Quadric Surface

quadric surface [′kwä·drik ′sər·fəs]

(mathematics)

A surface whose equation is a second-degree algebraic equation.

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Quadric Surface 

a surface such that the rectangular Cartesian coordinates of its points satisfy the second-degree algebraic equation

(*) a_{1 1}x² + a_{2 2}y² + a_{3 3}z² + 2a_{1 2}xy + 2a_{2 3}xy + 2a_{1 3}xz + 2a_{1 4}x + 2a_{2 4}y + 2a_{3 4}z + a_{4 4} = 0

If equation (*) does not define a real geometric locus, in order to preserve generality we say that it defines an imaginary quadric surface. Depending on the values of the coefficients of equation (*), it can be transformed by means of translation and rotation of axes into one of the 17 canonical forms listed below. A specific class of quadric surfaces corresponds to each of these forms. Five principal types of such surfaces are distinguished:

(1) Ellipsoids:

(2) Hyperboloids:

(3) Paraboloids (p > 0, q> 0):

(4) Quadric cones:

(5) Quadric cylinders:

The quadric surfaces listed above are classified as irreducible quadric surfaces. The reducible quadric surfaces are

The invariants of the general equation of a quadric surface— that is, expressions that are formed from the coefficients of equation (*) and that do not change in value upon translation or rotation of the axes—are of great importance in the investigation of the general equation of a quadric surface. For example, if

then equation (*) defines degenerate quadric surfaces: quadric cones, quadric cylinders, and reducible quadric surfaces. If the determinant is

then the surface has a unique center of symmetry called the center of the quadric surface and is said to be central. If δ = 0, then the surface either does not have a center or has infinitely many centers.

Affine and projective classifications have been established for quadric surfaces. Two quadric surfaces are considered to belong to the same affine class if they can be transformed into each other by an affine transformation; projective classes of quadric surfaces are defined in a similar manner. To each affine class there corresponds one of the 17 canonical forms of the equation of a quadric surface. Projective transformations permit the relation between different affine classes of quadric surfaces to be established because for these transformations the special role of infinitely distant elements of space vanishes. For example, hyperboloids of two sheets and ellipsoids, which are different from the affine point of view, belong to the same projective class of quadric surfaces.

REFERENCES

Aleksandrov, P. S. Lektsii po analiticheskoi geometrii…. Moscow, 1968.
Il’in, V. A., and E. G. Pozniak. Analiticheskaia geometriia, 2nd ed. Moscow, 1971.
Efimov, N. V. Kvadratichnye formy i matritsy, 5th ed. Moscow, 1972.

A. B. IVANOV