Suppose that we have the ellipsoid $\frac{x^2}{\left( \frac{\sqrt{c}}{2}\right)^2}+\frac{y^2}{(\sqrt{c})^2}+\frac{z^2}{\left( \frac{\sqrt{c}}{3} \right)^2}=1$.
Then the semi-axes are these ones: $\frac{\sqrt{c}}{2}, \sqrt{c}, \frac{\sqrt{c}}{3}$.
What do the semi-axes represent? What information do we get from them?
Also why is the set $\{ (x,y,z,t) | t=4x^2+y^2, z=0\}$ a paraboloid of revolution?
Best Answer
They represent half maximum dimensions ( like diameter) separately in three dimensions. For a sphere three of them are same, for a surface of revolution two axes are same.
ELLIPSOID
Please note coefficients of x,y,z are of different degree in ellipsoid and paraboloid:
$$ y^2+z^2= 4 f \cdot ( x-t)\, for\, x > t ; $$
represents a paraboloid having all intersections with same constant focal length $f$