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> The easiest way for me to conceptualize it is to think of it as orientation + rotation. 3 dims for the orientation vector to get the object facing/pointing th

> The easiest way for me to conceptualize it is to think of it as orientation + rotation. 3 dims for the orientation vector to get the object facing/pointing the right way, then a further 1 dim/var for rotation about that axis. For a total of 4 variables/dimensions Something's missing. Orientation in 3D space is a two-dimensional quantity; you would never need three dimensions to express it. The third dimension has to be providing some additional information, like a magnitude.

nighthawk454 2y ago
Only if you’re describing orientation as two orthogonal rotations. I’m saying think of it like a ‘pointing’ vector that defines the axis of rotation. And such a vector does require 3 components in 3d space
- creativenolo 2y ago
  It’s possible to point in 3D space with two rotation components. For example, the first two components of a UV texture. But I agree it is helpful to think of quaternions as direction and spin.
- thaumasiotes 2y ago
  Yes, a three-dimensional vector is a combination of a 3D orientation and a magnitude. An orientation by itself doesn't have three dimensions. The surface of a sphere is a two-dimensional space. > Only if you’re describing orientation as two orthogonal rotations. No, the space has the dimensionality it has. You may choose to describe a 3D orientation with more than two numbers, but you won't stop it from being a two-dimensional quantity that way. If you use more than two numbers, those numbers will fail to be independent of each other.
  - oasisaimlessly 2y ago
    Orientation would conventionally be a member of SO(3), which is a 3-dimensional manifold. Your comment is essentially correct if you replace the word "orientation" with "direction", though.
xeonmc 2y ago
Direction in 3D has two degrees of freedom, orientation has three.
vrighter 2y ago
I think a better word than orientation is "axis". They are conceptually similar to axis-angle, with some added sugar to make an algebra

nighthawk454 2y ago

Only if you’re describing orientation as two orthogonal rotations. I’m saying think of it like a ‘pointing’ vector that defines the axis of rotation. And such a vector does require 3 components in 3d space

creativenolo 2y ago
It’s possible to point in 3D space with two rotation components. For example, the first two components of a UV texture. But I agree it is helpful to think of quaternions as direction and spin.
thaumasiotes 2y ago
Yes, a three-dimensional vector is a combination of a 3D orientation and a magnitude. An orientation by itself doesn't have three dimensions. The surface of a sphere is a two-dimensional space. > Only if you’re describing orientation as two orthogonal rotations. No, the space has the dimensionality it has. You may choose to describe a 3D orientation with more than two numbers, but you won't stop it from being a two-dimensional quantity that way. If you use more than two numbers, those numbers will fail to be independent of each other.
- oasisaimlessly 2y ago
  Orientation would conventionally be a member of SO(3), which is a 3-dimensional manifold. Your comment is essentially correct if you replace the word "orientation" with "direction", though.

xeonmc 2y ago

Direction in 3D has two degrees of freedom, orientation has three.

vrighter 2y ago

I think a better word than orientation is "axis". They are conceptually similar to axis-angle, with some added sugar to make an algebra