Quadric

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In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of zeros of a quadratic polynomial. In coordinates \{x_0, x_1, x_2, \ldots, x_D\}, the general quadric is defined by the algebraic equation [1]

\sum_{i,j=0}^D Q_{i,j} x_i x_j + \sum_{i=0}^D P_i x_i + R = 0

where Q is a (D + 1)-dimensional matrix and P is a (D + 1)-dimensional vector and R a constant. The values Q, P and R are often taken to be real numbers or complex numbers, but in fact, a quadric may be defined over any ring. In general, the locus of zeros of a set of polynomials is known as an algebraic variety, and is studied in the branch of algebraic geometry.

A quadric is thus an example of an algebraic variety. Every projective variety can be shown to be isomorphic to the intersection of a set of quadrics. For the projective theory see quadric (projective geometry).

The normalized equation for a three-dimensional (D=3) quadric centred at the origin (0,0,0) is:

\pm {x^2 \over a^2} \pm {y^2 \over b^2} \pm {z^2 \over c^2}=1.

Via translations and rotations every quadric can be transformed to one of several "normalized" forms. In three-dimensional Euclidean space there are 16 such normalized forms, and the most interesting, the nondegenerate forms are given below. The remaining forms are called degenerate forms and include planes, lines, points or even no points at all. [2]

Ellipsoid

Elliptic Paraboloid

Hyperbolic Paraboloid

Hyperboloid of One Sheet

Hyperboloid of Two Sheets

Cone

Elliptic Cylinder

Hyperbolic Cylinder

Parabolic Cylinder
ellipsoid {x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,
    spheroid (special case of ellipsoid)   {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,
       sphere (special case of spheroid) {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,
elliptic paraboloid {x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,
    circular paraboloid {x^2 \over a^2} + {y^2 \over a^2} - z = 0 \,
hyperbolic paraboloid {x^2 \over a^2} - {y^2 \over b^2} - z = 0 \,
hyperboloid of one sheet {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,
hyperboloid of two sheets {x^2 \over a^2} - {y^2 \over b^2} - {z^2 \over c^2} = 1 \,
cone {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,
elliptic cylinder {x^2 \over a^2} + {y^2 \over b^2} = 1 \,
    circular cylinder {x^2 \over a^2} + {y^2 \over a^2} = 1 \,
hyperbolic cylinder {x^2 \over a^2} - {y^2 \over b^2} = 1 \,
parabolic cylinder x^2 + 2ay = 0 \,

In real projective space, the ellipsoid, the elliptic paraboloid and the hyperboloid of two sheets are equivalent to each other up to a projective transformation; the two hyperbolic paraboloids are not different from each other (these are ruled surfaces); the cone and the cylinder are not different from each other (these are "degenerate" quadrics, since their Gaussian curvature is zero).

In complex projective space all of the nondegenerate quadrics become indistinguishable from each other.

  1. ^ [1], Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).
  2. ^ Stewart Venit and Wayne Bishop, Elementary Linear Algebra (fourth edition), International Thompson Publishing, 1996.
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