Scrabble?! HYPERSURFACES

Definitions of HYPERSURFACES in various dictionaries:

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.