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 [tex]\mathbb{R}^4[/tex]. I'm well aware of how to derive the transform for both [tex]\mathbb{R}^2[/tex] and [tex]\mathbb{R}^3[/tex], 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 [tex]\mathbb{R}^3[/tex] case: [tex]x = r \sin(\theta)\cos(\psi)[/tex], [tex]y = r \sin(\theta)\sin(\psi)[/tex], [tex]z = r \cos(\theta)[/tex]. 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: [tex]w = r\sin(\theta)\sin(\psi)\cos(\phi)[/tex], [tex]x = r\sin(\theta)\sin(\psi)\sin(\phi)[/tex], [tex]y = r\sin(\theta)\cos(\psi)[/tex], [tex]z = r\cos(\theta)[/tex].