Quadric surface ($x^{T}Ax+b^{T}x+c=0$)

A quadric surface in three-dimensional Euclidean space is the zero set of a real polynomial of degree two. Its quadratic part can be written with a symmetric matrix $A$, since the antisymmetric matrix part contributes zero to $x^{T}Ax$. An orthogonal diagonalization and a translation expose its geometry: examples include an ellipsoid, an elliptic cone, and a hyperboloid. This affine geometric description allows singular surfaces such as a cone's vertex.

A projective quadric is the zero set of a nonzero homogeneous quadratic polynomial in projective space. Over a field of characteristic different from two, represent it by a symmetric matrix $B$, up to a nonzero scalar. The gradient at a representative $v$ is $2Bv$, so the singular locus is the projectivization of $\ker B$. Thus the quadric is nonsingular exactly when $B$ is invertible. In the complex projective plane, ranks three, two, and one give respectively a smooth projective conic, two distinct projective lines, and a double projective line.

The equation of a quadric surface after orthogonal diagonalization of its symmetric matrix and appropriate translation. The spectral theorem for real symmetric matrices removes mixed quadratic terms, and completing squares removes linear terms in directions with nonzero eigenvalues. Linear terms in zero-eigenvalue directions can remain and generate elliptic paraboloids or hyperbolic paraboloids. The signs, zero coefficients and constant distinguish ordinary surfaces, degenerate conics in plane sections, and empty real loci.

A quadric surface obtained by extending a parabola along a fixed direction outside its plane. For example, $x=c-y^2/(4c)$ with arbitrary $z$ is a parabolic cylinder parallel to the $z$ axis. Each cross-section of fixed $z$ has vertex $(c,0)$ and focus $(0,0)$ in its $xy$ plane. It occurs as a coordinate surface in parabolic cylindrical coordinates.

For a real symmetric matrix $A$, every stationary squared distance $x^{T}x$ on $x^{T}Ax=1$ occurs in a positive-eigenvalue eigenspace. The Lagrange multiplier equation is $x=\Lambda Ax$; multiplying by $x^T$ gives $\Lambda =|x{|}^2>0$, hence $Ax=|x{|}^{-2}x$. If $A$ is a positive-definite matrix, the nearest and farthest squared distances are $1/{\lambda }_{\max }$ and $1/{\lambda }_{\min }$. With a repeated eigenvalue, the stationary set may contain an entire sphere within its eigenspace. For an indefinite or singular form the surface may be unbounded, and a stationary point need not be a global extremum.

Let $A$ be a real positive-definite matrix and let the columns of $U$ be an orthonormal basis of a two-dimensional linear subspace in three-dimensional Euclidean space. Points in that central plane are $x=Uq$, with $|x{|}^2=|q{|}^2$. Restricting the ellipsoid $x^{T}Ax=1$ gives $q^T(U^{T}AU)q=1$. The compressed matrix $U^{T}AU$ is a positive-definite matrix, and its extreme eigenvalues give the distance formulas in the title by the Rayleigh quotient. The two extremal axes are the images under $U$ of the corresponding eigenvectors. This restriction reduces a problem with two Lagrange multipliers to a two-dimensional quadratic form.

For $a,b>0$, this quadric surface is independent of its third coordinate and is formed by translating a hyperbola along that direction. Its two connected components correspond to $u\ge a$ and $u\le -a$. In elliptic cylindrical coordinates, fixing a generic angular coordinate selects a half-branch of such a cylinder because the signs of both planar coordinates are fixed by that angular value.

For $a,b>0$, an elliptic cylinder is a quadric surface with the displayed equation in suitable Cartesian coordinates, independent of the third coordinate. It is formed by translating an ellipse parallel to a line perpendicular to its plane. The generator direction is the zero-eigenvalue direction of its quadratic part, while the transverse semiaxes are $a,b$. It is circular if $a=b$.

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 $A{u}^2+B{v}^2=1$. The continuous limit $C\to 0$ 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.

An elliptic cone is a real quadric surface consisting of the two nappes $u=\pm \sqrt{(b{v}^2+c{w}^2)/a}$, joined at the vertex. Its intersections with planes $u=u_0\ne 0$ are ellipses. It is the zero level of an indefinite quadratic form of signature $(1,2)$. Changing the level to a positive value produces a two-sheeted hyperboloid; changing it to a negative value produces a one-sheet hyperboloid, when the positive coefficient is assigned to the axial coordinate $u$.