In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center. It is a manifold of codimension one, i.e. with one dimension less than that of the ambient space. As the radius increases the curvature of the hypersphere decreases; in the limit a hypersphere approaches the zero curvature of a hyperplane. Hyperplanes and hyperspheres are examples of hypersurfaces.
Some spheres are not hyperspheres: if S is a sphere in Em where m < n and the space has n dimensions, then S is not a hypersphere. Similarly, any n-sphere in a proper flat is not a hypersphere. For example, a circle is not a hypersphere in three-dimensional space, but it is a hypersphere in the plane.