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 = [画像:[a h g; h b f; g f c]]

(2)

E = [画像:[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

[画像: |a-x h g; h b-x f; g f c-x|=0. ]

(7)

Also define

[画像: 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).

surface equation rho_3 rho_4 sgn(Delta) k

coincident planes x^2=0 1 1

ellipsoid (imaginary) (x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=-1 3 4 + 1

ellipsoid (real) (x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1 3 4 - 1

elliptic cone (imaginary) (x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=0 3 3 1

elliptic cone (real) (x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=0 3 3 0

elliptic cylinder (imaginary) (x^2)/(a^2)+(y^2)/(b^2)=-1 2 3 1

elliptic cylinder (real) (x^2)/(a^2)+(y^2)/(b^2)=1 2 3 1

elliptic paraboloid z=(x^2)/(a^2)+(y^2)/(b^2) 2 4 - 1

hyperbolic cylinder (x^2)/(a^2)-(y^2)/(b^2)=-1 2 3 0

hyperbolic paraboloid z=(y^2)/(b^2)-(x^2)/(a^2) 2 4 + 0

hyperboloid of one sheet (x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1 3 4 + 0

hyperboloid of two sheets (x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=-1 3 4 - 0

intersecting planes (imaginary) (x^2)/(a^2)+(y^2)/(b^2)=0 2 2 1

intersecting planes (real) (x^2)/(a^2)-(y^2)/(b^2)=0 2 2 0

parabolic cylinder x^2+2rz=0 1 3

parallel planes (imaginary) x^2=-a^2 1 2

parallel planes (real) x^2=a^2 1 2

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).

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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.

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Quadratic Surface

Cite this as:

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

Quadratic Surface

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