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hypersurfaces

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Definitions of hypersurfaces in various dictionaries:

HYPERSURFACES - In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic...

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Plural form of hypersurface.

Hypersurfaces description
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. * 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* * * * * x * * 1 * * * 2 * * * + * * x * * 2 * * * 2 * * * + * ⋯ * + * * x * * n * * * 2 * * * − * 1 * = * 0 * * * {\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}-1=0} * 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.

Hypersurfaces description

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.
* 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*
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* {\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}-1=0}
* 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.

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