Ellipsoids and hyperboloids

S is a symmetric matrix. Its three principal axes are perpendicular, drawn as the gold, ice-blue and coral lines, and its eigenvalues on those axes, λ₁, λ₂ and λ₃, are on the three sliders. With all three positive, xᵀSx = 1 is an ellipsoid whose semi-axis along each principal axis is 1/√λ: drag a number up and the ellipsoid shortens along that axis. Drag λ₃ to zero and the ellipsoid stretches along the coral axis into an elliptic cylinder; take it negative and it opens into a hyperboloid of one sheet, with a narrow waist, reaching off into the distance. Take λ₂ through zero as well and, passing through a hyperbolic cylinder, it splits into two bowls opening away from each other, a hyperboloid of two sheets; with all three negative there is no point on it at all. Only the part inside a fixed sphere is shown: whatever reaches further is cut off on it, and a surface can lie entirely outside it, so check the corner to see whether there is nothing there. The corner solves for the three eigenvalues from the nine entries of S and counts how many are positive and how many negative: which surface it is depends only on how many are positive, negative and zero. Show the asymptotic cone xᵀSx = 0 to see the hyperboloids open out along it.