Diffeomorphism: surface x4 + y6 + z2 = 1

[blerg]

I'm trying to figure out what the surface x⁴ + y⁶ + z² = 1 looks like.
I want to say that it is diffeomorphic to the sphere because (x²)² + (y³)² + (z)² = 1
but i can't seems to actually construct the diffeomorphism (I am having problems with the x² being invertible).
Please let me know if I'm on the right track (if I'm even right)

 

[blerg]

I tried (x,y,z)|-->(x²,y³,z) but then only positive x are mapped to (and twice)
So then I tried (x,y,z)|-->(sgn(x)x²,y³,z) which is bijective but isn't smooth (it doesn't have a second derivative when x=0)
Any suggestions?

 

[Hurkyl]

You could just graph it. What are its cross sections in planes parallel to the x-y plane?

Or, you could just use your mapping to understand the points away from the points where where x=0, and find some other means of understanding the subspace of points of small x.

What is the application? Is it really not enough to simply have a homoeomorphism?

P.S. my instinct is to map radially from the origin. Descartes rule of signs proves this is well defined for all points not lying in a coordinate plane.,,,

 

[blerg]

I was hoping to find the euler characteristic for it, so i suppose a homeomorphism would be sufficient. In this case my second construction would be suitable correct?
I graphed the level curves and they are rounded off squares which is much like i anticipated. (also it suggests your idea of mapping out radially would probably work)