# 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](https://en.wikipedia.org/wiki/3-sphere) space)?

* The four Euclidean coordinates for *S*3 are redundant since they are subject to the condition that $$x\_0^2 + x\_1^2 + x\_2^2 + x\_3^2 = 1$$**.**
* As a 3-dimensional manifold one should be able to parameterize *S*3 by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as [latitude](https://en.wikipedia.org/wiki/Latitude) and [longitude](https://en.wikipedia.org/wiki/Longitude)). 
* Due to the nontrivial topology of *S*3 it is *impossible to find a single set of coordinates* that cover the entire space. 
* Just as on the 2-sphere, one must use *at least* two [coordinate charts](https://en.wikipedia.org/wiki/Coordinate_chart). 
* Some different choices of coordinates are given below.
  * Hyperspherical coordinates
  * Hopf coordinates
  * Stereographic coordinates

References: [Coordinate systems on the 3-sphere](https://en.wikipedia.org/wiki/3-sphere#Coordinate_systems_on_the_3-sphere) space.

Some quick notes followed:

* [Stereographic projection](https://en.wikipedia.org/wiki/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](https://math.stackexchange.com/questions/1934931/bijection-between-so3-and-s2-times-s1) 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](https://www.gathering4gardner.org/g4g13gift/math/BickfordNeil-GiftExchange-WhyDoTheUnitQuaternionsDoubleCoverTheSpaceOfRotations-G4G13.pdf) 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.