algebraic surface

in threedimensional space, a surface the equation of which is f(x, y, z) = 0, with f(x, y, z) a polynomial in x, y, z. The order of the surface is the degree of the polynomial equation. If the surface is of the first order, it is a plane. If the surface is of order two, it is called a quadric surface. By rotating the surface, its equation can be put in the formAx^{2} + By^{2} + Cz^{2} + Dx + Ey + Fz = G.If A, B, C are all not zero, the equation can generally be simplified to the formax^{2} + by^{2} + cz^{2} = 1.This surface is called an ellipsoid if a, b, and c are positive. If one of the coefficients is negative, the surface is a hyperboloid of one sheet; if two of the coefficients are negative, the surface is a hyperboloid of two sheets. A hyperboloid of one sheet has a saddle point (a point on a curved surface shaped like a saddle at which the curvatures in two mutually perpendicular planes are of opposite signs, just like a saddle is curved up in one direction and down in another).If A, B, C are possibly zero, then cylinders, cones, planes, and elliptic or hyperbolic paraboloids may be produced. Examples of the latter are z = x^{2} + y^{2} and z = x^{2} − y^{2}, respectively. Through every point of a quadric pass two straight lines lying on the surface. A cubic surface is one of order three. It has the property that 27 lines lie on it, each one meeting 10 others. In general, a surface of order four or more contains no straight lines.
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Universalium. 2010.
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