather than computation.
@@camrouxbg basically on on the prjojective line then... Not that interesting in my opinion.
@Andrea Merlo suit yourself, but you're definitely missing some of the beautiful stuff. But hey, if it's beneath you then who am I to say otherwise.
@@camrouxbg projective geometry is very basic stuff and not such a deep idea. Imho is not that interesting. Especially if you limit yourself to biratios. It’s mathematics for the elementary school.
Beautiful.
MORE!!
Very interesting. I like this "math for artists" stuff a lot. Thank you!
Captivating video! I'm a PhD student in math finance which is a heavily applied field, but I found this deeply interesting!
The Fermat’s Last Theorem cameo hit me like a ton of bricks
wow! that's so cool! this is the most intuitive and "clean" way to deal with infinity as a number I've ever seen!
Loved your videos a few years back. I am now looking to do a PhD in pure mathematics.
Thanks for the videos!
Wow, is this some kind of apocryphal and forbidden part of mathematics? It somehow reminds me of what our teacher told us about x and y axes that those are not lines but circles with an infinity radius. No one from 150 attending students cared about that fact except me with my friend who just laughed about it like it was joke. Teacher didn't explain it further maybe because we were mere future engineers.
Wonderful animations, thank you! Helpful for understanding elliptic curves
Your enunciation is very clear and easy to focus on, keep my attention. Subscribed!
This video has made so many things so much clearer to me.
It is nice to see you are posting videos again. I’m an alumni from your GHP game theory class. i always enjoyed your teachings. :)
What a perfect introduction for this subject! Thank you.
Please continue this you are real mathemacian and math teacher
this is so good, I've always wondered how logarithmic scales could be constructed, and now I have my answer
I have been thinking about mathematics, specifically graphs, just like this (primarily: "There is only one unsigned infinity") all through my school life. Now, with everything put together, really hit a sweet spot...
That was super cool! I can't believe I was able to follow along with like 95% of that, math is only a hobby for me. I managed to explain the complex plane to my brother yesterday, and the mandelbrot set. I was so happy
This is amazing. I had no idea such interesting mathematics existed
I find Bill's explanations quite soothing. Great jpb!
Great to see your back
Awesome presentation! The link with art, love this part. Thanks
What a great exposition! Well done!
Bill is back! Welcome back mr Shillito!
Mind opening! Well done 🙏
Absolutely amazing video. Good job! 👍🏻
Thanks for this beautiful video!!!
Very nice !! especially the choice of the order on witch concepts are intruduced! The principles notion and subtilities are presented with clarity , pedagogy in a rigourous way. The beginer might have to use the "pause" bouton quite a lot, in order to get the worth of this lesson, but this will be a great benefit because this is not vulagarisation but real maths😍
Estoy sorprendido. . .gracias por su magnífico aporte en la transición del espacio proyectivos relajado al espacio euclidiano. . .
hats off to this marvellous video
the great return after few years
excellent production, introduction, deduction, induction, tiontion
I rarely write comments but it hit me well!
Back in school I finished art classes and perspective was always something intuitive but I tried to describe it mathematically. I ended up with massive formulas for even simple things and now it turns out mathematicians created a more convenient language for that.
It would be great if you reveal RP3 (which as I understand represents how we see the world in 3D). For example, imagine you have a cube in 3D perspective. How to find the coordinates of an inscribed sphere? In usual geometry we just say that the sphere intersects the cube at middlepoints of cube surfaces, however in perspective it is not the case.
Anyway, thank you!
dayum im not even in high school but you explained this in such a way that it piques my interest AND is able to make me understand most of it
Some of my favorite music from my child hood was created by a math teacher...Astounded.
Nice presentation, clear and well done. Love to see what other people do with math. I use these for computer vision and graphics
This was strange. Just 20 minutes ago, I was imagining making a video where a curve that looks like a parabola up close is actually an ellipse when you zoom out. And then I get recommended this!
imagining a point that is infinitely far away from everything is already hard but now i also gotta imagine how it wraps around to the other side
I always thought about how to properly prove the foci of a parabola (for orbital mechanic purposes). I know the equation, it is simple, but I just couldn't get my head around the "other foci being at the infinite point". This video has showed me how that works and I feel GREAT now. Thanks!
The fact that Parabolas are just ellipses stretched to infinity may not be as a surprise if we remember about the conic sections.
Circles are when the plane cutting it is paralel to the base. Ellipses are when they are oblique, but not parallel to the side, and parabolas are when the plane is parallel to the side of the cone, so it is the first ellipse that "couldn't find" the other plane to close on itself.
Hyperbolas are when the plane is orthogonal to the base.
I didn't know hyperbolas were parabolas in projective geommetry, but it makes sense, since they are orthogonal, they would only be a parabola in the extreme case where the angle of the cone is orthogonal, but in that case it wouldn't be a cone, but a cylinder(in which, in projective geommetry, it would be a cone at infinity). Pretty cool!
Oh wow oh wow oh wow! I had given up hope that you were going to do more videos. So glad to see that I'm wrong. You have such a clear way of explaining things.
Thank you, that means a lot! I really have been wanting to get back into making these videos, but they take quite a while to produce. I need to come up with a better process...
@@BillShillito The polished graphics and animations are lovely, but in my opinion what's special about your videos is that you present things so clearly. I'd happily watch videos of you lecturing at a whiteboard, especially if it would mean we could get more videos. (N J Wildberger has been doing exactly that for years, and his videos are quite popular.) But that's just my suggestion. If having a high production quality makes the process more rewarding for *you*, I'll be patient. At least, I'll try to be patient. ;^)
@@amydebuitleir Now, how did you like the infinity talk of mr Norman Wildberger? Personally I think mr Wildberger is spreading false teachings confusing especially the minds of children. So it would be cool if there would be a field of mathematics to prove him wrong, especially for the sake of these children. The problem with mr Wildberger is in my opinion that he appears to me like a modern mr Pythagoras who did not believe in irrational numbers like the square root of 2, or infinity all by itself.. He also claims that the rules of arithmetic break down with big numbers so you can not be certain about anything over there, yet he is clever enough to hide just beyond the reach of the modern calculator, so the kids have no way to prove him wrong with their calculators. Yet what's even more freaky that there is a whole bunch of people that agree with him and they call themselves "purists", just like with the Pythagorean sect... And so you can discuss infinity or the sqare root of 2 endlessly over there... While no concensus whatsoever ever comes out of this. At one time I thought I was going to write him a letter, but I never did, because I was getting this impression that mr Wildberger is not confused nor mentally sick at all, but that he is doing all this on purpose to have a false scientific island for himself where he can enjoy infinite glory while he is hiding with his examples just out of reach of a hand calculator, confusing children. In my opinion. I filed a complaint about him at the University in Australia that he was affiliated with, but no one responded. And so I have not listened to mr Wildberger in years, but now we have seen infinity in this lecture, maybe we can use this to make mr Wildberger stop confusing children, if that is what he is still doing. Still, for me, the most freaky part is all these followers of mr Wildberger, who actually appear to agree with him, maybe for their own glory, but as you may have guessed this already, I am not one of those. So Ok, putting infinity into perspective that would really be something that needs to be done over there in Australia, that is, if this in my opinion, craziness of mr Wildberger is still going on, especially when it confuses children with examples just beyond the reach of their calculators....
@@OndrejPopp I have only watched a few dozen of his videos, so I may not have seen the same ones you did. I was initially a bit concerned by his opinions on infinity and irrational numbers. However, in the videos I watched he made it very clear whenever his view differed from the consensus, and why, so I personally didn't feel misled. I only watched the videos on more advanced topics such as algebraic topology, and I felt that students at that level can benefit from considering alternative views as long as they are labelled as such. His absolute insistence on rigor and clarity leads him to reject real numbers, but those same qualities make him good at explaining complex ideas. I didn't watch any of his videos for children, so I can't comment on how he presents his ideas to them.
@@amydebuitleir Hi Amy. There is no problem expressing alternative views, as long as they are valid ofcourse... Because mathematics is not a religion or is it? And that's kind of an issue here. My concern about the children is that some of mr Norman Wildberger's followers are school teachers, so teachers who admire his ideas, and so mr Wildberger's alternative views may creep into the heads of children in this way. Anyway it is an endless discussion, but the best indication I got is that mr Wildberger's examples are just a little bit out of reach for a calculator... I don't know if you ever saw that one, this pyramid number 10 to the power of 10 to the power of 10 ... to the power of 10 and so on, and apparently so claims mr Wildberger, these pyramid numbers are so big that normal rules of arithmetic do not apply.. And you can not calculate them either because they do not fit into a calculator and some school teachers love this.... and are discussing the possibilities how to introduce this in schools and to the kids...
This is awesome!
this came up on my recommended a while ago, and i didn't even know you were the musician that made deltaMAX; love your works
Very cool! Mindblowing!
I'll admit, I struggled to follow a lot of this, but the reveal at 17:12 was so satisfying!
Nice, thanks. Fascinating.
Very nice. Thanks!!!
Awesome Job!
Fantastic video.
Great video!
This video is awesome.
18:33 hahahahah
also omg u made the music at the beginning love this so much
Many thanks!
Woah, this video is really good, I like this guy, subcribed
Excellent video thanks
very well explained!
So beautiful!
This is so cool!
This is the first time I've see someone use the set up I use in my work. Though I manly used it for art, it was really to understand the nature of distortion. Have also considered the horizon as infinity, but I also used localized infinity in light projections, and if you use this to generate a copy of the whole picture, in the picture in perspective in perspective, you can find infinity with the new finite infinity perseved in the images image of its self in it'd self. Also, was looking into how to transfer the governing lines of infinity to localized light infinity without disturbing an object, hard to explain what I mean by that part.
This is so so satisfying to generalize FTA even futher!
Never heard of this stuff before, but its really cool
Its 3 am and my brain is truning into a fine mushy paste as the cycle of trying to comperhend the contents of the video and failing repeats itself every second
I don't understand a thing of what this video said. But it reminds me of when I heard that anything is possible at infinity.
Thank you very much!!!!!
Mind blown!
I have absolutely no clue what is happening, and I am here for it
Great video.
Excellent.
Great video
Excellent!
Holy shit I'm blown away. That's nuts.
Rock on! Let's get this guy more views!
This was better than most movies I watched
I would not mind more videos on this.
Mind blown.
Awesome.
god this is fantastic
how have I not see this before, this is an amazing piece, TIL the fundamental theorem of algebra is only a special case of bezout's theorem 😲it's made so obvious here
thank you 🙂