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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 O {\displaystyle {\mathcal {O}}} are:
In projective geometry an oval is a point set in a plane that is defined by incidence properties. The standard examples are the nondegenerate conics. However, a conic is only defined in a pappian plane, whereas an oval may exist in any type of projective plane. In the literature, there are many criteria which imply that an oval is a conic, but ...
The term oval when used to describe curves in geometry is not well-defined, except in the context of projective geometry. Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. In particular, these are common traits ...
This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.
The term "projective geometry" is used sometimes to indicate the generalised underlying abstract geometry, and sometimes to indicate a particular geometry of wide interest, such as the metric geometry of flat space which we analyse through the use of homogeneous coordinates, and in which Euclidean geometry may be embedded (hence its name ...
Print/export Download as PDF; Printable version; In other projects ... Example geometry Example finite subgroups; O(3) Full symmetry of the sphere
An ovoid of () (a symplectic polar space of rank n) would contain + points. However it only has an ovoid if and only n = 2 {\displaystyle n=2} and q is even. In that case, when the polar space is embedded into P G ( 3 , q ) {\displaystyle PG(3,q)} the classical way, it is also an ovoid in the projective geometry sense.
An example of an ovoid is the elliptic quadric, the set of zeros of the quadratic form x 1 x 2 + f(x 3, x 4), where f is an irreducible quadratic form in two variables over GF(q). [f(x,y) = x 2 + xy + y 2 for example]. If q is an odd power of 2, another type of ovoid is known – the Suzuki–Tits ovoid. Theorem. Let q be a positive integer, at ...
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