Hyperboloid generatix rotation

A one-sheeted hyper­boloid of rota­tion — is a surface, formed by a rota­tion of a hyper­bola around its virtual axis (axis of symmetry, perpen­dic­ular to a segment with ends in focuses).

There are two straight lines, which fully lie on a hyper­boloid and go through every point of it. Each of them сovers all the surface during a rota­tion around the axis of a hyper­boloid. Conse­quently, a one-axis hyper­boloid can be received by rota­tion of a straight line around an axis, crossing to it.

Joining these views on a hyper­boloid of rota­tion is a base of spec­tac­ular and illus­tra­tive models, where a straight rod passes through a curved hole — hyper­bola.

A show­piece, where a recli­nate segment, being a part of a hyper­boloid gener­a­trix, is settled on a rotating disc, is the most inter­esting. During the rota­tion of the disc a segment passes through both branches of a hyper­bola, not touching edges.

In a show­piece, which we often meet in science museums, a tube, which repre­sents a straight line, crossing a rota­tion axis, is firmly connected by segment with a socket on a rota­tion axis.

Para­me­ters $a$ and $b$ of hole-hyper­bola, intended by an equa­tion $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ in a screen plane (in case of a natu­rally imple­mented coor­di­nate system) can be easily counted by two typical posi­tions of a model.

When a segment, which forms a gener­a­trix and an axis, lies in the plane of the screen, its end coin­cides with a “vertex” of a hyper­bola. It means that a para­meter is simply length of this segment.

When a gener­a­trix is parallel to the screen, its projec­tion on the screen is an asymp­tote for a hyper­bola-slash. Conse­quently, a para­meter can be defined from the rela­tion $\tg \alpha = \frac{a}{b}$, where $\alpha$ is an angle of gener­a­trix to a vertical axis.