4d Cartesian to Polar Transform

[chasehusky]

Howdy everyone,

I'm on a quest for something that is proving a bit elusive at the moment: a Cartesian to polar transform (along with its inverse) for $$\mathbb{R}^4$$. I'm well aware of how to derive the transform for both $$\mathbb{R}^2$$ and $$\mathbb{R}^3$$, as it is just a matter of looking at the angles made, with respect to the origin and appropriate coordinate axes, for the vector in question; e.g., for the $$\mathbb{R}^3$$ case: $$x = r \sin(\theta)\cos(\psi)$$, $$y = r \sin(\theta)\sin(\psi)$$, $$z = r \cos(\theta)$$. Unfortunately, as with all high-dimensional spaces, visualizing these angles becomes much trickier. If anyone can help me with this, I'd greatly appreciate it.

 

[chasehusky]

Well, after toying around a bit, it appears that the conversion would go something like: $$w = r\sin(\theta)\sin(\psi)\cos(\phi)$$, $$x = r\sin(\theta)\sin(\psi)\sin(\phi)$$, $$y = r\sin(\theta)\cos(\psi)$$, $$z = r\cos(\theta)$$.