Geometric Algebra - Linear and Spherical Interpolation (LERP, SLERP, NLERP)

[@prakharnigam9757]

why is this so underrated channel please never stop making videos

[@TheAramil]

I found this video using imprecise search terms. After 30 seconds it was clear this wasn't the answer to my question and in fact I already knew everything that would be covered in the video. After a minute it was also clear that I was going to watch the whole thing anyways because the presentation is just so clear and pleasant.

[@Bubatu7]

Great to see another video covering GA!

[@bestbuildbuildinginspectio4170]

Thankyou for the concise and clear explanations.

[@RalphBohn]

Your videos are very, very excellent. Thank you.

[@wesNeill]

Very interesting. I think it would be nice to make a video of this exact topic from the perspective of interpolating function values rather than points on a line or sphere. I know there is a connection, but it is hard to see it from watching this.

[@samirk.2104]

This is exactly what I was looking for

[@alkalait]

A quicker way to derive βs is by symmetry: in the equation for αs, swap vi with vf, and 1-s with s

[@MrPetoria33]

Absolutely beautiful! Thank you!

[@vishalramadoss668]

Simply loved it.

[@veteatomarporculo100]

Could you please make a video about the rotors and its natural association with the wave function in quantum mechanics?

[@BigDBrian]

Perhaps a dumb question, but considering the affine property is two things adding to 1, what about something like Vs = sin²(s)Vi + cos²(s)Vf ? What would that look like? Because originally, after you went from linear to spherical interpolation, that was my first guess for what iwas going to be. Since sin²(s) + cos²(s) = 1, and we already know this also relation defines the unit circle...

[@TinBryn]

If you substitute sin²(s) for (1-cos²(s)) according to the Pythagoras identity you will end up with Vs = (1-cos²(s))Vi + cos²(s)Vf, Now I'm going to define s' = cos²(s), the result is Vs' = (1-s')Vi + s'Vf and we see that it's just a Linear interpolation. The intuition I have with affine systems is that they tend to keep lines as lines, so if you have something that satisfies the affine conditions you should expect it to keep lines as lines, as we have shown here.

[@BigDBrian]

Ah, that makes sense. So you have to put the members with the affine relation inside a function to make it adifferent shape from just a line.

[@SlipperyTeeth]

The shape may be a line, but the way it moves on that line is different.

[@jatinsaini7790]

smoothly explained!

[@thezipcreator]

with the SLERP formula, what if θ=π? then the expression is undefined, so what do you do then?

[@zemoxian]

I'm also wondering if you have considered covering splines for (S|N)LERP?

[@javierfilippini2167]

Excellent video. Thanks!

[@alfonshomac]

Long time no see!

[@camwg]

Oh he's been getting around. Watch the livestream Mathoma did with Friended.

[@alfonshomac]

In all honesty, I skip the theism v atheism videos. I've become exhausted by the topic over the past 7 years and came to the conclusion that it's very similar to the political spectrum. I'd been called a "pragmatic deist" by turbo atheists and a vitriolic new atheist by turbo theists. The tag I'm given depends on my conversation partner. So I think what I think, I don't mind what you think so long as you're not being subversive or coercive and peace ensues. Nobody gets locked in the weird psychotic loop those conversations get into unless you're as gifted in speech as Hitchens and Dawkins or Peterson and Shapiro.

Perfect Lecture! Thanks!

[@SlipperyTeeth]

If this is used for graphics, wouldn't a parabolic interpolation be useful? Or is it easier to just approximate one with a series of LERPs? If so, why do we have SLERP instead of also approximating it?

[@SlipperyTeeth]

Sounds like LERP is more powerful because it can interpolate any 2 vectors. SLERP requires two vectors with the same magnitude. Are there equations for a SLERP-type of interpolation where the line connecting the two tips of random vectors is a chord of a circle, and the interpolation moves along the circle?

[@pronounjow]

Phoenix Fire I'm thinking the opposite. IIUC, the vectors are only supposed to provide coordinate independence. SLERP would maintain the shape of the object being transformed, whereas LERP looks like it would distort the object during the transformation and leave it distorted between v-sub-i and v-sub-f. It would be really awkward to watch an object undergo LERP based on my understanding.

[@zemoxian]

Jo Reven I'd interpret LERP as interpolation of a translation and SLERP as an interpolation of a rotation.

[@SlipperyTeeth]

Jamal Wills I wasn't even thinking about applications like graphics. I was completely in pure math world, where this takes two vectors and we pick a path between them; in which case SLERP limits which two vectors we can pick.

[@SlipperyTeeth]

Jamal Wills I guess then my suggestion would be a rotation and a translation at the same time.

[@zemoxian]

Phoenix Fire There’s a way you can do that with dual quaternions. Instead of a real angle you pass a dual number like theta + d epsilon. Where epsilon is a unitary base that commutes with the other bases and epsilon^2 = 0. It’s interesting that GA already has bases that square to 1 and -1. So a single 8-dimensional unit dual quaternion can act as a translation and a rotation. I think it’s called s screw if the translation is along the axis. And you use the same sandwich product to produce the change. I’ve played with it a little but still get confused with the details in some calculations.

[@Sfaxx]

Can you make a videos about graph theory? I think that it's an interesting topic

[@Songfugel]

Me as an engineer at the middle of the video: *imagine the sweat scene from Airplane* Also me when I see the nlerp:"No this is more like it, and close if enough for me! all I need is to just get a close enough algorithm (or an extremely dirty preset set of values) to change those 0-> 0.5 values to map correctly the between the nlerp input and slerp outputs and I am set for life :D Thank God we don't have to actually do these calculations every time, and can just use a function call to do them for us :)

[@davidste60]

And once you have Vi, Vf and Vs* you can just divide wedge terms instead of the sine terms as in the slerp formula. (because wedges are sines of thetas when using normalised vectors).

[@zemoxian]

I've been wondering if there's a GA version of dual quaternions? Is there a e^2=0 unit for GA?

[@pronounjow]

Differential forms, like dx^dy, maybe? That's all I can think of based on my non-rigorous study of this stuff...

[@zemoxian]

Jo Reven Actually I’ve found out about conformal geometric algebra. It has “origin” and “infinity” vectors that square to zero. So reflection, rotation, translation, scaling can all be represented by sandwich products.

[@zemoxian]

Jo Reven One meat feature is that you can represent a circle with 3 points. A line is just a circle with a point at infinity and is represented with the infinity unit vector. That gives it an infinite radius. Likewise a sphere is created by 4 points and a plane is an infinite sphere.

[@angeldude101]

Projective Geometric Algebra is a subset of CGA that only has flat objects, but it's also more efficient. In PGA3D, the even subalgebra is exactly the dual quaternions just like how 3D vanilla GA had the quaternions as the even subalgebra.

[@dogman12345]

*MEEP* Good stuff mate! Love the review *MEEP*

[@rasraster]

Took me forever to realize that in SLERP you had vi and vf originate from the center of the unit sphere. I kept thinking they were placed on the surface and was wayyyy confused.

[@datnd-21403]

Great❤

[@turner7777]

Hi, i've been researching what lerp is for a computer science project and i've come across alot of videos (And im extremely bad at math, thats why i was researching what this is :D) So at first i found this formula used in this video on stackoverflow: L=(1-S)M+S*X (Just like in your video just with changed letters) But i found this formula aswell in some other video: (X-M)*N+M (X is the maximum, M is the minimum, and N is like that S variable of yours, goes from 0 to 1) Which i tested with the above one and got the exact same answer. The question is can i use them interchangeably ? Because the latter on is more performant to calculate. Im sorry if i wrote something stupid or messed up the formula, agian im extremely bad at math.

[@wyboo2019]

changing the N to an S, we can get the first formula from the second: (X-M)S+M = XS-MS+M = M-MS+XS = (1-S)M+SX so they are indeed interchangeable, if you wanted to know 2 years late

[@wyboo2019]

also don't ever apologize for "saying something stupid or writing something wrong." the only way to learn is by making mistakes and then correcting them. if you never make mistakes then you can never improve

[@turner7777]

@@wyboo2019 haha thanks for the reply, i even forgot what was i working on to need this 😆

[@danpal6737]

as a poor unemployed obsolete student surviving with beans thank u

[@Math_oma]

Beans are too expensive. I'm a pasta and rice man.

[@beatsbymerks6048]

i meant to type sleep but this works too

[@stevenlu7324]

I worry that this will make game devs implement slerp in suboptimal ways using multiple sin() calls if they fail to see the next video in the series!!

[@ethansmusic9898]

What is the proper way then? I'm going to be a computer science major this fall so I'd like to know what the proper formula is

[@qm3ster]

NLERP: SLERP, but you get some easing for free :v