Rotation of a ... forms a hyperboloid in which the hyperbola becomes a — Hyperboloid

"[F]or the bare hyperboloid, both real and Lambertian sources are in the same plane. This is a limitation in certain applications; however, we can manipulate these positions with lenses to place the sources at more convenient locations. ...this is an ideal concentrator. ...we have postulated an ideal lens while the hyperboloid is an ideal concentrator, albeit operating on a virtual source. The design considerations readily follow from the geometry of the hyperbola."

"One way to deal with the matters... is to work with a set of numerical values. I can do this with the help of Balls Cartesian equation for the hyperboloids of reguli of the 3-system (1900). This equation is quoted and proved by Hunt (1978)... I shall choose the three principle screws of a 3-system of motion... I have, (a) accorded with convention, (b) ensured that the pitch quadric will be real... Balls equation... clearly represents a series of concentric quadric surfaces... This circumstance of there being none of the concentric hyperboloids coaxial with one of the principal axes is a characteristic of the 3-system. ...within a certain, central zone of the system, the intersections among the hyperboloids are complicated and not easy to understand. Outside that zone, however... the hyperboloids appear in relation to one another... Each successive hyperboloid is wholly outside its predecessor (or wholly inside as the case may be), and no intersections are apparent. ...outside a certain, central zone, only one real screw can be found to pass through a generall chosen point..."

"Suppose we start with a disk radiator and we place a mirror in the form of a hyperboloid of revolution coincident with a set of flow lines... We truncate the mirror at some distance so the open end is a circle... Considering the inside as a mirror, this forms a nonimaging concentrator with unusual properties. The foci of the hyperbolas in this section are at... the ends of the diameter of the original disk. Then all rays entering the aperture... and pointing somewhere inside the disk will be reflected by the mirror so as to strike, eventually, the inner disk... Thus, the concentrator takes all rays from the virtual source... which can pass the entry aperture... and concentrate them into an exit aperture. This result is easily proved for rays in the meridional section... the extreme angle rays emerge from the exit aperture but only after an infinite number of reflections. ...rays at angles inside the extreme angles all emerge. Thus, in the meridional plane this is a concentrator of maximal theoretical concentration. This property holds for skew rays, although this is not quite so obvious. ...When used in reverse, the same design produces a virtual ring that fills the space between a Lambertian source and the larger diameter... [disk]. The visual effect produced is striking."

"The equation of the tangent plane at the point (a, 0, 0) of the conicoid \frac{x^2}{a^2} \pm \frac{y^2}{b^2} \pm \frac{z^2}{c^2} = 1 is x = a; this meets the surface in straight lines whose projection on the plane x = 0 are given by the equation \pm \frac{y^2}{b^2} \pm \frac{z^2}{c^2} = 0. These lines are clearly real when the surface is an hyperboloid of one sheet, and imaginary when the surface is an , or an hyperboloid of two sheets. Hence the hyperboloid of one sheet is a . The hyperbolic paraboloid is a particular case of the hyperboloid of one sheet; hence the hyperbolic paraboloid is also a ruled surface. This can be proved at once from the equation of the paraboloid. For, the tangent plane at the origin is z = 0, and this meets the paraboloid ax^2 + by^2 + 2z = 0 in the straight lines given by the equations ax^2 = by^2 = 0, z = 0; the lines are clearly real when a and b have different signs, and are imaginary when a and b have the same sign. Hence an hyperboloid of one sheet (including an hyperbolic paraboloid as a particular case) is the only ruled conicoid in addition to a cone, a cylinder, and a pair of planes."

"If we write the equation of the hyperboloid in the form \frac{x^2}{ a^2} - \frac{z^2}{c^2} = 1 - \frac{y^2}{b^2}, \qquad (1)It is evident that (1) is the product of two equations\begin{array}{lcl} \frac{x}{a} - \frac{z}{c} = k_1(1 - \frac{y}{b}), \\ \frac{x}{a} + \frac{z}{c} = \frac{1}{k_1}(1 + \frac{y}{b}), \qquad (2) \end{array}for any value of k_1. But (2) are the equations of a straight line... Moreover this straight line lies entirely on the surface, since the coordinates of every point of it satisfy (2) and hence (1). As different values are assigned to k_1, we obtain a series of straight lines lying entirely on the surface. Conversely if P_1( x_1, y_1, z_1) is any point of (1), \frac{\frac{x_1}{a} - \frac{z_1}{c}}{1 - \frac{y_1}{b}} = \frac{1 + \frac{y_1}{b}}{\frac{x_1}{a} + \frac{z_1}{c}}Therefore P_1 determines the same value of k_1 from both equations (2). Hence every point of (1) lies in one and only one line (2). We may also regard (1) as the product of the two equations\begin{array}{lcl} \frac{x}{a} - \frac{z}{c} = k_2(1 + \frac{y}{b}), \\ \frac{x}{a} + \frac{z}{c} = \frac{1}{k_2}(1 - \frac{y}{b}), \qquad (3) \end{array}whence it is evident that there is a second set of straight lines lying entirely on the surface, one and only one of which may be drawn through any point of the surface. Equations (2) and (3) are the equations of the rectilinear generators, and every point of the surface may be regarded as the point of intersection of one line from each set."

"A surface which may be generated by a moving straight line is called a . The plane, the cone, and the cylinder are simple examples... the hyperbolic paraboloid is a . ...[T]he unparted hyperboloid \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{ c^2} = 1 is a ruled surface having two sets of rectilinear generators, i.e., ...through every point of it two straight lines may be drawn, each of which shall lie entirely on the surface."