In geometry, a hypersurface is a generalization of the concepts of hyperplane and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space.
Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation. When this equation is a multivariate polynomial the hypersurface is an algebraic hypersurface.
For example, the equation
defines an algebraic hypersurface of dimension n − 1 in the Euclidean space of dimension n. This hypersurface is a submanifold of the Euclidean space, which is also called an hypersphere or a (n – 1)-sphere.
Hypersurfaces occur frequently in multivariable calculus as level sets.
In Rn, every closed hypersurface is orientable. Every connected compact hypersurface is a level set, and separates Rn in two connected components, which is related to the Jordan–Brouwer separation theorem.
In algebraic geometry, a hypersurface in projective space of dimension n is an algebraic set (algebraic variety) that is purely of dimension n − 1. It is then defined by a single equation f(x1, x2, ..., xn) = 0, a homogeneous polynomial in the homogeneous coordinates.
Thus, it generalizes those algebraic curves f(x1, x2) = 0 (dimension one), and those algebraic surfaces f(x1, x2, x3) = 0 (dimension two), when they are defined by homogeneous polynomials.
A hypersurface may have singularities, and hence is not necessarily a submanifold in the strict sense. "Primal" is an old term for an irreducible hypersurface.