# Hypersurface

In geometry, a **hypersurface** is a generalization of the concepts of hyperplane, plane curve, 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.[1]
Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally.

A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface.

For example, the equation

defines an algebraic hypersurface of dimension *n* − 1 in the Euclidean space of dimension *n*. This hypersurface is also a smooth manifold, and is called a hypersphere or an (*n* – 1)-sphere.