Quadratic Surface

A second-order algebraic surface given by the general equation
ax^2+by^2+cz^2+2fyz+2gzx+2hxy+2px+2qy+2rz+d=0.
(1)
Quadratic surfaces are also called quadrics, and there are 17 standard-form types. A quadratic surface intersectsevery plane in a (proper or degenerate) conic section. In addition, the cone consisting of all tangents from a fixed point to a quadratic surface cuts every plane in a conic section, and the points of contact of this cone with the surface form aconic section (Hilbert and Cohn-Vossen 1999, p. 12).
Examples of quadratic surfaces include the cone, cylinder, ellipsoid, elliptic cone, elliptic cylinder, elliptic hyperboloid,elliptic paraboloid, hyperbolic cylinder, hyperbolic paraboloid, paraboloid, sphere, and spheroid.

Define
e=[a h g; h b f; g f c]
(2)
E=[a h g p; h b f q; g f c r; p q r d]
(3)
rho_3=rank e
(4)
rho_4=rank E
(5)
Delta=det E,
(6)
and k_1, k_2, as k_3 are the roots of
|a-x h g; h b-x f; g f c-x|=0.
(7)
Also define
k={1 if the signs of nonzero ks are the same; 0 otherwise.
(8)
Then the following table enumerates the 17 quadrics and their properties (Beyer 1987).
surfaceequationrho_3rho_4sgn(Delta)k
coincident planesx^2=011
ellipsoid (imaginary)(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=-134+1
ellipsoid (real)(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=134-1
elliptic cone (imaginary)(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=0331
elliptic cone (real)(x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=0330
elliptic cylinder (imaginary)(x^2)/(a^2)+(y^2)/(b^2)=-1231
elliptic cylinder (real)(x^2)/(a^2)+(y^2)/(b^2)=1231
elliptic paraboloidz=(x^2)/(a^2)+(y^2)/(b^2)24-1
hyperbolic cylinder(x^2)/(a^2)-(y^2)/(b^2)=-1230
hyperbolic paraboloidz=(y^2)/(b^2)-(x^2)/(a^2)24+0
hyperboloid of one sheet(x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=134+0
hyperboloid of two sheets(x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=-134-0
intersecting planes (imaginary)(x^2)/(a^2)+(y^2)/(b^2)=0221
intersecting planes (real)(x^2)/(a^2)-(y^2)/(b^2)=0220
parabolic cylinderx^2+2rz=013
parallel planes (imaginary)x^2=-a^212
parallel planes (real)x^2=a^212
Of the non-degenerate quadratic surfaces, the elliptic (and usual) cylinder, hyperbolic cylinder, elliptic (and usual) coneare ruled surfaces, while the one-sheeted hyperboloid and hyperbolic paraboloid are doubly ruled surfaces.

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