Quadric
In mathematics a quadric, or quadric surface, is any D-dimensional hypersurface 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 the following:
| ellipsoid | <math>x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 \,<math> |
| spheroid (special case of ellipsoid) | <math> x^2/a^2 + y^2/a^2 + z^2/b^2 = 1 \,<math> |
| sphere (special case of spheroid) | <math>x^2/a^2 + y^2/a^2 + z^2/a^2 = 1 \,<math> |
| elliptic paraboloid | <math>x^2/a^2 + y^2/b^2 – z = 0 \,<math> |
| circular paraboloid | <math>x^2/a^2 + y^2/a^2 – z = 0 \,<math> |
| hyperbolic paraboloid | <math>x^2/a^2 – y^2/b^2 – z = 0 \,<math> |
| hyperboloid of one sheet | <math>x^2/a^2 + y^2/b^2 – z^2/c^2 = 1 \,<math> |
| hyperboloid 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> |
| elliptic cylinder | <math>x^2/a^2 + y^2/b^2 = 1 \,<math> |
| circular cylinder | <math>x^2/a^2 + y^2/a^2 = 1 \,<math> |
| hyperbolic cylinder | <math>x^2/a^2 – y^2/b^2 = 1 \,<math> |
| parabolic cylinder | <math>x^2 + 2y = 0 \,<math> |
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.
External links
- http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html, Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).