Mathematics and Origamipaper hyperboloid showing its generatrices, whereas Fig. 4 is a wire-work vision. Note that
the hyperbolas which are just visible laterally in Fig. 3, are heavily marked in Fig. 4.
It is recommended that the conic angle of the auxiliary cones will be close to that of the
asymptotic cones not shown in Fig. 4. Fig. 6 shows both cones with the traces left in them by
the hyperboloid, as well as the transversal section and development (to a different graphic
scale). See Point 19 on how to construct a cone.
If the starting ∆ABC has sides AB = 95; AC = 99,3658; BC = 12, the resulting cones
will have these measures: generatrix = 65; altitude = 41,5331; conic angle = 100,57º. Thus the
radius of the cone base is 50 against 40,2570 which is the extreme radius of the hyperboloid.
This bears the consequence that both cones surpass the hyperboloid as can be seen in Fig. 7,
in which the superposition of hyperboloid and lower cone is simplified.
Let ́s see now some geometric questions with regard to the hyperboloid we have just
constructed. We could be interested, e.g., in the parameters of the outlined hyperbola of Fig.
4, from the measures of ∆ABC and its inclination of 50º.
In the first place we should recall Point 17.2 (a conoid of paper) on ruled warped sur-
faces. Here we are in front of one of them; therefore Fig. 1 is not the unfolding of the hyper-
boloid surface, since it is undevelopable (it rather is a virtual surface consisting in straight
generatrices): Fig. 1 is only the folding diagram that enables its construction.
When forming Fig. 4, in fact we have got two hyperboloids: one showing to the outside
the mountain generatrices (convex paper from the exterior); the other exhibiting another set of
generatrices, also mountain fold and also convex, but in this case, as seen from the interior of
the hyperboloid: they are the valley creases of Fig. 1. For the time being we shall refer only to
the former hyperboloid; the other set of generatrices has served just as ancillary to facilitate
the hyperboloid construction.
A one-sheet ruled hyperboloid of revolution can be generated this way (Fig. 8): let the
hyperbola h in a vertical plane with one focus at F, be the directrix curve, and a horizontal cir-
cumference c moving vertically and resting on h, be the generatrix (of course adopting the di-
ameter that corresponds to each position). The locus of the centers of these circumferences is
the vertical axis OZ that is also the axis of the hyperboloid and of the hyperbola.
The minimum radius of the generatrix circumference occurs when it rests on the hyper-
bola ́s vertices; then it receives the name of neck circumference.
Conversely, rotating the directrix hyperbola h around axis OZ (Fig. 10) can also gener-
ate the hyperboloid: then, all the former circumferences (the neck one included) will rest on
those various hyperbolas.
Here it is the equation of a hyperbola like that of Fig. 9 (two symmetrical branches, see
Point 8.2.8.6):
6