Lengths of radii of ellipsoid cap

Question

Given $a,b,c$ and $h$. What would be the calculations for $d$ and $e$?

Answer 1

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$

$x=0\rightarrow \frac{y^2}{b^2}+\frac{z^2}{c^2}=1$

$z=h\rightarrow \frac{(y=d)^2}{b^2}=1-\frac{h^2}{c^2}$

Therefore :

$d=\pm\frac bc\sqrt{c^2-h^2}$

Similarly:

$y=0\rightarrow \frac{x^2}{a^2}+\frac{z^2}{c^2}=1$

$z=h\rightarrow \frac{(x=e)^2}{a^2}=1-\frac{h^2}{c^2}$

Therefore :

$e=\pm\frac ac\sqrt{c^2-h^2}$

Answer 2

From

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ and $z=h$, you draw the equation of the section,

$$\frac{x^2}{a^2\left(1-\dfrac{h^2}{c^2}\right)}+\frac{y^2}{b^2\left(1-\dfrac{h^2}{c^2}\right)}=1.$$