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.