Let $$S^2=\{ (x,y,z) | x^2+y^2+z^2=1 \} $$ and $$E=\{ (x,y,z) | \frac{x^2}{a^2} +\frac{y^2}{b^2}+\frac{z^2}{c^2} =1 \} $$ Let the map $f: S^2 \to E$ be defined as $$f(x,y,z)=(ax,by,cz)$$ I don't understand how this map takes an element of $S^2$ and sends it to $E$, visually I can see that this map just takes a point in the unit sphere and scales it, but algebraically I can't see how $f$ does that.
$f$ takes a point $(x,y,z)$ such that $x^2+y^2+z^2=1$ to the point $(ax,by,cz)$ such that $$\frac{a^2x^2}{a^2} +\frac{b^2y^2}{b^2}+\frac{c^2z^2}{c^2} =x^2+y^2+z^2=1$$ But that's just the unit circle, I obviously have made a terrible mistake in my reasoning but I can't spot it, can you explain?
My apologies if this is a trivial question.