Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$.

Question

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$.

I've been looking for an diffeomorphism between a sphere in $\mathbb R^3$ and an ellipsoid of the form $$\{ (x,y,z) \in \mathbb R^3 \mid \frac {x^2} {a^2} + \frac {y^2} {b^2} + \frac {z^2} {c^2} = 1\}$$

I know from linear algebra such an ellipsoid can be represented as a matrix product $x^T A x = 1$, but how can this be used to get a diffeomorphism ?

Can such a diffeomorphism be found for a general ellipsoid ?

Answer 1

Denote by $E$ your ellipsoid, and by $S^2$ the sphere $\{x \in \mathbf R^3 \mid x_1^2 + x_2^2 + x_3^2 = 1\}$. Define a map $f \colon E \to S^2$ by $$ f(x) = \left(\frac{x_1}a, \frac{x_2}b, \frac{x_3}c \right) $$ Then $f$ is a diffeomorphism.