Hyperboloids

open centric surfaces of the second degree. Two types of hyperboloids are distinguished: hyperboloid of one sheet (Figure 1) and hyperboloid of two sheets (Figure 2). They represent two of the five principal types of

Figure 1. Hyperboloid of one sheet

Figure 2. Hyperboloid of two sheets

second-degree surfaces and, on intersection with various planes, give all the conic sections—the ellipse, hyperbola, and parabola—as well as pairs of straight lines (in the case of a hyperboloid of one sheet). A hyperboloid comes infinitely close to a conic surface (the so-called asymptotic cone). The hyperboloid of one sheet is a ruled surface. In the corresponding coordinate system (see Figures 1 and 2), the equations of the hyperboloids have the form