In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension . Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid are: Any line intersects in at most 2 points, The tangents at a point cover a hyperplane (and nothing more), and contains no lines. Property 2) excludes degenerated cases (cones,...). Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...). An ovoid is the spatial analog of an oval in a projective plane. An ovoid is a special type of a quadratic set. Ovoids play an essential role in constructing examples of Möbius planes and higher dimensional Möbius geometries. Definition of an ovoid In a projective space of dimension a set of points is called an ovoid, if (1) Any line meets in at most 2 points. In the case of , the line is called a passing (or exterior) line, if the line is a tangent line, and if the line is a secant line. (2) At any point the tangent lines through cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension ). (3) contains no lines. From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because For an ovoid and a hyperplane , which contains at least two points of , the subset is an ovoid (or an oval, if ) within the hyperplane . For finite projective spaces of dimension (i.e., the point set is finite, the space is pappian), the following result is true: If is an ovoid in a finite projective space of dimension , then . (In the finite case, ovoids exist only in 3-dimensional spaces.) In a finite projective space of order (i.e. any line contains exactly points) and dimension any pointset is an ovoid if and only if and no three points are collinear (on a common line). Replacing the word projective in the definition of an ovoid by affine, gives the definition of an affine ovoid. If for an (projective) ovoid there is a suitable hyperplane not intersecting it, one can call this hyperplane the hyperplane at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to . Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space. Examples In real