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:

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.

In the case of ${\displaystyle |g\cap {\mathcal {O}}|=0}$, the line is called a passing (or exterior) line, if ${\displaystyle |g\cap {\mathcal {O}}|=1}$ the line is a tangent line, and if ${\displaystyle |g\cap {\mathcal {O}}|=2}$ the line is a secant line.

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