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Representation of 3-sphere
- The four Euclidean coordinates for S3 are redundant since they are subject to the condition that.
- Due to the nontrivial topology of S3 it is impossible to find a single set of coordinates that cover the entire space.
- Some different choices of coordinates are given below.
- Hyperspherical coordinates
- Hopf coordinates
- Stereographic coordinates
Some quick notes followed:
- Stereographic projection can mapto, where N is the North Pole. This is one coordinate chart. If we pick another point (anyone but not North Pole) on S2 and project again, we can obtain another coordinate chart. With these two charts, we can cover the entire S2 space.
- The full space in 4D space is.
- The space of unit quaternion(which is a 3D surface embedded in 4D space) double covers the space of rotation SO(3). See this note for more information.
- Basically the unit quaternioncan represents a counterclockwise rotation by the anglearound the normalized axis. If we set the rotation to be, the corresponding quaternion representation will become; if we set the rotation to be, the corresponding quaternion representation will remain. That said, for each rotation we will have exactly two representations in quaternion.