a pair of foci is a constant. Many theorems relating to planar conic sections also extend to spherical conics. Intersection of a sphere with a more general surface If a sphere is intersected by another surface, there may be more complicated spherical curves. Example sphere – cylinder The intersection of the sphere with equation and the cylinder with equation is not just one or two circles. It is the solution of the non-linear system of equations (see implicit curve and the diagram) Generalizations Ellipsoids An ellipsoid is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an affine transformation. An ellipsoid bears the same relationship to the sphere that an ellipse does to a circle. Dimensionality Spheres can be generalized to spaces of any number of dimensions. For any natural number , an -sphere, often denoted , is the set of points in ()-dimensional Euclidean space that are at a fixed distance from a central point of that space, where is, as before, a positive real number. In particular: : a 0-sphere consists of two discrete points, and : a 1-sphere is a circle of radius r : a 2-sphere is an ordinary sphere : a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for are sometimes called hyperspheres. The -sphere of unit radius centered at the origin is denoted and is often referred to as "the" -sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space. In topology, the -sphere is an example of a compact topological manifold without boundary. A topological sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere (an exotic sphere). The sphere is the inverse image of a one-point set under the continuous function , so it is closed; is also bounded, so it is compact by the Heine–Borel theorem. Metric spaces More generally, in a metric space , the sphere of center and radius is the set of points such that . If the center is a distinguished point that is considered to be the origin of , as in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a