|
I think that's a good motivation why we would study quaternions, but it's kinda hiding the big difference between 2D and 3D under the rug. In 2D, we have a nice, global coordinate system for the space of all rotations: what we call the angle. (Actually, it's a coordinate system for the "universal cover" of the space of rotations since angle X and angle X + 2pi give the same rotation, which mostly doesn't really matter.) Meanwhile, in 3D, there is no global coordinate system for the space of rotations! Euler examples uniquely specify a rotation, but the problem of Gimbal lock [1] means that they break down as coordinates at some point (i.e. there's no inverse to go from rotation in 3D to its corresponding Euler angles, which there is in 2D, with the caveat already mentioned). This is analogous to the problem of finding a coordinate system for the globe: specifying latitude and longitude tells you were you are, but there's a degeneracy at the poles. And no possible coordinate system can solve this problem entirely. Contrast this to the situation of giving a coordinate system for the circle, which we do with it's angle. This isn't quite a coordinate system, due to the problem we already encountered that X and X + 2pi are the same, but that's OK because the these two points are separated from each other. On the sphere, the latitude/longitude pair (pi/2, x) gives the north pole for any value of x, even ones that are arbitrarily close together. That maps not even locally invertible! You suggest we think of points on the circle as point in 2D space that happen to lie on the circle (i.e. cos and sin of the angle corresponding to that point). Analogously, we can think of points on the sphere as points in 3D space that happen to lie on the sphere (like some point (x,y,z) with x^2 + y^2 + z^2 = 1). And analogously, we can think of rotations of 3D space as a point in 4D space (that happens to satisfy some conditions), and the quaternions give that 4D point. This is fantastic and convenient in both 2D and 3D! But in 2D we didn't need to do this, but could if we wanted to. For 3D rotations, we do need to, or else we have this terrible degeneracy that never rears its head in 2D. In that sense, 2D and 3D are very different! [1] https://en.wikipedia.org/wiki/Gimbal_lock |
There are plenty, it's just that you can't have a 3-dimensional one without singularities.
https://en.wikipedia.org/wiki/Hairy_ball_theorem