HYPERBOLIC FUNCTIONS: see TRIGONOMETRY. HYPERBOLOID, in geometry, either of two open centric surfaces of the second degree (conicoids, quadrics). The gen eral equation of such centrics takes the form = I, where A, B, C, may have four orders of signs : + + + (ellip soid), and (a nowhere ellipsoid, with no real points) ; the other two, + + and + , yield hyperboloids of one nappe or two nappes. The central rectangular equations may be written = I and = I Planes parallel to X Y cut the hyperboloid in similar ellipses, = I, and the hyperboloid of two nappes in similar hyperbolas = I likewise for sections parallel to the other coordinate planes. It is simplest to consider the limit ing cases, when a = b = c. Then hyperboloid is plainly the "revolute" of the rectangular hyperbola = rotated round the Z-axis, each point (x, z) tracing a circle For z=o, = becomes minimal circle. The general surface (fig. i) is found, as in the ellipsoid (q.v.), by compression (or expansion) of all y's and z's in the ratios b/a and c/a respectively, which yields I. The "revolute" has as tangent at co the central cone = o, which remains tangential in the form + = o, after the affine transformation. Rotating the same hyperbola = round X yields the conoid = whose asymptotic cone is On compression (or expansion) parallel to Y or Z in the ratio b/a, c/a respectively, there results the general hyperboloid of two nappes y2/b2 = r, with its asymptotic cone = o (fig. 2) .
Surfaces of second degree are all ruled, i.e., traceable by a right line moving in a definite way. For, combining the two equations of a line and the one of such a surface, we may eliminate two coordinates and obtain a resultant equation in one coordinate, and this equation, which is of the second degree, vanishes identi cally when its three coefficients reduce each to o; and since the equations of the line have four parameters, this reduction is possi ble in an co of ways; hence there are co many lines lying on a conicoid, like line-elements forming a cone. In the ellipsoid and the hyperboloid of two nappes these lines are imaginary, but in the hyperboloid of one nappe they are real, forming two sys tems, each line of each system meeting all lines of the other system but none of its own (fig. 3). See MATHEMATICAL MODELS.