PGA Ep 1 : The Reflection Menace

[@howdy832]

The joke that you calmly slid across the table at 3:50 had me floored. I love it

[@leokovacic707]

Finally someone who understands that taking basis multiplication tables as a basic tool is just a milder form of coordinitis. That was hestenes is original wiew, one that he stressed out far to infrequently, in hopes of conforming to the standard literature.

[@rotgertesla]

Can you elaborate your point further please? Do you mean that using the e1, e2, e3 in geometric algebra is a bad idea?

[@leokovacic707]

@@rotgertesla not always, for example to relate data to a computer screen you need some kind of a basis or coordinates for easy inputs, but generally any kind of a basis is arbitrary and subjective, and GA shows us that a theoretical scientific method is possible, and that was the original Clifford's wiev that math is the gate of science, and science has to be done objectively. If you assume a basis multiplication table at the start, you will pay with complexity and problems due to arbitrarines afterwards in any theoretical endeavor including concrete calculations.

[@minma02262]

THANK YOU VERY MUCH. VERY VERY MUCH!

[@karltraunmuller7048]

That’s my cup of tea, I’m paying attention.

[@idobenamram3743]

incredible!

[@MiguelVega-kh7dq]

Amazing! Thank you !

[@UliTroyo]

This is awesome, thank you!

[@buttonasas]

Hmm. I could follow for most of the traditional 3D Geometric Algebra without the element that squares to zero and then the 2019 presentation made duals understandable in R*3,0,1 but this video was definitely harder to grok. Seems like something is skipped over or in the wrong order but I can't put my finger on it.

[@bivector]

Hi @That Scar. Perhaps it is the general 'transformations first' approach that I could have mentioned more specifically. I may have been to focused on having the animations and buildup explain this main point that I forgot to also actually say it. If you start to build up geometry from the elements (as we usually do), you have to make important representational choices early on (e.g. you start axiomatically from vectors are points), that can end up not being ideal w.r.t. their impact on the representation of the transformations. If instead, you start with the transformations (and remember to include the reflections!) - the representation of elements naturally rolls out. It turns out that if you do it this way some things are exactly opposite to how we usually do it - but as it also turns out - those differences are key to an intuitive dimension independent treatment.

[@markvp71]

@@bivector I got lost around 19:30, where you mention that aaã = +-a, without explaining what a and ã represent here. I also don't understand why a^2 can be -1, I would think the square of a reflection is always 1. Around 21:40 you don't explain how to add 1 to ab if a and b are geometric numbers, neither how to normalize.

[@bivector]

@@markvp71 , these details get handled in episode 2. In short it's also possible to add timelike generators, and the twiddle is notation for the reverse.

[@MrPuppyPuppet]

22:38 "We will call 'e' the 'eye point', that's the point you're looking at. (...) We will call 'c' the 'center', that's the point where you want the camera to sit at". Isn't it supposed to be the other way around, at least in the terminology of classic gluLookAt? "eye" should be the new origin, where the camera ("eye") will be placed, and "center" is the thing the camera will be looking at, right?

[@MrRyanroberson1]

27:26 when you calculate all of the intermediate orange arrows, does the program find the transform for the smallest change and just repeat that, or do you need to calculate each one independently, and how?

[@bivector]

Call the constructed transformation R, then the orange arrows are exp(log(R)*t) for t in 16 discrete steps between 0 and 1.

[@MrRyanroberson1]

@@bivector amazing! in that case this is therefore also a good tool for smooth camera transitions between two endpoints, rather than just a snap-to tool

[@MDNQ-ud1ty]

Could GA be used in AI?

[@randyhelzerman]

Are there any course notes/associated papers?

[@bivector]

We are finalizing notes, expect them in the first half of November. 2021.

[@bivector]

writeup at bivector.net/PGADYN.html demos and implementation details at enki.ws/ganja.js/examples/pga_dyn.html

[@randyhelzerman]

@@bivector Thanks!

[@blue5659]

​@@bivector thank you

[@pizamora]

​@@bivectora big thank you!

[@guidosalescalvano9862]

What does the aa~a = +- a mean at 20:23 ?

[@bivector]

That means that if you reflect an object in itself, you'll always end up with ± the same object. In PGA any element that represents a geometric entity (point, line, plane), parametrizes also a reflection that leaves exactly that element invariant. Reflect in a plane and that plane stays in place, same for reflecting in a line or in a point.

[@buttonasas]

Why do you say that PGA is "chirality agnostic"? To me, it seems to involve chirality just as much as classical methods - there's a chosen convention and some minus signs flip in a couple places depending on it. And it seems it would be problematic if you changed the convention mid-calculation either way. For example, when saving a triangle mesh in one application and restoring it in an another application that has a different handedness.

[@bivector]

Classically the difference between left handed and right handed is specified in the text around the formulas. In GA it is specified in the formula (using e.g. e21 instead of e12). As such the info is included and someone using e21 will still be able to process data from someone using e12 (it'll automatically get the minus sign).

[@MahendraReddyTambela]

where are the notes?

[@bivector]

bivector.net/PGADYN.html

[@pastaferian5798]

Subtitles language is wrongly identified as dutch. PLease fix.

[@bivector]

Fixed. Thanks for the heads-up!

[@scififan698]

That's because of KZbin's accent detection. Lol

[@blue5659]

​@@bivector not fixed . i still get dutch.

[@NuYiDao]

You should spend much more time on fundamentals of what is a 'geometric' number. I get that it is a solid base but only from the fact that so much is immediately built on top.

[@bivector]

I did cover this in some more detail in the SIGGRAPH2019 and GAME2020 Dual Quaternion lectures.