Quadric

In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). If the space coordinates are <math>\{x_1, x_2, ... x_D\}</math>, then the general quadric in such a space is defined by the algebraic equation

<math>

\sum_{i,j=1}^D Q_{i,j} x_i x_j + \sum_{i=1}^D P_i x_i + R = 0 </math> for a specific choice of Q, P and R.

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

<math>

\pm {x^2 \over a^2} \pm {y^2 \over b^2} \pm {z^2 \over c^2}=1 </math>

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 are following:

• ellipsoid: <math>x^2/a^2 + y^2/b^2 + z^2/c^2 = 1</math>
• spheroid - special case of ellipsoid
• sphere - special case of spheroid: <math>x^2/a^2 + y^2/a^2 + z^2/a^2 = 1</math>
• elliptic hyperboloid
• elliptic paraboloid: <math>x^2/a^2 + y^2/b^2 - z = 0</math>
• hyperbolic paraboloid of one sheet: <math>x^2/a^2 + y^2/b^2 - z^2/c^2 = 1</math>
• hyperbolic paraboloid of two sheets: <math>x^2/a^2 - y^2/b^2 - z^2/c^2 = 1</math>
• cone: <math>x^2/a^2 - y^2/b^2 - z^2/c^2 = 0</math>
• cylinder: <math>x^2/a^2 + y^2/b^2 = 1</math>

In real projective space, the ellipsoid, the elliptic hyperboloid, and the elliptic paraboloid are not different from each other; 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.

External links:

• http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html, 16 Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).