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In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections.
The two families of lines on a smooth (split) quadric surface. In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine ...
For example, a hyperboloid of one sheet is a quadric surface in ruled by two different families of lines, one line of each passing through each point of the surface; each family corresponds under the Plücker map to a conic section within the Klein quadric in .
Similarly, [3] if C is a smooth curve on the quadric surface P 1 ×P 1 with bidegree (d 1,d 2) (meaning d 1,d 2 are its intersection degrees with a fiber of each projection to P 1), since the canonical class of P 1 ×P 1 has bidegree (−2,−2), the adjunction formula shows that the canonical class of C is the intersection product of divisors ...
A surface of general type with the same Betti numbers as a minimal surface not of general type must have the Betti numbers of either a projective plane P 2 or a quadric P 1 ×P 1. Shavel (1978) constructed some "fake quadrics": surfaces of general type with the same Betti numbers as quadrics. Beauville surfaces give further examples.
Attempting to rotate a QGA quadric surface may result in a different type of quadric surface, or a quadric surface that is rotated and distorted in an unexpected way. Attempting to rotate a QGA point may produce a value that projects as the expected rotated vector, but the produced value is generally not a correct embedding of the rotated vector.
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane section of a paraboloid made by a plane parallel to the axis of symmetry is a parabola.
For more examples see the list of algebraic surfaces. The first five examples are in fact birationally equivalent. That is, for example, a cubic surface has a function field isomorphic to that of the projective plane, being the rational functions in two indeterminates. The Cartesian product of two curves also provides examples.
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