Hypersurface is a related term of hyperplane.

As nouns the difference between hyperplane and hypersurface

is that hyperplane is an n-dimensional generalization of a plane; an affine subspace of dimension n-1 that splits an n-dimensional space. (In a one-dimensional space, it is a point; in two-dimensional space it is a line; in three-dimensional space, it is an ordinary plane. while hypersurface is a n-dimensional surface in a space (often a Euclidean space) of dimension n+1.