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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 intersects every 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 a conic 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) cone are ruled surfaces, while the one-sheeted hyperboloid and hyperbolic paraboloid are doubly ruled surfaces.

A curve in which two arbitrary quadratic surfaces in arbitrary positions intersect cannot meet any plane in more than four points (Hilbert and Cohn-Vossen 1999, p. 24).


See also

Cone, Confocal Quadrics, Cubic Surface, Cylinder, Doubly Ruled Surface, Ellipsoid, Elliptic Cone, Elliptic Cylinder, Elliptic Paraboloid, Hyperbolic Cylinder, Hyperbolic Paraboloid, Hyperboloid, Plane, Quadratic, Quartic Surface, Ruled Surface, Surface

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 210-211, 1987.Hilbert, D. and Cohn-Vossen, S. "The Second-Order Surfaces." §3 in Geometry and the Imagination. New York: Chelsea, pp. 12-19, 1999.Mollin, R. A. Quadrics. Boca Raton, FL: CRC Press, 1995.

Referenced on Wolfram|Alpha

Quadratic Surface

Cite this as:

Weisstein, Eric W. "Quadratic Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/QuadraticSurface.html

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