Not detracting from this post, but has anyone else noticed there's a front page post about Quaternions or Kalman filters on about a monthly cadence? Wonder why that is?
I have noticed too that HN has an enduring fascination with quaternions, and they seem to reach front page surprisingly frequently (considering that there's not a huge amount of discussion of other geometry topics). I'm certainly not complaining though - I love quaternions too!
At least this is a topic that the average HN reader is likely to be able to understand and which is closely related to software. Several years ago there was a period where there were periodic posts about things like homotopy type theory and research-level algebraic geometry which inevitably spark only inane misunderstandings and uninformed speculation in the comments. I can only attribute it to some kind of fetish for the frontiers of pure math.
If I had to describe Quaternions to someone I would first try to explain a Plane (Ax + By + Cz + D = 0) to them. ABC being a (normal) direction that the Plane is pointed towards and D being the distance from the origin.
A Quaternion from what I believe is just the same but instead of distance it just encodes the rotation around that direction as a fixed axis. (Instead the angle stored is half etc).
Feel free to correct me if I'm wrong, I'm not a math-heavy person.
Think of a fraction as a pizza. The bottom number is how many slices the pizza has, and the top number is the slices that you are going to eat. Sally will eat the rest, so you need to leave some for her.
> These discontinuities are not just an artifact of poor implementation; it can be proven that any representation of 3D rotations using only three values must contain discontinuities.
This is a bit pedantic - and the blog post actually does clarify this - but the problem isn't that a 3D representation of representations has "discontinuities" as such, it's that it's not orientable in Euclidean 3D space. It is similar to the Klein bottle - the mathematical description is continuous, but any Klein bottle made of real-world glass has to intersect itself. Or likewise that a Mobius strip can be demonstrated in 3D but can't be built in Flatland without a 3D entity doing the copy-pasting. Reality just has the wrong topology to represent the full group of rotations. Hence the discussion about projective space later in the blog post.
So adding a fourth gimbal is really tantamount to a correctly-oriented embedding of 3D rotations onto a 4-torus (that is, [0,2pi]^4).
Gimbal lock also relates to another issue of continuity, related to the "plate dance."[1] Rotations themselves have a sense of continuity (infinitesimal differences in either the angle or axis of rotation, aka they are Lie groups), but Euler angles fail to respect the equivalent fundamental theorem of calculus: adding a bunch of infinitesimal changes might say you are at the identity rotation according to Euler angles, but in reality you have flipped the meaning of the right-hand rule and the overall state of the system is not at the identity. In a robotics context, the robot's hand might have done a complete rotation, but its arm is twisted without the robot "knowing." I believe robotic arms used to have a serious problem with this, either overrotating and breaking the arm, or swinging dangerously fast in the opposite direction. Using quaternions / a fourth gimbal / etc. there would be a measurable phase or pole indicating the true state of the system, and letting the robot know how to rotate its arm without malfunctioning. So,
like the Apollo mission, the need for a fourth dimension
to keep track of that stuff - and even the quaternion multiplication structure - comes about pretty naturally without ever thinking about abstract math.
Not detracting from this post, but has anyone else noticed there's a front page post about Quaternions or Kalman filters on about a monthly cadence? Wonder why that is?
I have noticed too that HN has an enduring fascination with quaternions, and they seem to reach front page surprisingly frequently (considering that there's not a huge amount of discussion of other geometry topics). I'm certainly not complaining though - I love quaternions too!
Because quaternions are awesome? :) it definitely feels like a discovery when you learn about them first.
Maybe because it intersects deeply with game dev?
yes, that's the module i learnt about them in during college.
IMHO I think for me they're interesting because it felt like mathematically they straddle the line of being nearly impossible for me.
I could just barely do them and felt very accomplished when I could. Anything else was either impossible or easy by comparison.
assuming my maths skills are average here then most people have similar experiences with them.
At least this is a topic that the average HN reader is likely to be able to understand and which is closely related to software. Several years ago there was a period where there were periodic posts about things like homotopy type theory and research-level algebraic geometry which inevitably spark only inane misunderstandings and uninformed speculation in the comments. I can only attribute it to some kind of fetish for the frontiers of pure math.
If I had to describe Quaternions to someone I would first try to explain a Plane (Ax + By + Cz + D = 0) to them. ABC being a (normal) direction that the Plane is pointed towards and D being the distance from the origin.
A Quaternion from what I believe is just the same but instead of distance it just encodes the rotation around that direction as a fixed axis. (Instead the angle stored is half etc).
Feel free to correct me if I'm wrong, I'm not a math-heavy person.
If you a explaining this to the average American, you are going to have to start a bit further back than a "Plane".
Think of a fraction as a pizza. The bottom number is how many slices the pizza has, and the top number is the slices that you are going to eat. Sally will eat the rest, so you need to leave some for her.
> These discontinuities are not just an artifact of poor implementation; it can be proven that any representation of 3D rotations using only three values must contain discontinuities.
This is a bit pedantic - and the blog post actually does clarify this - but the problem isn't that a 3D representation of representations has "discontinuities" as such, it's that it's not orientable in Euclidean 3D space. It is similar to the Klein bottle - the mathematical description is continuous, but any Klein bottle made of real-world glass has to intersect itself. Or likewise that a Mobius strip can be demonstrated in 3D but can't be built in Flatland without a 3D entity doing the copy-pasting. Reality just has the wrong topology to represent the full group of rotations. Hence the discussion about projective space later in the blog post.
So adding a fourth gimbal is really tantamount to a correctly-oriented embedding of 3D rotations onto a 4-torus (that is, [0,2pi]^4).
Gimbal lock also relates to another issue of continuity, related to the "plate dance."[1] Rotations themselves have a sense of continuity (infinitesimal differences in either the angle or axis of rotation, aka they are Lie groups), but Euler angles fail to respect the equivalent fundamental theorem of calculus: adding a bunch of infinitesimal changes might say you are at the identity rotation according to Euler angles, but in reality you have flipped the meaning of the right-hand rule and the overall state of the system is not at the identity. In a robotics context, the robot's hand might have done a complete rotation, but its arm is twisted without the robot "knowing." I believe robotic arms used to have a serious problem with this, either overrotating and breaking the arm, or swinging dangerously fast in the opposite direction. Using quaternions / a fourth gimbal / etc. there would be a measurable phase or pole indicating the true state of the system, and letting the robot know how to rotate its arm without malfunctioning. So, like the Apollo mission, the need for a fourth dimension to keep track of that stuff - and even the quaternion multiplication structure - comes about pretty naturally without ever thinking about abstract math.
[1] https://en.m.wikipedia.org/wiki/Plate_trick
Related a few weeks ago https://news.ycombinator.com/item?id=42880242