Quadric

In mathematics a quadric, or quadric surface, is any D-dimensional hypersurface defined as the 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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

for Q in M(C, D), P in CD and R in C.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 are the following:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Hypersurface: In mathematics, a hypersurface is some kind of submanifold....

Real projective space: In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. The case n = 1 gives the real projective line which is topologically equivalent to a circle. This case n = 2 is called the real projective plane, RP2. The space RPn is a compact, smooth manifold of dimension n. I...

Up to: :This article is about the sense of "up to" used in mathematics. See for other definitions....