Putting Algebraic Curves in Perspective

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.
* Coxeter, H. S. M. Projective Geometry. New York: Springer, 2003.
* Hisel, Jordan. “Addition Law on Elliptic Curves." 2014.

Image credits:

* Albert Durer – Public Domain
* Hans Vredeman de Vries – Public Domain

Music: DM Ashura vs. Enoch – Chaotic White

[kidredglow]

"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.

[theflaggeddragon]

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.

[auroraaustralis]

you can tell he did this all in one take by the breath, what a legend

[thezipcreator]

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

[bwn]

I remember combing Youtube 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.

[NoNTrvaL]

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.

[NonTwinBrothers]

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

[DankePrime]

Bro, math is so much cooler than people give it credit for 😢

[jojojorisjhjosef]

This man makes me realize I had an intuition for infinity.

[julianlastname]

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!

[antonseoane]

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

[Waffle_]

projective algebra is the coolest thing ive ever seen

[reidflemingworldstoughestm]

New and exciting ways to confuse flat earthers.

[incommunicado_o]

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!

[robertmatthews]

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.

[zanityedpo]

I always thought 0÷0 should be ANYTHING other than nothing, but my math teachers always told me, "You just can't divide by 0, shut up." So later in the year, when she brought in a REAL mathematician, SHE said, "You can't divide by zero because zero is invalid on the bottom of a fraction, shut up."

Hearing you say that 0÷0 is just *not meaningful* as it can represent any number fills me with such glee.
I don't care that I was wrong thinking 0÷0=1 now, because you treated the question seriously and gave me an answer that doesn't treat 0 ÷ 0 = "ERROR" as dogma

[luker.]

Last quarter of this video was legitimately mind-blowing. Thanks for inspiring me and no doubt many others!

[mystifiedoni]

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.

[camrouxbg]

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!

[RV]

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.