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A locus in 3-dimensional space that can be represented in a Cartesian coordinate system by a polynomial equation in x, y and z of the second degree; that is, an equation of the form

ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy + 2ux + 2vy + 2wz + d = 0,

where the constants a, b, c, f, g and h are not all zero. When the equation represents a non-empty locus, it can be reduced by translation and rotation of axes to one of the following canonical forms, and hence classified:(i) Ellipsoid:(ii) Hyperboloid of one sheet:(iii) Hyperboloid of two sheets:(iv) Elliptic paraboloid:(v) Hyperbolic paraboloid:(vi) Quadric cone:(vii) Elliptic cylinder:(viii) Hyperbolic cylinder:(ix) Parabolic cylinder:(x) Pair of non-parallel planes:(xi) Pair of parallel planes:(xii) Plane:(xiii) Line:(xiv) Point:Forms (i), (ii), (iii), (iv) and (v) are the non-*degenerate quadrics.

(i) Ellipsoid:

(ii) Hyperboloid of one sheet:

(iii) Hyperboloid of two sheets:

(iv) Elliptic paraboloid:

(v) Hyperbolic paraboloid:

(vi) Quadric cone:

(vii) Elliptic cylinder:

(viii) Hyperbolic cylinder:

(ix) Parabolic cylinder:

(x) Pair of non-parallel planes:

(xi) Pair of parallel planes:

(xii) Plane:

(xiii) Line:

(xiv) Point:

Subjects: Mathematics.

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