PSI - Issue 78

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

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planes passing through the Oz axis are its ‘meridians’, i.e. the curves whose rotation around the z axis originates the surface itself.

Fig. 2.

By intersecting the surface [2.8] with the plane Oxz, i.e. with the plane y = 0 see Fig. 2 and placing Y = 0 in equation [2.8], we obtain the equation of the sundial:

2.10. Equation form 3

2 2 − 2 2 =  For  = 1 it is the perbola i1, for  = -1 it is the hyperbola i2 and for  = 0 it is the pair of incident lines i3. Through their rotation they give the one-pitch rotation hyperboloid, the two-pitch rotation hyperboloid is the right circular cone, see Figure 3, respectively.

Fig. 3.

Generic one-pitch hyperboloids, two-pitch hyperboloids and second-order cones are held from the considered rotation surfaces by an expansion of the Oxz plane with coefficient b/a. If one compares equations [2.5] and [2.6] with equation [2.11]:

2.11. Equation form 4

2 2 + 2 2 = 2