ALGEBRAIC SURFACE

in three-dimensional space, a surface the equation of which is f(x, y, x) = 0, with f(x, y, x) a polynomial in x, y, x. 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 form Ax2 + By2 + Cz2 + Dx + Ey + Fz = G. If A, B, C are all not zero, the equation can generally be simplified to the form ax2 + by2 + cz2 = 1. This surface is called an ellipsoid (q.v.) if a, b, and c are positive. If one of the coefficients is negative, the surface is a hyperboloid (q.v.) 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 at which the curvatures in two mutually perpendicular planes are of opposite signs). 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 = x2 + y2 and z = x2 - y2, respectively. Through every point of a quadric pass two straight lines (possibly imaginary) 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.