Manifold or algebraic variety of dimension n in a space of dimension n+1
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.
^Lee, Jeffrey (2009). "Curves and Hypersurfaces in Euclidean Space". Manifolds and Differential Geometry. Providence: American Mathematical Society. pp. 143–188. ISBN 978-0-8218-4815-9.
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...
In relativity and in pseudo-Riemannian geometry, a null hypersurface is a hypersurface whose normal vector at every point is a null vector (has zero length...
In algebraic geometry, a Coble hypersurface is one of the hypersurfaces associated to the Jacobian variety of a curve of genus 2 or 3 by Arthur Coble....
geometry, a Dupin hypersurface is a submanifold in a space form, whose principal curvatures have globally constant multiplicities. A hypersurface is called a...
differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different...
variables x1, x2 and x3. For higher values of n, the level set is a level hypersurface, the set of all real-valued roots of an equation in n > 3 variables....
In algebraic geometry, given a projective algebraic hypersurface C {\displaystyle C} described by the homogeneous equation f ( x 0 , x 1 , x 2 , … ) =...
(quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension...
_{i=1}^{n}a_{i}{x_{i}}^{m}\ } for some given degree m. Such forms F, and the hypersurfaces F = 0 they define in projective space, are very special in geometric...
can be viewed as the equation of an hypersurface, and the solutions of the equation are the points of the hypersurface that have integer coordinates. This...
mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three...
In mathematics, a Dold manifold is one of the manifolds P ( m , n ) = ( S m × C P n ) / τ {\displaystyle P(m,n)=(S^{m}\times \mathbb {CP} ^{n})/\tau }...
coordinate system we may speak of coordinate planes. Similarly, coordinate hypersurfaces are the (n − 1)-dimensional spaces resulting from fixing a single coordinate...
an equivalent way to describe the shape operator (denoted by S) of a hypersurface, I I ( v , w ) = ⟨ S ( v ) , w ⟩ n = − ⟨ ∇ v n , w ⟩ n = ⟨ n , ∇ v w...
of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space...
edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles...
problem with two classes, a decision boundary or decision surface is a hypersurface that partitions the underlying vector space into two sets, one for each...
even-dimensional phase space of a mechanical system or constant-energy hypersurface, which, being codimension one, has odd dimension. Like symplectic geometry...
polynomial F is non-differentiable is called its associated tropical hypersurface, denoted V ( F ) {\displaystyle \mathrm {V} (F)} (in analogy to the vanishing...
Global Information