= Signature transition from an ellipsoid to an elliptic cylinder and a one-sheet hyperboloid
{title2=$A u^2+B v^2+C w^2=1,\quad A,B>0$}
After an <orthogonal diagonalization>, the sign of the third coefficient classifies the displayed <quadric surface>. If $C>0$, it is an <ellipsoid> with semiaxes $A^{-1/2},B^{-1/2},C^{-1/2}$. If $C=0$, it is an elliptic cylinder parallel to the $w$ axis, with elliptical transverse section. If $C<0$, it is a <one-sheet hyperboloid> with axis $w$ and waist ellipse $Au^2+Bv^2=1$. The continuous limit $C\to0$ sends the associated axial length scale to infinity. Repeated positive eigenvalues make the corresponding principal-axis choices nonunique within their eigenspace, without changing the surface.
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