PSI - Issue 78

Mauro Mazzei et al. / Procedia Structural Integrity 78 (2026) 1649–1656

1653

It can be seen that it represents the surface obtained from the rotation of the parabola x2 = 2°2cz around the Oz axis. This is the so-called rotation paraboloid see Fig. 4. The generic elliptic paraboloid [2.5] is obtained from the rotation paraboloid with an expansion of the plane Oxz.

Fig. 4.

Finally, surface [2.6] must be studied in a different way, namely by examining its intersections, which result in hyperbolas with the z = h planes. A horizontal projection of surface [2.6] is shown in Fig. 5, with another position of the axes I have considered this surface in Fig. 1. It is saddle-shaped as seen in Fig. 4 and is called a hyperbolic paraboloid. Its intersections with the planes parallel to the plane Oxz are equal parabolas. Its intersections with planes parallel to the Oyz plane give the same result. It is curious but not obvious that one-pitch hyperboloids and hyperbolic paraboloids, similarly to conical and cylindrical surfaces, can be obtained from the motion of a straight line. In the case of the hyperboloid, it is sufficient to demonstrate this property for the one-pitch rotational hyperboloid:

2.12. Equation form 5

2 2 + 2 2 − 2 2 =1

Since the generic one-step hyperboloid is obtained from this by uniform dilation from the Oxz plane, and any straight line in such dilations is again a straight line. We intersect the rotation perboloid with the y = a plane, parallel to the Oxz plane, substitute y = a and obtain: 2 2 + 2 2 − 2 2 =1 that is: 2 2 − 2 2 =0 This equation, together with y = a, gives a pair of incident lines: x/a - z/c = 0 and x/a + z/c = 0. In this way we find that a pair of incident lines lies on the hyperboloid. If we rotate the hyperboloid about the Oz axis, each of these lines draws the entire hyperboloid, see Figure 5, thus we obtain two families of lines, each of which covers the entire hyperboloid. 2.13. Equation form 6

Fig. 5.