Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Algebraic Geometry for Computer Graphics

courses.cs.washington.edu

Post reply on HN
Print view View original

31–40 of 44 posts

Re: Algebraic Geometry for Computer Graphics

#31

Post by kevinventullo »

dhosek wrote: post #29
mindcrime wrote: post #11

Algebraic Geometry as studied by Mathematicians often deals with very abstract and general issues that I admit that I don’t completely understand myself. Nice to see this kind of honesty and/or modesty from an instructor! Reminds me of the time I took Discrete Math at a local community college. We all showed up at the room for the scheduled first session, and found no instructor. We waited around about 20 minutes, an…

Yeah, I was looking over the materials and thinking, where’s the Algebraic Geometry? I don’t think that Algebraic Geometry (as opposed to algebraic geometry in the sense that the author here is using it) has any applications to computer graphics, but definitely can be applied to other aspects of computer science (I can imagine it being potentially useful in cryptographic applications), although my own Algebraic Geome…

Projective space, syzygies, Bezier curves, cubic curves aren’t algebraic geometry?

Re: Algebraic Geometry for Computer Graphics

#32

Post by auggierose »

SideQuark wrote: post #27
Sharlin wrote: post #20

Earlier quoted context omitted.

To be fair, it’s Jim Blinn , one of the pioneers of computer graphics, so his admission could be read as a polite way of saying that AG as studied by mathematicians is needlessly abstract and lacking any real-world applications :)

AG as studied by mathematicians has produced an incredible amount of applications that would not exist without that level of abstract tooling. And it's highly nontrivial to learn those tools (as someone that has worked through Hartshorne... And then put 20 years into applications).

What are some of those applications?

Re: Algebraic Geometry for Computer Graphics

#33

Post by auggierose »

aaaaaaaaaaab wrote: post #23
ncmncm wrote: post #14

I thought that algebraic geometry was really cool when I first encountered it. But people with a lot of experience insist that for computational processes it is slower than more familiar methods. So it is convenient and intuitive for one-off transformations, but not preferred in computational kernels. I see room for both, and for quaternions, which I gather are also a bit slower.

Geometric algebra is not the same as algebraic geometry.

The article mentions Clifford algebras, so while not the same, it is certainly related.

Re: Algebraic Geometry for Computer Graphics

#34

Post by npalli »

Wow, Jim Blinn a legend in the Computer Graphics field. Looking forward to reading these lectures. Ivan Sutherland, Turing award winner and himself a legend in the Computer Graphics field (Sketchpad and other works) had this to say about Jim Blinn

"There are about a dozen great computer graphics people and Jim Blinn is six of them.”

https://www.fxguide.com/fxfeatured/founders-series-industry-...

Re: Algebraic Geometry for Computer Graphics

#35

Post by 6gvONxR4sf7o »

I read the first slide deck, and by total coincidence I’ve been getting really into using tensor diagrams lately. Once you grok grouping and splitting, a lot of things you might otherwise do with block matrices becomes so much more straightforward. Deriving gradients gets a lot easier too. I’d recommend anyone who regularly has to compute gradients to play around with it.

Re: Algebraic Geometry for Computer Graphics

#36

Post by kcexn »

kevinventullo wrote: post #31
dhosek wrote: post #29

Earlier quoted context omitted.

Yeah, I was looking over the materials and thinking, where’s the Algebraic Geometry? I don’t think that Algebraic Geometry (as opposed to algebraic geometry in the sense that the author here is using it) has any applications to computer graphics, but definitely can be applied to other aspects of computer science (I can imagine it being potentially useful in cryptographic applications), although my own Algebraic Geome…

Projective space, syzygies, Bezier curves, cubic curves aren’t algebraic geometry?

I'd say they're more algebraic representations of regular geometries.

Algebraic Geometry as a mathematical field is interested in solving highly non-trivial geometric problems (think, from differential geometry, functional analysis, etc.), using tools from Abstract Algebra (think Galois Fields, Lie Groups etc.)

It saddles a bridge between traditional geometry, and abstract algebra, and allows insights from one field of mathematics to be applied to the other. As such, it allows practitioners skilled in these tools to make many useful inferences about incredibly complicated systems.

It's also incredibly dense. In part because many of the tools of algebra are incredibly involved. But also, in part because to define an algebraic object in a way that is equivalent to a geometric object, sometimes requires a fairly complicated definition.

Re: Algebraic Geometry for Computer Graphics

#37

Post by edflsafoiewq »

kcexn wrote: post #36
kevinventullo wrote: post #31

Earlier quoted context omitted.

Projective space, syzygies, Bezier curves, cubic curves aren’t algebraic geometry?

I'd say they're more algebraic representations of regular geometries. Algebraic Geometry as a mathematical field is interested in solving highly non-trivial geometric problems (think, from differential geometry, functional analysis, etc.), using tools from Abstract Algebra (think Galois Fields, Lie Groups etc.) It saddles a bridge between traditional geometry, and abstract algebra, and allows insights from one field…

Algebraic geometry existed before Grothendieck lol. It's still algebraic geometry even it looks boring next to etale cohomology of infinity stacks or whatever.

Re: Algebraic Geometry for Computer Graphics

#38

Post by matt-noonan »

auggierose wrote: post #32
SideQuark wrote: post #27

Earlier quoted context omitted.

AG as studied by mathematicians has produced an incredible amount of applications that would not exist without that level of abstract tooling. And it's highly nontrivial to learn those tools (as someone that has worked through Hartshorne... And then put 20 years into applications).

What are some of those applications?

Here are a few off the top of my head, as a mathematician-turned-programmer who never has been an algebraic geometer.

- Elliptic curve cryptography (https://en.wikipedia.org/wiki/Elliptic-curve_cryptography)

- Grobner bases, with many applications. Example domains: coding theory, robotics, signal processing... (https://math.stackexchange.com/questions/32421/applications-...)

- Physics [solitons] (https://kasmana.people.cofc.edu/SOLITONPICS/)

- Physics [string theory] (https://royalsociety.org/~/media/people/new-fellows-2014/Pre...)

- Automata theory, via "tropical" algebraic geometry (https://link.springer.com/article/10.1007/s00233-019-09999-8)

This is not even considering applications of AG to other areas of pure mathematics, which are extensive.

Re: Algebraic Geometry for Computer Graphics

#39

Post by kcexn »

edflsafoiewq wrote: post #37
kcexn wrote: post #36

Earlier quoted context omitted.

I'd say they're more algebraic representations of regular geometries. Algebraic Geometry as a mathematical field is interested in solving highly non-trivial geometric problems (think, from differential geometry, functional analysis, etc.), using tools from Abstract Algebra (think Galois Fields, Lie Groups etc.) It saddles a bridge between traditional geometry, and abstract algebra, and allows insights from one field…

Algebraic geometry existed before Grothendieck lol. It's still algebraic geometry even it looks boring next to etale cohomology of infinity stacks or whatever.

Even before Grothendieck's scheme theory, algebraic geometry was already pretty focused on varieties in rings of polynomials and other such constructions.

The heart of the domain is still using abstract algebraic arguments to solve geometric problems.

E.g.; Euclid's method for finding the midpoint of a line is to draw two concentric circles centred at the vertices with radius the length of the line. The straight line that passes through the two intersection points of the circles, also passes through the midpoint of the line.

This is the same as saying the midpoint of a line is the intersection of an algebraic variety with a root at one vertex, and another algebraic variety with a root at the other vertex.

You don't need schemes, or projective curves, or local rings to prove it.

Re: Algebraic Geometry for Computer Graphics

#40

Post by raghudotcc »

timostrating wrote: post #5
raghudotcc wrote: post #2

Are there lecture recordings?

If you are looking for general advanced graphics video lectures then this might help. The lectures at the university of Utrecht (one of the top universities of the Netherlands) are now often recorded and published sinds covid started. There advanced graphics lecture recordings, slides and exercises from last year are published on this website c http://www.cs.uu.nl/docs/vakken/magr/2021-2022/index.html

Thanks a lot.. Been watching Keenan Crane’s videos on YouTube. This would be really helpful.

For anyone else interested in Alg geo, Ravi Vakhil’s videos are very good and accessible.

Post reply on HN