Encyclopedia > Quadric

  Article Content

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:



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Father Damien

... (leprosy) who lived on the island of Molokai, Hawaii. He was born in Tremeloo, Belgium, the son of a farmer. He entered the novitiate of the Fathers of the Sacred ...

 
 
 
This page was created in 67.4 ms