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

Related Topics:
Mathematics - Hypersurface

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

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

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

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pm {x^2 over a^2} pm {y^2 over b^2} pm {z^2 over c^2}=1

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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:

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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).

Related Topics:
Real projective space - Up to - Projective transformation - Ruled surface - Gaussian curvature

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In complex projective space all of the nondegenerate quadrics become indistinguishable from each other.

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