51–60 of 118 posts
This is fine; there are lots of similar "intro to quaternions" pages around. One thing I haven't seen yet is someone showing how to interpolate between multiple quaternions over time, smoothly. (Hint: it's not slerp.) It's a non-trivial problem.
What is slerp?
Earlier quoted context omitted.
>> Quaternions are hypercomplex numbers of the form > >I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. Think it is right. >> >> Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. > >Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1…
Thank you, this was very helpful. > The axis of rotation is actually the vector "u". Ah, I thought the vector was what we were rotating.
Earlier quoted context omitted.
10-12 years ago, the consensus used to be that this singularity simply does not show up in the wild in the vast majority of scenarios, and the performance hit of using `slerp()` is just not worth it unless truly needed. It was essentially the same reasoning as for using `-ffast-math`. Yeah it's not "correct", but users can't tell the difference and it has a measurable performance benefit. To be clear, I'm not questio…
Yeah, I understand. I just wanted to show that one can understand the exact nature of the approximation in 2D geometry. I think it's useful to think of it as a separate category of "wrong" than -ffast-math, which is "wrong" because two expressions that are mathematically equivalent over real numbers, like a + (b + c) and (a + b) + c, are not equivalent over floating-point numbers. Using lerp in place of slerp is like…
This is fine; there are lots of similar "intro to quaternions" pages around. One thing I haven't seen yet is someone showing how to interpolate between multiple quaternions over time, smoothly. (Hint: it's not slerp.) It's a non-trivial problem.
> Hint: it's not slerp. What is slerp?
This is fine; there are lots of similar "intro to quaternions" pages around. One thing I haven't seen yet is someone showing how to interpolate between multiple quaternions over time, smoothly. (Hint: it's not slerp.) It's a non-trivial problem.
In my opinion, Geometric Algebra and bivectors ( https://bivector.net ) in particular is a much better (both practically and pedagogically) approach compared to quaternions and significantly more elegant. It's a real shame that people continue to focus so much on quaternions in this day and age.
I've read linear and geometric algebra and it's amazing for linear operations. Unfortunately it gets more complex when you want to use it for nonlinear operations (such as translation) - now you need to use CGA or PGA or something. Still, seems like it would be amazing for computer graphics, physical simulation, etc. especially if someone can figure out a good strategy for efficient compilation (e.g. representing a multivector with exactly the minimal number of non-zero entries, with non-zero basis elements tracked at compile time).
tldr: Simply explained without demonstrations: Quaternions are hypercomplex numbers of the form w + xi + yj + zk Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: pqp' where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(…
> Quaternions are hypercomplex numbers of the form I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. > Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most wel…
Earlier quoted context omitted.
> Quaternions are hypercomplex numbers of the form I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. > Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most wel…
It seems to me that i, j, and k are three things that have the same property, not three identical things.