Shame, he should have more time for these kind of presentations. His work is priceless.

What a legend He doesn't even waste time to tell me what his name is. So I had to go back and look it up to look up his other talks.

Excellent talk, especially when you pause it and code it live as well. Thank you!

the more i rewatch this, the more valuable it appears to be to me.

These math talks are the best!

Squirrel Eiserloh was my game programming professor at SMU Guildhall. He was the best teacher I ever had. The most amazing person, who explained complex concepts in such a cool, interesting and easy-to-understand manner. Everyone has their No. 1 teacher that they remember fondly for the rest of their lives. For me, that teacher is Squirrel Eiserloh.

Quadradic is 2nd degree... because it comes from quadrado(square). Thats not a bad nomenclature, the confusion is caused by english mixing germanic and latin words...

Excellent! I was exactly looking for some of these functions, and the performance update on power with mix is really nice to know. I am doing procedural generation for creating galaxies, and I needed something very specific (the Arch namely). The added toolkit will be very helpful in general, as I use all sorts of such functions a LOT. Plus now I know inserting 2 inverse-functions (1-x) is actually not obscene.

Really good channel. Happy that I followed you guys, good work GDC.

Great talk as always with this host

lovely accurate, effective efficient math for curves, thanks ^^ its hard to find better concepts than old dated ones ^^

Excellent talk! Real eye opener.

good math stuff, helped me with 1D non-linear transforms

Dude, this is great for figuring various jump arcs for a 2D(or 3D I guess) Platformer inorder to implement Variable Gravity. You just implement where you are in _dt in the jump over all and you would get your perfect Gravity value so your player moves in whatever arch you define. Very nice talk.Simple and easy enough that a cave man like me could understand it.

This talk is excellent.

What great timing this got recommended to me. Just finished programming almost all the linear transformation stuff for my game and a non-linear tutorial appears

Good talk, timeless!

you guys should check out the 'tunable normalized sigmoid function'. It can represent most of these shapes with just a change in a constant. Can be very usefull

Very useful topic. Saving this for later reference, since I've made a custom Tweening class and these will come in handy.

12:05 didn't noticed this last time I watched. Now I understood the purpose of this talk.

The animation is awesome.

this is what i call a good talk

Damn, wish there was more examples like his, I'm on the hunt, but this stuff is greattt

This is juicy. Helped me understand a few things a little more.

Small error in the BellCurve6 tween - you need to Flip(SmoothStop3) instead of just using SmoothStop3 otherwise you get a cumulative function (an S curve looking thing).

It is surprising how much personality non-mathematicians often attach to such abstract concepts as multiplication or power functions. This is an interesting perspective on how to achieve fast and ‘good-looking’ transformations, but I am slightly worried about the attachment to particular cases for educational purposes and not recommending any general ideas as further reading or at least references. I would be happy to hear some remarks about division vs multiplication (powers are expensive, but what about reciprocals?) because this basic operation enriches the non-linear set a lot.

13:14 I think it should be opposite for higher powers (say >4). pow(x,n) function will do the calculation in O(logn) time but x*x*..x will do it in O(n) time.

Neat stuff

24:18 Looking at the composite I'm getting some Bernstein-vibes. It's just missing the outer functions to complete the picture.

Smoothstart2.2's relative error compared to x^2.2 just explodes as it narrows zero and is even for like 0.2 15% off. This wrecks total havoc with your color reproduction, in particular given the fact that the x^2.2 is already a quite poor approximation to the sRGB gamma transfer function, which already suffers huge relative errors.

As someone who comes from a Flash scene in the late nineties and especially in the early naughts, much before iPhone and other slick UX interfaces which got animated, all of this is really really old news for me. I'm incredibly glad for all you people to learn about this, because these things have directly inspired the modern trends in touch screens and gaming UIs. All thanks to people who had the knowledge and the nerve to use little gimmicks of animation in their buttons and frames, windows and palettes, in times when MS Excel was the sole authority on how a computer software should look & feel (you can see the effect it had in Civilization games prior to the fifth installment..). All thanks to the once revolutionary Flash player, which is now sadly banned from every online platform, mostly because of scene politics, dry business, and PR nonsense. But I liked how it married a technical mind with an artistic edge, and paved a way to brave pioneers to show us how a software made for humans should look like. I personally, as a developer, cannot fathom making a software that doesn't employ a tastefully tweaked tween animation since at least 2001. Of course, I'm a game developer, though I have moved from Flash to other graphics engines out there. However, I can't understand this guy coming along and cynically destroying the point of the terms that were invented, changed, reinvented and solidified much before he had any appreciation for it. Initially the thing was known as 'easing,' but we kinda learned to grow out of this programming lingo, as soon as we learned that animation already uses this technique, known as "betweening", for almost 100 years. It simply made more sense, animation isn't just a word that is used for traditional animation -- animation in it's primary definition is that thing that gives anything immobile, uncanny, or simply "inanimate," a lively, unpredictable, and natural quality of change. It literally means "the state of being full of life." With added animation you basically decorate a shift or change or motion with a series of subtle in-between frames, to enrich the experience, and make it more palatable. It made things pop, it made the interface seem more responsive, it made the transitions appear more fluid, or more organic, or naturally twitchy and bouncy, as opposed to synthetic jitter and instant state switching. For this reason, the whole point of animating stuck with us, this has become known as betweening, or tweening in short. He also got the point of in and out completely backwards: you don't stop a train *in* the station, the point of *in* is to reflect "getting into motion" i.e. letting go, and *out* as "getting out of motion" i.e getting something to stop. His mind is not in the right place and he's definitely not an authority on these things. Also when he talks about Quad, Cubic, Quartic, and Quintic, he's basically referring to the old 'easing' lingo that was too dry for most of us anyway, and thus abandoned, but seriously, these names are pretty old mathematical names for these exponential functions... What's he gonna do next, reinvent Bezier curves as well? I'm sorry, but in my practice, I'm perfectly well-acquainted with such names, when I see Quad and Cubic, I know pretty well what to expect, not to mention that in my native language we say "kvadrat" (pronounced as kvad-rut) which is a square, so these things run pretty deep. And in fact, English etymology for square says this "Middle English: shortening of Old French esquare (noun), esquarre (past participle, used as an adjective), esquarrer (verb), based on Latin quadra ‘square’." No one's gonna change how things are called based on how terribly your ancestors adopted their words from Old French. No excuse.

13:00 17:30 this was what i was looking for, awesome.

14:18 in Russian "quadrat[ic]" means "a square", so for us it doesn't actually sound weird, since X quadratic means literally "X squarish". I don't really know why it's called quadratic in English though, - since you call a square "square", not "a quadrat" like we do, so you should be actually calling it: x² = x squarish x³ = x cubic etc...

20:09 What does the t parameter do in the mix function? Looks like something is missing....

I get why he decided to use flip, power, flip, to explain smooth stop. But it feels very unmotivated. If you add a little rigor you would notice that SmoothStart has derivative 0 at 0 and derivative n for x^n. So we really want a function with the opposite derivatives. The constraints are f(0)=0, f(1)=1, f'(0)=n, f'(1)=0. So let f'(x) = n(1-x)^k obviously has these properties for f'. Then integrate using u-subsitution F(x) = -n/(k+1) (1-x)^(k+1) + C. Then solve the boundary condition and find that C=1 and n=k+1. So F(x) = 1 - (1-x)^n. Now you might say, "that is way harder". But it's also more general and doesn't just feel like trial and error.

Squirrel is kewl. i liked his tween joke.

22:15 SmoothStart2.2 isn't that 20% 2 and 80% 3? Not the other way around, so the BlendB value should be the inverse 1-BlendB.

What about easing using trigonometric function. Is it expensive too?

I don't quite understand the SmoothStart2.2 approximation with Mix function. How does that work?

like the math but i really expected to see some code with the math applied to some game examples. example, vectors, lerping, shooting projectiles, closest neighbor etc etc..

Links to the two talks referenced at 0:40 in my comments below.

I can't get the smoothstep5 to work... I did (1-t)(t^5)+t(1-(1-t)^5) but it comes out LESS smooth than smoothstep3... Anyone knows what I am doing wrong? 20:24

I really like SmoothStart compared to EasingIn

epic

24:41 BellCurve6(t) = SmoothStop3(t) * SmoothStart(t); Since there is no Flip(t)*t invovled, why it become looking like Bell(Normal Distribution) Shape?

i'm terrible at math but when doing animation, i've always thought it's like ease in=ease at start (when going "in" to animation) ease out=ease at end (going "out" of animation) and sine, cube, quad as the exaggeration of the ease instictually. plus it didn't take a crazy genius to realize that, you just try each one and see the animation play... p.s: i'm not a native english speaker

I believe t means inTerpolant. Like how i is iteration?

When flipping though. If x is 10, then 1-10 is -9 so wouldn't that change the whole value you're trying to smooth? Wouldn't it be better to multiply it by negative -1? -1(-1*x)" ?

For anyone who wants easy access to the link from the last slide: essentialmath.com/tutorial.htm

“Having t till you gotta p”. (I'm so sorry!)

The shape of this square root graph in 17:24 doesn't look correct. sqrt(x) has this upper-right end in the origin, the curve is more abrupt there, as opposed to this.

I dont get any of this... what foundational stuff am I missing?

Lol it's the guy that made the other dude move away :D

i dint get the Arch2. can anyone explain how to implement it

[0, 2pi] is a very important domain.

0:13 He is here to plant his seeds uwu

Too bad he was noticeably running out of time towards the end...

x*(2-x)

the naming t comes from newton 's low of physics and yes it stands for time.

So it is proved that Google also uses AfterEffects to create UI animations, rite!?

After watching this i now understand why so many games are riddled with issues.

This is why I use external dlls

I didn’t understand a word of this

14:21 Actually they are "terrible" names if you're not from places where the main language pulls from latin (or if you don't speak those languages). For example knowing Portuguese, which pulls from Latin a lot, those terms are very familiar. Most if not all prefixes in math terms come from Latin, so if you know any romance language you easily identify the number associated with it. Even tho English does pull from Latin in some aspects (in this case the terms are using Latin prefixes even) it's not the same degree as Romance languages Edit: Also regarding the EasingIn and EasingOut part, at the start of the presentation it was established that this functions could be called Easing Functions so int that context using EasingIn to identify the start of the function and EasingOut to mean the end of the function is not really an out there concept imo

Oof

Use Chebyshev polynomials. Goddamn game programmers.

If anyone can translate this to UNITY FOR ME PLEASE HELP!

use tween engines! (Dotween etc)

Super annoying that he decided that the terms literally everybody uses, every piece of software uses and every programming language uses aren't good enough for him so I have to go through the whole thing translating what he's saying. That's just petty and pointless. Sure, jellyfish aren't made of jelly and they're not fish. No, you don't get to use you're own name for them, that's what they're called, get over it.

Let me save you 30 mins: Go to wolphram and experiment with different X multiplications, exponentiation etc You won't get any practical tips from this talk.