PARABOLOID, in geometry, a non centric open surface of the second degree.
Its vertical axial equation is either The first form, the elliptic paraboloid, meets the X-Z and Y-Z planes in the parabolas = 4az and = 4bz respectively, and is traced by an ellipse moving parallel to X Y, its centre on Z, its vertices on these two parabolas, with parameters 4a, 4b. For a= b, this elliptic paraboloid becomes the "revolute," of round Z; conversely, on shortening all its y's in the ratio b/a, the "revolute" becomes the elliptic paraboloid (fig. 1). A plane parallel to XY cuts the surface in the moving ellipse ay' 4abz ; on turning this plane round the major axis of the ellipse through the angle 0, the minor axis grows indefinitely ; at some angle 0 (cos 0 = b/a) it equals the major axis, and the ellipse becomes a circle. Par allel planes make similar sec tions; clearly there are two such plane-directions, two sets of . _ circles and two cyclic points of tangency. The right lines lying on the surface are unreal.
The second paraboloid is hyperbolic Clearly XY cuts it in the pair of lines all parallel cuts are hyperbolas or an hyperbola and its conjugate according as z is positive or negative. Sections
made by YZ and ZX are the parabolas (fig. 2), -4bz and = 4az respectively. The par aboloid is the path of an hyper bola always parallel to XY, mov ing with its vertices on one of these parabolas, and with its asymptotes always parallel to - ay' = o. At X Y the moving hyperbola passes through this asymptote-form over into the con jugate hyperbola, its vertices passing over from one parabola on to the other. At any stage +z and -z yield an hyperbola and its conjugate, respectively. The surface is saddle-like, 4a and 4b are the parameters. 0 is the vertex, Z is the axis, and its cyclic planes and points are unreal. However, similarly to the simple hyperboloid (q.v.), it contains two systems of real right lines, each line of each system cutting all lines of the other system, but none of its own (fig. 3). Each such pair of intersectors fixes a tangent-plane cutting the surface, which is not developable, but is skew or a scroll. (See MATHEMATICAL MODELS.)