Ovoid

(Projective Geometry)

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Projective Geometry
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What is 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 are: 1. * Any line intersects in at most 2 points, 2. * The tangents at a point cover a hyperplane (and nothing more), and 3. * 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.

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