Quadric (algebraic geometry)
In algebraic geometry, a quadric, also known as a quadric hypersurface or a quadratic hypersurface, is a hypersurface defined by a polynomial equation of degree two. That is, it is the set of points in a projective or affine space that satisfy a quadratic equation.
More specifically, a quadric in n-dimensional projective space (denoted as Pn) over a field K is defined by a homogeneous polynomial equation of degree two in n+1 variables. Similarly, in n-dimensional affine space (An) over a field K, a quadric is defined by a polynomial equation of degree two in n variables.
Quadrics are generalizations of conic sections (circles, ellipses, parabolas, and hyperbolas) to higher dimensions. In three dimensions, they are known as quadric surfaces and include familiar shapes like ellipsoids, paraboloids, hyperboloids, and cones.
The classification of quadrics depends on the underlying field K and the space in which they are embedded (affine or projective). The classification is generally up to projective or affine transformations. Important invariants used in the classification include rank and signature. The rank of a quadric is related to the rank of the associated symmetric matrix. The signature, when the field allows it, provides further differentiation between quadrics.
Quadrics are fundamental objects of study in algebraic geometry and have applications in various fields, including computer graphics, physics, and cryptography. The study of quadrics often involves linear algebra, specifically the study of quadratic forms and their associated symmetric matrices. Understanding the intersections of quadrics with lines, planes, and other quadrics is a crucial aspect of their analysis.