Explicit homeomorphisim from Cylinder to Ellipsoid

Question

I am trying to create an explicit homeomorphisim from a cylinder A to a ellipsoid C defined by

A= {$(x,y,z) ∈ \mathbb R ^{3}, x^2+y^2= 4, |z|\leq$ 1}

C= {$(x,y,z) ∈ \mathbb R ^{3},x^2+y^2 + z^2 = 4, |z| \leq$ 1}

My answer was

$f(x,y,z)= ((4- y^2-z^2)^{(1/2)}, (4- x^2-z^2)^{(1/2)}, z) $

$f^{-1}(x,y,z)=((4- y^2)^{(1/2)}, (4- x^2)^{(1/2)}, z) $

But I believe this explicit homeomorphisim is wrong.

Answer 1

It's not just any ellipsoid! The very top and bottom parts have been cut off of a sphere with radius $2$.

Try finding a formula for the following equation:

given $f:B \to A, \,(x,y,z) \mapsto (rx,ry,z)$, for some $r \in \mathbb R_+$ that maps to $\{(x,y,z) \mid x^2+y^2=4\}$, which is basically just a sending a point along some vector that keeps the height constant, and finding where it intersects the cyllinder.

Keep in mind that for $(x,y,z) \in B$, we have that $x^2+y^2=4-z^2 \dots$ what should we multiply it by to get $4$? Why are we allowed to multiply by this value?