Editing Ellipsoid (section)

=== Steps of the construction ===
# Choose an ''ellipse'' {{mvar|E}} and a ''hyperbola'' {{mvar|H}}, which are a pair of [[focal conics]]: <math display="block">\begin{align}
E(\varphi) &= (a\cos\varphi, b\sin\varphi, 0) \\
H(\psi) &= (c\cosh\psi, 0, b\sinh\psi),\quad c^2 = a^2 - b^2
\end{align} </math> with the vertices and foci of the ellipse <math display="block">S_1 = (a, 0, 0),\quad F_1 = (c, 0, 0),\quad F_2 = (-c, 0, 0),\quad S_2 = (-a, 0, 0)</math> and a ''string'' (in diagram red) of length {{mvar|l}}.
# Pin one end of the string to [[vertex (curve)|vertex]] {{math|''S''<sub>1</sub>}} and the other to focus {{math|''F''<sub>2</sub>}}. The string is kept tight at a point {{mvar|P}} with positive {{mvar|y}}- and {{mvar|z}}-coordinates, such that the string runs from {{math|''S''<sub>1</sub>}} to {{mvar|P}} behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from {{mvar|P}} to {{math|''F''<sub>2</sub>}} runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance {{math|{{abs|''S''<sub>1</sub> ''P''}}}} over any hyperbola point is at a minimum. The analogous statement on the second part of the string and the ellipse has to be true, too.
# Then: {{mvar|P}} is a point of the ellipsoid with equation  <math display="block">\begin{align}
 &\frac{x^2}{r_x^2} + \frac{y^2}{r_y^2} + \frac{z^2}{r_z^2} = 1 \\
 &r_x = \tfrac{1}{2}(l - a + c), \quad
  r_y = {\textstyle \sqrt{r^2_x - c^2}}, \quad
  r_z = {\textstyle \sqrt{r^2_x - a^2}}.
\end{align}</math>
# The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.