In mathematics a quadric, orquadric surface', is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). If the space coordinates are {x1,x2,...xD}, then the general quadric in such a space is defined by the algebraic equation
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:
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.
External links
http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html, Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).
Quadric fitting is usually formulated as a nonlinear least squares problem, which is solved either using iterative methods for minimizing a nonlinear function or casting it as an eigenvalue problem which is solved directly and no approximate values for the parameters are needed.
31], the reconstruction of objects having quadric patches is improved by incorporating geometric constraints that fix feature relationships between the patches.
7], the parameters of a quadric are estimated from two quadratic curves fitted to the measured image coordinates of two stripes projected onto the object surface.
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