Hypersurface

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For differential geometry usage, see glossary of differential geometry and topology.
Ackley3.gif

Ackley's function of three variables, with time the 3rd variable.


In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface. Equivalently, the codimension of a hypersurface is one. For example, the n-sphere in Rn + 1 is called a hypersphere. Hypersurfaces occur frequently in multivariable calculus as level sets.

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, so not in fact be a submanifold in the strict sense. "Primal" is an old term for an irreducible hypersurface.

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