Math for Game Devs [2022, part 10] • Abstract Algebra, Procedural Animation & Splines

Primarily for my students at FutureGames - I will only read chat/superchats during breaks!

0:00:00 Intro
0:11:06 Lecture Start
0:18:24 Smoothing & Easing functions
0:54:00 Deriving Hermite Interpolation
1:20:00 Splines
1:29:12 i (imaginary unit)
1:39:12 Complex Plane
1:42:05 Complex Numbers
1:50:38 Euler’s Formula & Identity
1:56:00 e (the constant)
2:07:03 Dual Numbers
2:10:43 Why use quaternions in games?
2:17:27 Vector Algebra (an alternate perspective)
2:24:30 Multiplying vectors?
2:30:00 2D Bivectors & the wedge product
2:34:40 Geometric Product
2:43:00 Multivector & Bases
2:48:03 3D Geometric Algebra
2:52:49 Full 3D Multivector
2:57:56 The generality of GA
3:00:26 Quaternions (from a GA perspective)
3:05:05 4D Games
3:07:28 Why quaternions rotate well
3:09:21 Q&A Outro & how I animate in Unity

[mystergan]

i just woke up on 2 hours to your video idk what autoplay thinks but you can just think what i think

[tswdev]

Senior dev here, I just love how effortlessly you flow through the code while explaining. Your shortcut use is on point. I seriously need to up my game there.

[liamsmyth]

I always feel like I am more inspired by math from these streams than I ever was in years of formal education 😂 your meticulous, artful presentation and color coding really makes all the difference!

[vaskokovachev]

The fact that such content is given to us for free just blows my mind, what a world we live in. Thank you so much for your amazing work <3

[stephencurtis]

Years ago I tried to write a CAD program. It was (still is) based on vector graphics. Lines, Arcs and Circles were easy enough, and high school trigonometry was sufficient for much of it. I then attempted B-splines, Nurbs and elementary Beziers. What would have made it fully possible would have been a solid grounding in matrices. To format the solutions to various curves including Beziers or any of the medium complexity systems of equations I was dealing with, in matrix form, would have made it manageable. What I dug for myself, without that, was repetition and a coding rabbit hole. This content has rung so many bells for me, and to see the proper structure rigoursly applied is heartening and inspiring. Thank you Freya, for inviting us into your world and making these arcane ideas seem possible, graspable, doable, understandable. I knew my path was somehow not the right one; now I'm fully convinced and if I were to re-attempt it, I would set out the whole approach very differently and I feel, would get a lot further without digging a hole. Ignorance is not always bliss.

[kanamis]

hi, just wanted to say that your videos are great sleeping to (in the best way possible). hoping to rewatch these when I'm next awake lol

[monetmonteal]

After watching all the videos and solving all the questions for the past 3 months, I have arrived at the final part. I feel that I have evolved as a mathematician in this time period. Thank You

[TheRottenICE]

I've no idea how I ended up here, I'm not a game dev or good at Math, nor have I watched any of the other parts but this was really informative and a joy to watch I learnt a lot. Thank you

[FuckFistingAssosuFFosu]

My dad died May 1st, he was only 35 and it's pushed me to be more healthy and always be in shape (maybe out of fear lol). I've always been a skinny guy but I started eating healthier and running everyday (night running is so fun) and I'm looking at a gym membership for the first time! I love hearing about people getting back in the gym and going out in nature & I'm excited to start weight training and getting shredded along with you!

[mattwillis]

Awesome math series Freya! Thankyou so much for your amazing work ❤❤

[allemangd]

The wgarble meme actually touches on something interesting. Define a symbol ε, and declare that ε²=0. The numbers x + εy are the "dual numbers", which also have interesting applications like for automatic differentiation and projective coordinates. There are also the hyperbolic numbers which define a symbol j and declare that j²=1, but j≠1.

[gsuberland]

That coefficient constraint trick in Desmos made me say "woah" out loud. Did not know you could do that.

[clearly_c]

you just explained complex numbers better than my uni professor, you are amazing

[Daniel-gjqh]

​​Tuxedo purring ASMR. This video is leg.... wait for endary

[echoes]

"Quaternions are extremely annoying and complicated" I love them 😭 I love that quaternions are the double cover of a 2-sphere and I love how quaternions relate to geometric algebra and I find all of this stuff really beautiful. Actually tho I love this video I love your streams, I could spend all day listening to you talk about math.

[bagas-]

This content is actual distilled gold of knowledge. Thank you for sharing this with us

[eternaldoorman]

29:25 If that curve were expressed as a Bezier curve, ...

[leonard]

thanks for putting all the lectures on youtube
the api you showed for animating things in code at the end looked really nice, I think it would be really cool if you show it off some more
or even release it just like shapes
you said you didn't like 3blue1brown's animation tool, but did you take a look or took inspiration from the way he designed his api? and if so what did and didn't you like about it?
I really like your way of doing animation in code, could maybe even use it for video game cutscenes or scripted events

[zubairrahmaanugrah]

i don't know why this shows up on my yt recom, but it's dope, although my math is so terrible ahhahahaha

[charlesabramo]

I hope you enjoyed your experiment. I surely did.