Quadric

A quadric is a second-degree algebraic variety defined by a polynomial equation of degree two in several variables. In three-dimensional space, a quadric surface is described by a general second-degree equation that may include squared terms, cross terms, linear terms, and a constant.

The quadratic part of the defining equation can be represented by a symmetric matrix, and the surface is determined up to an affine change of coordinates. Quadrics are classified as nondegenerate when the associated quadratic form has full rank; degeneracies include objects such as cylinders, cones, and pairs of planes. By diagonalizing the quadratic form, quadrics can be brought to canonical forms that reveal their geometric type, including ellipsoids, hyperboloids, paraboloids, cones, and cylinders.

Examples of quadric surfaces include:

- Sphere: x^2 + y^2 + z^2 = r^2

- Ellipsoid: x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

- Hyperboloid of one sheet: x^2/a^2 + y^2/b^2 - z^2/c^2 = 1

- Hyperboloid of two sheets: -x^2/a^2 - y^2/b^2 + z^2/c^2 = 1

- Paraboloid: z = x^2/a^2 + y^2/b^2

- Cone: x^2/a^2 + y^2/b^2 - z^2/c^2 = 0

- Circular cylinder: x^2 + y^2 = r^2

In projective geometry, a quadric is the zero locus of a homogeneous degree-two polynomial and generalizes conic sections from the plane to higher dimensions. Quadrics arise in many areas of mathematics and its applications, including computer graphics, computer-aided design, and physics, where they model smooth, symmetric surfaces and their intersections.