hypersurface
Sign in to saveIn 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 , which is embedded in an ambient space of dimension , 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, at least locally (near every point), and sometimes globally.
~7 min read
Encyclopedic overview
7 sectionsContents
- Smooth hypersurface
- Affine algebraic hypersurface {{anchor|Algebraic hypersurface}}
- Properties
- Real and rational points
- Projective algebraic hypersurface{{anchor|projective hypersurface}}
- See also
- References
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 , which is embedded in an ambient space of dimension , 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, 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.
Excerpted from Wikipedia’s “hypersurface” article, available under the CC BY-SA 4.0 licence.