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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.

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