Ovoid (projective geometry)

In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid ${\displaystyle {\mathcal {O}}}$ are:

1. Any line intersects ${\displaystyle {\mathcal {O}}}$ in at most 2 points,
2. The tangents at a point cover a hyperplane (and nothing more), and
3. ${\displaystyle {\mathcal {O}}}$ 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.