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# Representation of 3-sphere

Question: In

$S^2$

we have latitude-longitude representation, though a sphere does not have the same topology as a plane. Do we have something similar to represent the $S^3$

space (i.e. 3-sphere space)?- The four Euclidean coordinates for
*S*3 are redundant since they are subject to the condition that$x_0^2 + x_1^2 + x_2^2 + x_3^2 = 1$**.** - Due to the nontrivial topology of
*S*3 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 map$S^2 \setminus N$to$\mathbb{R}^2$, 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.
- $SO(3)$and$S^2 \times S^1$are not the same space (not homeomorphic). See this question at StackExchange.
- The full space in 4D space is$R^4 \times S^3 \times S^2 \times S^1$.
- The space of unit quaternion$S^3$(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 quaternion$q = (cos(\theta/2), x sin(\theta/2), y sin (\theta/2), z sin (\theta/2))$can represents a counterclockwise rotation by the angle$\theta$around the normalized axis$\mathbb{n} = (x, y, z)^\top$. If we set the rotation to be$\theta + 2\pi$, the corresponding quaternion representation will become$-q$; if we set the rotation to be$\theta + 4\pi$, the corresponding quaternion representation will remain$q$. That said, for each rotation we will have exactly two representations in quaternion.

Last modified 5mo ago