# Euler Angles on $\mathbb{R}P^3$?

I am curious if anyone can point me to a reference where the Euler angle coordinates are visualized as a parametrization of $\mathbb{R}P^3$. Bonus points if there are visualization aids for the quaternion and axis-angle parametrization as well.

Thanks!

• Presumably you're asking about projective space because you're aware that $\mathbf{RP}^{3}$ is diffeomorphic to the rotation group $SO(3)$...? Commented Jun 26, 2017 at 20:44
• Yes, I am speaking of Real Projective 3-Space. As I understand it, unit quaternions provide a Cartesian parametrization of $S^3$, which we chop in half and identify antipodal points at the equator to get $\mathbb{R}P^3$.
– Evan
Commented Jun 26, 2017 at 20:45
• Not sure what you mean by "at the equator." We regard a vector $(x,y,z)$ in $\mathbb R^3$ as a "pure" quaternion, zero real part, $v =xi + yj + zk.$ Given a unit quaternion $q,$ with conjugate $\bar{q},$ our rotated vector is $q v \bar{q}.$ Short proof that this is also a pure quaternion. Meanwhile, $q$ and $-q$ give precisely the same rotation. Commented Jun 26, 2017 at 21:31
• I'm talking about the equator of $S^3$, which is a 2-sphere, $S^2$. Unit quaternions (with non zero real part) provide a Cartesian cover of $S^3$.
– Evan
Commented Jun 26, 2017 at 21:32
• My earlier comment was unclear. The questions were: 1. Are you happy with Euler angles and axis-angle descriptions of $SO(3)$? 2. Are you happy with the axis-angle description of the diffeomorphism between $SO(3)$ and $\mathbf{RP}^{3}$? and 3. Are you (therefore) looking for a description in the language of projective space rather than in terms of rotation matrices/unit quaternions? Commented Jun 29, 2017 at 0:33