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Quadric surfaces

A quadric, or quadric surface, is a 2 -dimensional surface in 3-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates x, x, x, the general quadric is defined by the algebraic equation

∑ i, j = 1 3 x i Q i j x j + ∑ i = 1 3 P i x i + R = 0. {\displaystyle \sum _{i,j=1}^{3}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{3}P_{i}x_{i}+R=0.}

Quadric surfaces include ellipsoids (including the sphere), paraboloids, hyperboloids, cylinders, cones, and planes.