Algebraic Geometry for Computer Graphics
courses.cs.washington.edu
courses.cs.washington.edu
"Robert McDermott and Jim Blinn are jointly responsible for the precise mathematics and computer programming which implemented the geometric concept Accomplishment of the highly sophisticated and complex mathematics brought to a close the long-sought-after geometric definition of the faceted Egg." [1]
Came across this little tidbit of information reading the information plaque when my dad was showing us around town. Imagine my surprise coming across the name of probably my first computer graphics books from university.
[0] https://en.wikipedia.org/wiki/Vegreville_egg
[1] http://www.ronresch.org/ronresch/the-egg/easter-egg-booklet/...
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, and were about to leave, when a grey-bearded gentleman who looked to be of the instructorly persuasion rocked up and started unlocking the door. So we follow him in, and he gets in front of the class and says something approximately like:
"Hi, so-and-so, who was scheduled to teach this class, had to back out at the last minute and so the school called me. I'll be your instructor. I was a Physics major 60 years ago, and can do this math, but I've never taught this before and don't really remember anything about it. So we'll all be learning it together as we go."
It was an, ahem, interesting experience, to say the least. Luckily for me I'd had Discrete before and was only taking it due to a technicality that kept my previous credit from transferring.
And it's highly nontrivial to learn those tools (as someone that has worked through Hartshorne... And then put 20 years into applications).
- 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.
Some of these are likely why Blinn decided to make an entire course on using it just in computer graphics alone.
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.
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.
"There are about a dozen great computer graphics people and Jim Blinn is six of them.”
https://www.fxguide.com/fxfeatured/founders-series-industry-...
Long answer: https://news.ycombinator.com/item?id=33133369#33134462
Blinn provides some notes as to where his material comes from (see the page). He also mentions that some of this material was presented previously in some of his Jim Blinn's Corner articles. In particular:
* the "Lines in Space" series: https://ieeexplore.ieee.org/abstract/document/7047281
* the "How many different rational parametric cubic curves are there?" series: https://www.oreilly.com/library/view/jim-blinns-corner/97815...
Part of the reason why I posted this to HN is so that others can hopefully drop some more information on this "field".
See also the author's site: https://www.jimblinn.com/
https://en.m.wikipedia.org/wiki/Geometric_algebra https://en.m.wikipedia.org/wiki/Algebraic_geometry
I understood maybe a third of what he’s talking about, but it’s fascinating.
He seems to be using similar slides to those on this page.
Wondered what he is doing these days then...
> "He is currently retired." - https://www.jimblinn.com/biography/
The hairiest issue here is when there's student participation. But you don't need each quarter to be online, just once every time there are significant course changes. The quarter that will be uploaded could be announced as such and students could consent to being recorded. But this isn't a big deal, MIT has a lot of open courseware with students being recorded. It'd be easy to survey and compare student experiences with/without recordings. But the benefit for the public could be enormous! Think about the public good of every publicly funded university's courseware being accessible online for free.
And even if some public universities are self-sustaining now, that doesn't lessen the fact that public money was the major cause that started that flywheel.
And regardless of funding, UW, like all public universities, has an explicit directive to operate for the "benefit of present and future citizens of the State of Washington"[2][3].
As for course lecture quality, if it's worthwhile to the students, it can be worthwhile for the public.
[0] https://www.washington.edu/opb/uw-data/fast-facts/ [1] https://educationdata.org/how-do-people-pay-for-college [2] https://www.washington.edu/regents/ [3] https://apps.leg.wa.gov/RCW/default.aspx?cite=28B.20.130
For anyone else interested in Alg geo, Ravi Vakhil’s videos are very good and accessible.
However, Blinn writes that some of this material was presented in more detail in some of his "Jim Blinn's corner" articles.
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.