Thanks for the explanation. I found this post that connects your explanation to an explanation of the “double cover.” I believe this is how it works:
Consider a point on the surface of a 3D sphere. Call it the “origin”.
From the perspective of this origin point, you can map every point of the sphere to a 2D coordinate. The mapping works like this: Imagine a 2D plane going through the middle of the sphere. Draw a straight line (in the full 3D space) from the selected origin to any other point on the sphere. Where the line crosses the plane, that’s your 2D vector representation of the other point. Under this visualization, the origin point should be mapped to a 2D “point at infinity” to make the mapping smooth. This mapping gives you a one-to-one conversion between 2D coordinate systems and points on the sphere.
You can create a new 2D coordinate system for sphere surface points using any point on the sphere as the origin. All of the resulting coordinate systems can be smoothly deformed into one another. (Points near the origin are always large, points on the opposite side of the sphere are always close to the 0,0,0, and the changes are smooth as you move the origin smoothly.)
Each choice of origin on the surface of the sphere (and therefore each 2D coordinate system) corresponds to two unit-length quaternions. You can see this as follows. Pick any choice of i,j,k values from a unit quaternion. There are now either 1 or 2 choices for what the real component of that quaternion might have been. If i,j,k alone have unit length, then there’s only one choice for the real component: zero. If i,j,k alone do not have unit length, then there are two choices for the real component since either a positive or a negative value can be used to make the quaternion unit length again.
Take the set of unit quaternions that have a real component close to zero. Consider the set of 2D coordinate systems created from those points. In this region, each coordinate system corresponds to two quaternions EXCEPT at the points where the quaternion’s real component is 0. This exceptional case prevents a one-to-one mapping between coordinate transformations and quaternion transformations.
As a result, there’s no “smooth” way to reduce the two-to-one mapping from quaternions to coordinate systems down to a one-to-one mapping. Any mapping would require either double-counting some quaternions or ignoring some quaternions. Since there’s a one-to-one mapping between coordinate systems and candidate origin points on the surface of the sphere, this means there is also no one-to-one mapping between quaternions and points on the sphere.
No matter what smooth mapping you choose from SU(2), unit quaternions, to SO(3), unit spheres, the mapping must do the equivalent of collapsing distinctions between quaternions with positive and negative real components. And so the double cover corresponds to the two sets of covers: one of positive-real-component quaternions over the sphere, and one of the negative-real-component quaternions over the sphere. Within each cover, there’s a smooth one-to-one conversion between quaternion-coordinates mappings, but across covers there is not.
Thanks for the explanation. I found this post that connects your explanation to an explanation of the “double cover.” I believe this is how it works:
Consider a point on the surface of a 3D sphere. Call it the “origin”.
From the perspective of this origin point, you can map every point of the sphere to a 2D coordinate. The mapping works like this: Imagine a 2D plane going through the middle of the sphere. Draw a straight line (in the full 3D space) from the selected origin to any other point on the sphere. Where the line crosses the plane, that’s your 2D vector representation of the other point. Under this visualization, the origin point should be mapped to a 2D “point at infinity” to make the mapping smooth. This mapping gives you a one-to-one conversion between 2D coordinate systems and points on the sphere.
You can create a new 2D coordinate system for sphere surface points using any point on the sphere as the origin. All of the resulting coordinate systems can be smoothly deformed into one another. (Points near the origin are always large, points on the opposite side of the sphere are always close to the 0,0,0, and the changes are smooth as you move the origin smoothly.)
Each choice of origin on the surface of the sphere (and therefore each 2D coordinate system) corresponds to two unit-length quaternions. You can see this as follows. Pick any choice of i,j,k values from a unit quaternion. There are now either 1 or 2 choices for what the real component of that quaternion might have been. If i,j,k alone have unit length, then there’s only one choice for the real component: zero. If i,j,k alone do not have unit length, then there are two choices for the real component since either a positive or a negative value can be used to make the quaternion unit length again.
Take the set of unit quaternions that have a real component close to zero. Consider the set of 2D coordinate systems created from those points. In this region, each coordinate system corresponds to two quaternions EXCEPT at the points where the quaternion’s real component is 0. This exceptional case prevents a one-to-one mapping between coordinate transformations and quaternion transformations.
As a result, there’s no “smooth” way to reduce the two-to-one mapping from quaternions to coordinate systems down to a one-to-one mapping. Any mapping would require either double-counting some quaternions or ignoring some quaternions. Since there’s a one-to-one mapping between coordinate systems and candidate origin points on the surface of the sphere, this means there is also no one-to-one mapping between quaternions and points on the sphere.
No matter what smooth mapping you choose from SU(2), unit quaternions, to SO(3), unit spheres, the mapping must do the equivalent of collapsing distinctions between quaternions with positive and negative real components. And so the double cover corresponds to the two sets of covers: one of positive-real-component quaternions over the sphere, and one of the negative-real-component quaternions over the sphere. Within each cover, there’s a smooth one-to-one conversion between quaternion-coordinates mappings, but across covers there is not.